/
Автор: Ladyzhenskay O.A. Solonnikov V.A. Uraltseva N.N.
Теги: mathematics
ISBN: 978-0-8218-1573-1
Год: 1988
Текст
MATHEMATICAL
MONOGRAPHS
VOLUME 23
O. A. Ladyzenskaja
V. A. Solonnikov
N. N. Uralceva
Linear and
Quasi-linear Equations
of Parabolic Type
American Mathematical Society
O. A. JlaALDKeHCKafl
B. A. COJIOHHMKOB
H. H. Ypajimeiia
JIHHEHHLIE
M KBA3HJIMHEHHbIE YPABHEHHfli
IlAPABO/IHMEOKOrO TMITA
M.3^aTCJii>CTBO «HayKa»
FjiaBHan Pe^aKiiHH
•^H3HK0-MaTeMaTH^ecK0ii JlHTepaTypu
MocKBa 1967
Translated from the Russian by S. Smith
2000 Mathematics Subject Classification. Primary 35--XX.
Library of Congress Card Number 68-19440
International Standard Book Number 978-0-8218-1573-1
International Standard Serial Number 0065-9280
Copying and reprinting. Individual readers of this publication, and nonprofit libraries
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Republication, systematic copying, or multiple reproduction of any material in this publication
is permitted only under license from the American Mathematical Society. Requests for such
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© Copyright 1968 by the American Mathematical Society. All rights reserved.
Reprinted with corrections 1988
Printed in the United States of America.
The American Mathematical Society retains all rights
except those granted to the United States Government.
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Visit the AMS home page at http://www.ams.org/
10 9 8 7 6 16 15 14 13 12 11
PREFACE
Equations of parabolic type are encountered in many branches of mathematics
and mathematical physics, and the forms in which they are investigated vary wide¬
ly. The equations encountered most frequently (and in adjoining fields of study
almost exclusively) are those of second order. Such equations (and certain classes
of systems of second order), both linear and quasi-linear, make up the subject of
investigation of the present book. Our study of these equations is concerned main¬
ly with the solvability of their boundary value problems and with an analysis of
the connections between the smoothness of the solutions and the smoothness of
the known functions entering into the problem.
A basic condition that is assumed to be fulfilled for all equations considered
is the condition of uniform parabolicity. For such equations we have managed to
give sufficiently complete answers to central questions on the solvability of the
above-indicated problems and to establish a series of exact dependences of the
properties of the solutions on the properties of the known functions in terms of ■
their mutual membership in the most commonly occurring function spaces.
For linear equations the solvability of the basic boundary value problems and
of the Cauchy problem depends only on the smoothness of the functions defining
the problem (i. e. the functions considered to be known in the problem, namely the
coefficients and the free terms of the equations, the functions assigning the in¬
itial and boundary conditions and the boundary of the domain in which the solu¬
tion exists). The smoother these known functions, the better behaved will be the
solution. Conversely, if one worsens the properties of the known functions in the
problem, then the differential properties of the solutions also become worse, where
the deterioration (as would equally be true with an improvement) has a local char¬
acter (for example, the smoothness of die solutions inside their domain of defini¬
tion is determined only by the smoothness of the coefficients and free terms of
the equation and does not depend on the smoothness of the boundary or of the in¬
itial and boundary functions). But one cannot arbitrarily worsen the properties of
the functions defining the problem (for example, admit in the coefficients singu¬
larities of high order). There exists a limit to admissible deteriorations, beyond
which such properties of the problems as uniqueness are lost. As in the analysis
vi
PREFACE
carried out by us for elliptic equations in the book [65 4] we begin by determin¬
ing this limit, for which we construct appropriate examples. With these examples
(and examples from [651, n, oj) we have managed to outline with sufficient accu¬
racy the limits of a possible theory of boundary value problems for equations with
discontinuous and, in general, unbounded coefficients and free terms, which is
later presented in Chapter III.
As a characteristic of such "bad” known functions we have selected their
membership in the spaces r(Qf)-D The solutions here fall into a certain func¬
tion space, the elements of which have derivatives of first order with respect to
x and of order % with respect to t. We then observe that the properties of these
solutions improve as the differential properties of the functions defining the equa¬
tion or problem improve.
A qualitatively different situation holds for nonlinear equations. For them the
smoothness of the solutions and the solvability "in the large” of the boundary
value problems and of die Caucby problem is determined not only by the smooth¬
ness of the known functions a-{x, t, u, p), a(x, t, it, p) making up the equation
but also by their behavior as u and p increase without limit, to § 3 of Chapter 1
we cite a number of examples elucidating certain restrictions on this behavior,
the nonfulfilment of which implies a nonsolvability of these problems “in the
large.” And in subsequent chapters (Chapters V, VI, VII) it is proved that these
restrictions, together with a certain not large smoothness, are on the whole also
sufficient for the unique solvability of the basic boundary value problems and of
the Cauchy problem for quasi-linear equations.
The general plan of the book is as follows. In Chapter 1 we present the basic
notation and terminology used in the book, a description of die main results proved
in it, and a number of examples indicating the exactness of these results; finally,
we give a brief historical survey. In Chapter II we have assembled propositions
that are used throughout the book and describe the properties, not of the solutions
of any differential equations, but of arbitrary functions belonging to various func¬
tion spaces or classes. It is perhaps better to treat this chapter as a reference on
l)For functions ulx, i) of a space riQf) the norm
is finite.
PREFACE
vii
its different assertions. The main text begins with Chapter III. It and Chapter IV
are devoted to linear equations. In Chapters V and VI we investigate quasi-linear
equations. Finally, in Chapter VII we examine linear and quasi-linear systems of
second order with common principal parts and give a survey of the results on gen¬
eral boundary-value problems for linear parabolic systems, the most general of
those considered up to die present time. The main contents of each chapter can
be understood independently of the others.
The contents of all the chapters, except Chapter IV and parts of Chapters II
and VII, are based on tbe work of O. A- LadyJfenskaja and N. N. Ural'ceva. These
chapters were written by them. Chapter IV and §§8-10 of Chapter VII were writ¬
ten by V. A. Solonnikov, who is responsible for many of the results in this part of
die book.
The authors are extremely grateful to Academician V. 1. Smirnov for having
looked over the manuscript of the entire book and having made a number of impor¬
tant critical remarks and suggestions. They were taken into account during the
final revision.
The authors express their heartfelt thanks to their colleagues and students
A. Treskunov, A. Oskolkov, M. Faddeev, I. Krol', V. Matveev and technician L. M.
Diku£ina for their help in the preparation of the book. A particularly large amount
of quite expert assistance was rendered by A. Treskunov, a graduate student at
Leningrad University, who worked with us throughout tbe writing of the book and
obtained during this time some interesting results on linear equations (see Bibli¬
ography).
Prefatory Note to tbe Translation
The active cooperation of tbe Russian authors has made it pos¬
sible to bring the present translation up-to-date and to improve it in
several respects. Slight additions and corrections have been made
throughout, and some of the material has been entirely rewritten,
most notably Chapter II §2 on embedding theorems. Chapter IV
§4 on certain supplementary theorems, and Chapter V §6 on solv¬
ability of the first boundary problem. The translator and the edito¬
rial staff wish to thank the Russian authors for their long-continued
and cordial assistance.
TABLE OF CONTENTS
Page
Preface v
Prefactory note to the Translation i*
Chapter I. Introductory material 1
§1. Basic notation and terminology * 2
§2. Classical statements of the problems. The maximum principle 11
§3. On admissible extensions of the concept of a solution 25
§4. Basic results and their possible development 42
Chapter II. Auxiliary propositions 57
§1. Some elementary inequalities 58
§2. The spaces IF* (0) and Embedding theorems 60
§3- Different function spaces depending on X and t. Embedding
theorems 74
§4. On averagings and cuts of elements of Lq{Q), Lq r(Q^) and
F‘'°«?r) 82
§5. Some other auxiliary propositions 39
§6. On estimates of max|u|. The class %{Qt, y, r, k, k) 102
§7. The class 8y, r, S, k) 110
§8. The function classes (QT (J T', • • •) and (J r\ • • • ) 122
§9. The function classes 128
Chapter HI. Linear equations with discontinuous coefficients 133
§1. Statement of the problem. Generalized solutions 134
§2- The energy inequality 139
§3- Uniqueness theorems 145
§4. Solvability and stability of the first boundary value problem in the
classes and W\-HQt) -153
§5. On the solvability of other boundary value problems. The Cauchy
problem 167
§6. On estimates in the space IPj’MOj.) and their consequences 172
§7. An estimate of maxpr|u|. The maximum principle 181
§8. Local estimates of max |u( 191
§9- Estimates of some norms of Orlicz for generalized solutions 194
xi
§10. An estimate of Holder’s constant. Harnack’s inequality 204
§11. An estimate of mait^,|ux| and <u 210
§12. On the dependence of the smoothness of generalized solutions
on the smoothness of the data of the problem 219
§13- On diffraction problems 224
§14. Functional methods for the solution of boundary value problems 233
§15. The method of continuity in a parameter 239
§16. Rothe’s method and the method of finite differences 241
§17. On Fourier's method 252
§18. On the Laplace transform method 255
Chapter IV. Linear equations with smooth coefficients 259
§ I. The heat equation and heat potentials 261
§ 2. Estimates of the heat potentials in Holder norms 273
§ 3. Estimates of the heat potentials in the norms of W*m'm 288
§ 4. Domains. Some auxiliary propositions 294
§ 5. Formulation of basic results on the solvability of tbe Cauchy
problem and boundary value problems for equations with variable
coefficients in Holder function classes 317
§ 6. Model problems in a half space 323
§ 7. On the solvability of problem (5.4') 328
§ 8. On the solvability of problem (5.4) 338
§ 9- The first boundary value problem in classes 341
§10. Local estimates of the solutions of problems (5.3) and (5.4) 351
§11. A fundamental solution of the parabolic equation of second order .... 356
§12. Some auxiliary inequalities for the function Q 364
§13- Estimates of the fundamental solution 376
§14. Solution of the Cauchy problem 389
§15. The single-layer potential 395
§16. Solution of the first boundary value problem 406
§17. On the estimates of S. N. Bernstein 414
Chapter V. Quasi-linear equations with principal part in divergence form 417
§ 1. Bounded generalized solutions. Holder continuity 418
§ 2. On the boundedness of generalized solutions 423
§ 3. Estimates of max^z |us| and (uxy<Q? 430
§ 4. An estimate of max jux| in the whole domain 438
xiii
§ 5- Estimate of and higher derivatives in an arbitrary sub¬
domain of the domain Qr 444
§ 6. The solvability of the first boundary value problem 449
§ 7. Other boundary value problems 475
§ 8. The Cauchy problem 492
§ 9. On the Stefan problem 496
§10. Another method of estimating the Holder constant for solutions $03
Chapter VI. Quasi-linear equations of general form 515
§ 1. A proof of the smoothness of generalized solutions of class ffi
and an estimate of (ux) 516
§ 2. An estimate of (ux) ^ 534
§ 3- The estimation of max |ux( 533
§ 4. Existence theorems 55^
§ 5- Equations with one space variable 56O
Chapter VII. Systems of linear and quasi-linear equations 571
§ 1. Generalized solutions of linear systems : 571
§ 2. On the boundedness of maxp^, |»| 574
§ 3- An estimate of |u|£f£ 579
§ 4. On estimates of lu^l^V and of other higher norms of the solutions 583
§ 5- Quasi-linear parabolic systems. Estimates of the norms
iu$*“\ I> 1> *n terms of max^ju, u^l 585
§ 6. An estimate of maxp^ (u^i 588
§ 7. An existence theorem for quasi-linear systems 596
§ 8. Linear parabolic systems of general form 597
§ 9- Statement of the boundary value problems and the Cauchy prob¬
lem for parabolic systems 604
§10. Basic results on the solvability of the Cauchy problem and of
the general boundary value problems for parabolic systems 615
Bibliography 631
EDITOR’S NOTE
For reasons of economics, most displayed formulas in this translation have been
inserted from the original Russian. This means that certain letters unfortunately have a
different appearance in formulas from their counterparts in the text. The principal in¬
stances are summarized in the following table:
Displayed formulas Text
JL a
$ $
e e
& s.
oM Jli
X K
CHAPTER I
INTRODUCTORY MATERIAL
This book is devoted to the basic linear and quasi-linear second-order partial
differential equations of parabolic type. For them the solvability of the basic
boundary value problems and of the Cauchy problem in various function spaces is
studied and investigations are carried out concerning the dependences of the
smoothness properties of the solutions of these equations on the known functions
making up the equations and on the properties of the other known functions in the
problems. We begin with a description of certain examples that permit one to out*
line with sufficient accuracy the contours of a possible theory for these questions,
and with an enumeration of the basic results of the present book. These sections
(§ § 3 and 4) may be usefully reread after making an acquaintance with Chapters
lH and IV. Preceding them (§2) is a description of tbe statements of the basic
problems for parabolic equations and an account of one of the basic properties
inherent in the solutions of parabolic equations of second order namely, in their
classical form, the maximum principle. Later on, both the statements of the problems
and the maximum principle acquire a different form, appropriate to the function
space containing the solutions being investigated. These modifications in the
classical form and the methods created for working with them led to success in
studying quasi-linear equations "in the large’* and linear equations with bad coef¬
ficients.
Although it can be read independently, the present book has much in common,
in regard to the methods of investigation, with the book by O. A. Ladyzenskaja
and N. N. Ural *ceva, "Linear and quasi-linear equations of elliptic type” [65*?].
All functions, arguments and parameters considered in this book are real. An
exception occurs with § §.4 and 18 of Chapter III and §8 and 9 of Chapter VII,
where Fourier and Laplace transforms are used.
1
2 I. INTRODUCTORY MATERIAL
•§1. BASIC NOTATION AND TERMINOLOGY
1. Abridged notation.
En is the n-dimensional euclidean space; * = (* j, • • •, *„) is an arbitrary
point in it.
£n+j is the (n + l)-dimensional euclidean space; its points are denoted by
(*, t), where * is in En and t is in (- ■»,
n is a domain in £fl, i.e. an arbitrary open connected set of points of En. In.
all chapters except IV, unless otherwise stated, H is considered to be a bounded
domain. In Chapter IV the letter f! denotes an arbitrary domain.
S is the boundary of 0.
S is the closure of 0, so that fi = fl (J S.
Kp is an arbitrary (open) ball in En of radius p, k„ = mes Kv and a>n is
the surface area of Ky
n, = *,nn.
Q j is tbe cylinder Q x (o, T), i.e. tbe set of points Or, t) of £Jtj with
*60,16(0, T).
Q‘ is an arbitrary open subset of Qj.
Sj. is the lateral surface of Qj; or more precisely the set of points (x, t)
of En +1 with * 6 S, t 6 [ 0, 7"].
r j. = Sr U K*, l): *60, (« Ol.
S„ > I(*, t): * 6 S, t = oi; P0 = {(*, <): * € fi, s -■ 0&.
^*1 >*2 = ^ ^ 1 < *2^'
Q(p, t) is an arbitrary cylinder of the form t(x, t): j* - < p, tQ< t < tQ + r|.
Q{k, p, r) is the set of all points (*, t) 6 Q(p, r) at which the investigated
function u (x, f) > k.
Qt (t) is the set of all points Qt »Dx(o<i< tj), at which u(*, t) > k.
n> ft S, Sk, 0, $k, y, <xf jS are positive cmstaots, with a being assumed
to belong to the interval (0, 1) •
!/(*) is a positive nonincreasing continuous function defined for t > 0.
fi{t) is a positive nondecreasing continuous function defined for t > 0.
dt is the Ktonecker delta symbol: *= 1, - 0 for i £ /.
§ 1. BASIC NOTATION AND TERMINOLOGY
P=(Pl Pnh
p*~\pP.
•s-kp- “V-=(s)2-
■CMJ'
a(x, /, a. /»)s=<t(.xi, .... xn, t, a,
a(x, t. u. uz) => a (x, x„, t, a, uXx °r„)-
ocs 1b(x); 01 is the oscillation of u (x) on 0, i.e. the difference between vrai
maxQU (x) and vrai rain j-j« (x). osc Su (x, t); Qj. i is defined analogously.
In the equations below we will encounter such expressions as
t, «(*, t), ax(x, /))],
which mean that in calculating the derivative d/dxi it is necessary to take into
account the presence of xf not only in the first group of arguments but also in the
other two, i.e. in the functions u(x, t) and u% (x, t), so that
JLla(x. t, *(*./>.0)1=-^- + ^^ + ^
Here and elsewhere pairs of equal indices imply a summation from 1 to n; in
particular,
»
da Vi da
Vc
Sometimes, when it does not cause confusion, the symbol for total differ¬
entiation d/dxi will be replaced by the more widely used symbol d/dx? For ex¬
ample, in a linear equation we usually write a teem such as
(d/dx.fta.-U, t)ux.(x, t)) ip the form (d/dx^i {a{.(x, t)u%. (x, t}), even though in
differentiating here oae must take account of x. in both a..(x, t) and u_. (x, t).
* */ */
n is the outward (from £1) unit normal to S at each of its points; d/do
denotes differentiation along n.
4
I. INTRODUCTORY MATERIAL
In Chapter VO the notation ys-n is also used; u is the inward unit normal
to S.
A function u (x) (or u (*, t)) is said to be finite in 0 (in Qj) if it is dif¬
ferent from zero only on some compact set that is separated from the boundary of
0 (of @7.) by a positive distance.
A function £(*) (or £(x, t)) is said to be a cutting function for the domain
0 (for Q f) if it is continuous in Q tin Qp)> has first-order piece wise-continuous
bounded derivatives, vanishes on the boundary of this domain (on Vy), and has
its values contained between zero and one.
2. Definitions of the basic function spaces. (ft) is the Baaach space con¬
sisting of all measurable functions on ft that are ?th-power (9 > l) summable on
fi. The norm in it is defined by the equalities
-L
I! “ ll?, Q = ^ J I« (x) I® dx'j q and II u !!„_ „ = vraHnax | u |.
Measurability and summability are to be understood everywhere in the sense
of Lebesgue. The elements of L (ft) are equivalence classes of the functions on
n.
Lqr (<?T) is the Banach space consisting of all measurable functions on Qy
with a finite norm
MI»,r.Qr = (j ( J !«(*• 01*dxj7d/'j' ,
where q > 1 and r > 1.
Lq,q be denoted by (Qj), and the norm |j • ||?1? by
fl *
Generalized derivatives are to be understood in the way that is now customary
in the majority of papers on differential equations. Different but equivalent defi¬
nitions and their basic properties can be found, for example, in [112. Vol.Vj
and [HJa}.
Wq (ft) for I integral is the Banach space consisting of all elements of
Ly (ft) having generalized derivatives of all forms up to order I inclusively, that
§1. BASIC NOTATION AND TERMINOLOGY 5
are ^th-power summable on I). The norm in (Si) is defined by the equality
!i“ii£0=i d-i)
where
((“yy/’a ~ 2 I! D'u ||,_ a. (1.2)
The symbol denotes any derivative of u {x) with tespect to * of order /,
while j denotes summation over all possible derivatives of u of order /. For
domains with "not too bad” boundaries IP* (0) coincides with the closure in the
norm (1.1) of the set of all functions that are infinitely differentiable in fi. This
will be true, for example, for domains with piecewise-smooth boundaries (a defi¬
nition of which is given below). Sometimes IF* is written in place of IT* (Q),
particularly if the domain 0 is subject to a further refinement.
IT* (0) is the set of elements of W* (0), that are finite in SI.
Ot i
f. (fl) is the subspace of W (Q) in which the set of all functions that are
9 r 9 *■
infinitely differentiable and finite in ft is dense. It is known that r (0) C
q
(Qj) for I integral (q > l) is the Banach space consisting of the ele¬
ments of Lq (Q-f) having generalized derivatives of the form D* with any r
and s satisfying the inequality 2r + s < 21- The norm in it is defined by the
equality
ii“iror=|o««»<fv (i.3)
where
I. V
The summation *s ta*cea over aH nonnegative integers r and s satis¬
fying the condition 2? + s = /.
Spaces (Q) and with nonintegral I will be used in Chapters
IV and VII. The former space is defined in §2 of Chapter II, and the latter in
§ 3 of Chapter II.
6 I. INTRODUCTORY MATERIAL
In addition to Iwe will encounter two spaces witb different ratios
of die upper indices:
n^’°((?r) is the Hilbert space with scalar product
(“' t'V‘.«(Cr)= / (“v+u^Jdxdt
Of
and is tbe Hilbert space with scalar product
(«. (Qr)— J (uv + 4- utv,) dx dt.
Or
V2 (Qjl is the Banach space consisting of all elements of having
a finite norm
|aj0r=vralomaxr||«(Jr. Oil,, 0+IKH* ^ (I.5)
where here and below
■/7<
r qt
ll“j:lls.Qr=l/ I “IdXdt.
Vl'°(QT) is the Banach space consisting of all elements of V2(QT) that
are continuous in t in die norm of Z>2 w*t*> norm
|B,<?r“o<f<rl,B(j(- f>^.a+H“Jk0r- (1.6)
The continuity in 1 of a function u (x, t) in the norm of L.(Cl) means that
juU, t + At) - u(x, *)|_ jj —i> 0 for At —» 0. The space I'j is obtained by
completing the set in the norm of (Qj.).
Vl'X(Qr) is the subset of those elements u(x, /) of V^,0(Qy) for which
t~h
J J A'1 {*(*, <+•*) —U(x. Wdxdt-^+O.
0 Q
A zero over ij'MC,.), V2(QT), V^iQf), V\'A(QT) “eans
that only those elements of these spaces are taken which vanish on Sj~
We now define spaces consisting of functions that are continuous in the
sense of Hdlder.
§1. BASIC NOTATION AND TERMINOLOGY 7
We will say that a {unction u(x) defined in O satisfies a Holder condition in
x with exponent a, a € (0, 1), and Holder constant in the domain SI if
sup p-° osc (a; Q'} ss < oo, (1.7)
where the supremum is taken over all connected components 0^ of all with
p< p0. If the boundary of the domain SI is "not too bad” (for example, piecewise*
smooth without double points), then (“) can also be defined in another way, as
sup j,«W-«(^)l. = <Bya>.
x,x-\a Ijt— *'1® ' •'
ix-jr1 K(,
For domains with a two-piece boundary, for example for the domain
l(*j, *2): |*j| < 1, )*2| < 1 and *2 ?£ 0 for |*j| < %|, definitions (1.7)and (1.8)
are not equivalent. In such cases we will adhere to the first definition.
We proceed to define the Holder spaces (fi) and In them I is
always a nonintegral positive number.
Hlm is the Banach space whose elements are continuous functions u(%) in
S having in 12 continuous derivatives up to order [2] inclusively andf a finite
value for the quantity
111
!“!!’ = W + a-9)
where <«>£> = I u |<°> = max I u j.
<«w> -si du g, <«><»=s (oirV rw’-
Equality (1.9)defines the norm l“l^ in
is the Banach space of functions u (*, l) that are continuous in
Qj., together with all derivatives of the form D£ for 2r + s < I, and have a
finite norm
Whete <<-l< = m«|al.
' Qr fr+7Zui) *'*><*'
I. INTRODUCTORY MATERIAL
JL
0<t-3r-s<2 t. Or
«r*- »F o<«<'. <1.11)
(*. O.W. <)€Qr I"* — * i
IJr-JT’ l<«i
.... !**(.*, tf) a (jc,/') | a ^ ~ ^ t
'*«. flr P - .W<0<I. (1.12)
(j:, 0. m. <*K I* — * I
!/-«' I<0«
For boundaries S with double points, instead of (1.11) we take another defi¬
nition for (u}^Qj" ***** ** an*logous to (1.2), namely
<“)l?0r = ^Pr| sup p~« CSCX {« (x, t); C4).
Here the second sapremum is taken over all connected components of all
with p> Pq, while
oscx {«(x, t); fi£} = vrai max a (x, f) — vrai mln a (x, t).
For / < 1 the spaces tf*(fi) and T will most frequently be denoted
by Ha(Q) and Ha' ^(Q -p). The norms defined by us in (0) and in HQ p)
depend on pQ, but for different p0> 0 they are equivalent, and therefore their
dependence on pQ will not be noted in the sequel.
All of tbe function spaces just described are complete.
H^(Q) is the set of functions belonging to H*{S') for any closed subdomain
ST' cQ.
2{Qj.) is the set of functioos belonging to Hl’^2(Q') for any closed
subdomain Q' CQr
C (0) (C(Q)) is the set of all functions that are continuous in fi (in 0).
C (Qj-) and C (Qj-) are defined analogously. C*(0) (C*(fl)) is the set of all
continuous functions in fl (fl) having continuous derivatives in fl (0) up to
order I inclusively.
C2'1 (Qj-) (C2’1^^)) is *e set of all continuous functions in Qj- (in Qj.)
having continuous derivatives ux, axx, Kj in Qj- (in Qj-).
c^HQt) is the set of all continuous functions in Qj- satisfying
§ 1. BASIC NOTATION AND TERMINOLOGY
9
i Lipschitz condition in * and a Holder condition in t with exponent %,
0l (A) (I - 1, 2) is tbe set of all continuous functions in H haying continuous
derivatives in O up to order 2-1, with the derivatives of order 2-1 having a
first differential at each point of Q and the derivatives of order I being bounded
in 5.
02,1{Qy) (02,1 (Qj-)) is the set of all continuous functions in Qj, (in Qj,)
having at each point of Qf (of Q-p) derivatives <i% and uf with the ux being
continuous in x and having a first differential with respect to x at each point of
Qr (of Q-p) and the functions u%, u(, uxx being bounded in Qf (in
3. On domains, their boundaries, and functions prescribed on these bound*
aries. Throughout tbe book we limit our considerations to domains having
"piecewise-stnooth boundaries with nonzero interior angles'* or, more briefly,
'’piecewise-smooth boundaries.” By this restriction we will understand a domain
whose closure can be represented in the form fl - fij (J • ■ ’ U Ojy, Slj 0^ = 0,
where each of the 0^ can be homeomorphically mapped onto tbe unit ball or cube
by means of functions satisfying a lipschitz condition in and such that the
Jacobians |d(z*)/d(x)| of these transformations are bounded from below by a
positive constant.
We will say that the boundary S of a domain fl (or a part Sj of it) satisfies
condition (A) if there exist two positive numbers aQ and 6q such that the
inequality
tnesQp-<(l —60) mes K0
holds for any ball with center on S (on Sj respectively) of radius p < “q
and any of the connected components fl^ of the intersection of the ball
K with Q.
P h n
Let * sc U j, • • •, *Jj) be any point of the boundary S of a domain SI. We
will call (y^, • - *, jn) a local Caitesian coordinate system with origin at if
y and x are connected by the equations y(- = - xj?), i = l, ■ -•, n, where the
aik form an orthogonal numerical matrix and the yn axis has the direction of the
outward (with respect to fl) normal to S at x®.
We will say that a surface S belongs to class H/ > 1 (or C\ or 0\
I > I) if there exists a number p > 0 such that the intersection of S with a ball
Kp of radius p with center at an arbitrary point 6 S is a connected surface,
10
I. INTRODUCTORY MATERIAL
the equation of which in the local coordinate system (yj, •••,/„) with origin at
has the form yn - • • • , y„_j)> where <a(yj, ■ • • , y„_j) a function of
class Hl (C* or 0* respectively) in the domain D that is the projection of Don the surface yn = ft
Suppose a function 0(s) is given on a surface S of class ij > 1
(C* 1 or 0*1). We will say that <ji(s) is a function of class tf*{S), I <1^, if as a
function of yj, • • • , yB_j it is an element of #*(D). As the norm l<6|^ we take
the largest of the norms |^(y)|^ calculated for all points *° of S. Functions of
classes C*(5) and 0*(S) are defined analogously.
If ^ is given on all of fl and <j>(x) € (fi) (C* (Q) or 0* (fi)), then on the
boundary S (of Q) belonging to class Hl 1 (C*1 or 0*1) with > maxll, l\ it
defines a function tft(s) = <^(*)|x_.sgj of class Hl(S) (C*(5) or 0*(S) respec¬
tively). The converse is also true: if <f>{s) £ tf*(S) and S € fl*, I > I, then
<£(s) can be continued onto all of the domain fl in such a way that the continued
function rf>(x) belongs to And what is more, this continuation can be
performed for all functions of #*(S) by means of one and the same construc¬
tion, so that the norms |^(s)|j and |<£(*)$> will be equivalent. It is precisely
this continuation of <f>(s) onto fl that we will have in mind in formulating bound¬
ary conditions by means of a function Analogous facts hold for the spaces
C* and 0* with I > 1. But for I < 1 all of what has been said concerning func¬
tions on S that belong to #* (S) (C* (5) or 0* (S)) is not true. Of two possible
but different definitions it is more convenient for us to take the one that preserves
the equivalence of the norms |$ Cs) ||- and |<£U)|q . For this reason we define
l$(s)|j*^ in the following manner:
|<p\f = max|<p(s)|H-sup(p~“osc {<p(s); S')), (1.13>
where the supremum is taken over all connected components 5^ of all K^f]S
with p < pQ.
All that we have said concerning surfaces 5 in En and functions <f>is) on
them carries over in a natural way to surfaces Sy. in En+^ and functions
<f>i.x9 t) prescribed on them.
§2. CLASSICAL STATEMENTS OF THE PROBLEMS 11
§2. CLASSICAL STATEMENTS OF THE PROBLEMS.
THE MAXIMUM PRINCIPLE
It is expedient to divide linear and quasi-linear second-order equations of
parabolic type into (our groups: linear equations of tbe form
&u = ut — -^{au(x. t)uX) + at(x, t)a)
+ bt(x. t)uXi + a(x, t). (2.D
linear equations of the form
£>u ssu, —a,; {x. t)uX[Xf + at(x. t) aX(
+ a (x, <)« — /(•*. 0. (2.2)
quasi-linear equations with principal part in divergence form
JS’u^ul — ~-i{al{x. t. a, ax)) + a(x, t, u, ax) = 0 (2.3)
and quasi-linear equations of general form
J^ussut — atj (jc, t. a, ux) uXlXj + a (x. t, u, ax) = 0. (2.4)
In regard to these equations we assume, unless otherwise stated, that they are
uniformly parabolic. For equations (2.1) and (2.2) this means fulfilment of the
condition
v|*<ati(x. Hi2. <*. t)€QT. (2.5)
where here and everywhere below u and fi are fixed positive numbers, and
if = (£j, is an arbitrary real vector; for equations (2.3) and (2.4) it means
fulfilment of the conditions-
v(M) (2-6)
and
v(MH*<%(.*. t, u, ^)|{^<(i(|a|)l! (2.7)
respectively for arbitrary u and p and (x, t) 6 Qj. The functions >/(r) and fi(r),
as indicated in §1, are certain continuous functions of r> 0 that assume only
positive values.
12
I. INTRODUCTORY MATERIAL
V Che functions o,y, a, and fj are differentiable, equation (2.1) can be written
in the form of (2.2) and, conversely, equation (2.2) can be put in the form of (2.1).
The most common problems involving equations (2.1)—(2*4) are the Cauchy
problem, which consists in determining a function u (x, t) in the half-space
f |*| < oo, t > Ol satisfying the corresponding equation for t > 0 and the initial
condition
“l».=o = >i>o(-*0 (2.8)
for t = 0, and the boundary value problems (or, what is the same thing, mixed
boundary value problems) in cylinders of form Qj. = fl x (0, T), which consist in
determining functions u(x, l) in Qj. satisfying the corresponding equation in Qy,
the initial condition (2.8) on the lower base of Qj, and one of the boundary con¬
ditions on the lateral surface Sj. , for example, the first
“Ur = ♦(*. *). (2*9)
or the second
+ o«|Srs= atjux.cos(n, J:,) + m|Sj. = i|i(s. 0- (2.10)
We will confine ourselves on the whole to these problems, although much of what
will be said will also be applicable to boundary value problems in which Qy is
not a cylinder.
In the classical formulation of tbe Cauchy problem it is required that the
solution be continuous at all points Gt, t) 6 1|*| < t > Ol. that it have con¬
tinuous derivatives u(, ux and u%x and satisfy the equation at all interior points
(x, t) (i.e., for |z| <■»,(> o), and that it remain bounded for |*| — •*> (or, more
generally, that it not exceed some given function). In tbe classical formulation of
the first boundary value problem it is required that the solution be continuous in
Qj, have continuous derivatives u[t ux, uxx in Qj, satisfy the equation at all
points of Qj, and satisfy conditions (2-8) and (2.9) for I = 0 and (*, t) € Sj.. If
Q is an unbounded domain, then to these requirements must be added a condition
on tbe behavior of the solution for |x| —> oo. For the second boundary value
problem it is required in addition that the derivatives ux, exist and be continuous
in Qj. (J Sj. (sometimes this requirement is weakened to the existence ol the
derivatives du/dNj^j, and their continuity under certain approaches to Sj.). For
§2. CLASSICAL STATEMENTS OF THE PROBLEMS
13
the nonlinear equations (2.3) and (2.4), id addition to tbe conditions just enumer¬
ated, one sometimes requires the continuity in Qj. of the derivatives u% or even
of all the derivatives u(, ux and u (these requirements will be specially
stipulated).
For all of the problems just enumerated, the uniqueness theorems in their
classical statement are valid as long as the functions making up the equations
have a certain smoothness.
They ate proved with the help of the well-known maximum priociple. In its
simplest form it consists in the fact that the solutions it of equation (2.2) witb
a m { s 0 assume their least and greatest values in Qj, on the boundary Pj.. In a
more general form it finds its expression in the estimates of a s f a 0 given below
by Theorems (2.1-2.3). It is based on the following fact: at interior extremum
points of uix, t) the expressions uf — a^ux.x. and u(- (daj/d>ix)ux.x., which
form the principal part of the parabolic operators £u of (2.1)—(2.4), are either both
nonnegative or both nonpositive.
Theorem 2.1- Suppose u(x, t) is the classical solution of problem (2.2),
(2.® , (2.9), in the cylinder Qj., where the coefficients and free term in (2-2)
are bounded functions and
alj(•*. 0hlj>0. (2.11)
Then for any t j 6 [o, T] the estimate
sup mln { 0; mtn (<)>«*■ <<,-<)); mln (fek 1
x> a, I r. * — a° Q, I
inf maxfo; maxr—- max(2 \7\
x>«, I r. *-“• «, J K ’
M *1
for u(*, t) is valid, where the number a^ is. equal to maXq^ (- a (x, t)) - -
maa (*, t), while tjj coincides with ip0(x) of (2.8) on VQ, and with
xfi(s,t) of (2.9) on Sj-.
We recall that we agreed to denote the cylinder fl x (0, *), by Qt , and the
sum of its lateral surface 5( and lower base Pq by
For the proof of (2*12) we make a transformation which will frequently be
encountered in the sequel; namely, we pass from u(x, t) to a new function v(x, t)
connected with it by the equality
a(xt t)*zv(x, <)***. (2.13)
where A is for the time being an arbitrary number. The function v, as is easily
14
1. INTRODUCTORY MATERIAL
seen, satisfies tbe equation
Vi — a//Vjr{x^H~a<*jrf‘4-(a + A)t>*E: }e~u (2-14)
resulting from (2.2). Take an arbitrary 2j from (0, T). There are three possible
cases: either tifce, t) is nonpositive in Qti’ or least positive value of t>(x, t)
in Qt is assumed on r((> or its greatest value is assumed at some point
(x°, l^eflx (o, t j]. In the first case max^ v (x, l) < 0; in the second case
0 < maXgt vix, t) < maxp^ v; and in the third case 0 < max^ ti (*, t) <
v (x°, t®), with the relations
»<> 0, vX{ — 0 and — at j-vXiXj > 0 (2.15)
and equation (2.14) being satisfied at the point (*°, £°). The validity of the last
of the relations (2.J 5) is due to the fact that at a maximum point tbe pure second
derivatives dy^dy^ are nonpositive in any direction y^ -
(det \<^i\ / 0) while the expression —<*ijvx.x. is equal to - \vyjtyk *°r *“ orth°g°nal
matrix Ojp where Aj, • • •, A„ are the characteristic numbers of tbe quadratic
form (and consequently are nonnegative by (2.11)). By virtue of
(2.15) and equation (2.14) the inequality
[a(x°. <°)4-Mw(*°. fi)*-*-
holds at the point (*°, t°). We will assume that A > max^ [-«(*, «)] = aQ. Then
in all cases the estimate
maxi
Q,
is valid, and hence
axw(x, <)-=Cmaxl 0; max®; max
\ \ r/, Qt, + * j '
e
a(jc, /,)< ex'imax fo; max(ue~u)\ )~_a max(fe~KI)|. (2.17)
I r‘< * Qt, J
Analogously, by considering a point of least nonpositive value of the function
t> (*, e), we arrive at the estimate of u (x, t) from below indicated in inequality
(2.12).
An estimate of u in the form
|#(x. *,)!< maxlilieMf.-OI 4-f, max|/«*«M-0| (2.18)
r<, \
§2. CLASSICAL STATEMENTS OF THE PROBLEMS
15
is also valid. This follows from the fact that the functions
± u(x,
where
jjt, = max | , fi2 = max)
r<,
ate nonnegative in Qt by virtue of the equations that they satisfy.
Corollary 2.1. Suppose the conditions of Theorem 2.1 are satisfied. Then the
solution u(x, t) is nonpositive in Qj. for «|pr < 0 and f <0, and is nonnegative
in Qj. for “Ip^. > 0 and f > 0. Also, if aix, t) s fix, t) & 0, then
min a < «(*, fjXmaxa.’ (2.19)
r, r.
for any tj £ [0, T\.
The first two assertions follow directly from (2.12). For a proof of (2-19) we
take the function wix, t) *> u (x, t) - u (x^, o), where is a point of S, and,
since it satisfies die same equation £u> = 0 as nix, t), apply estimate (2-12) to
it. This gives
sup min |0; min j(«—a(jr0, 0))ex<''~'>]\<B(jc, t,)—u(x0, 0)
*1
< ^inf^ max j0; max [(u — a(jc0, 0))e'«i-0j j.
But the left side of this inequality is equal to minp^ u - u (x^, o), while the
right side is equal to maxp( u - u(xq, 0)> and thus the inequalities (2.19) are
valid for u (x, t j).
We now consider for equation (2.2) a boundary value problem in Qj with
boundary condition
6, (x. t) a,( + b(x, t)«Isr = 1|> (s, f), (2.20)
in which the vector b «■ (i> j, • • ■ , £>n) forms with the direction of the outet normal
o to Sj. at the same point (*, t) € Sj. an angle not exceeding it/2. The following
two theorems are valid for this problem.
Theorem 2.2. Suppose u (x, t) is a classical solution of problem (2.2),
(2.8), (2.20) in the cylinder Qj. and suppose the coefficients of equation (2.2)
and the functions f, , b and if/ are bounded, with the a- being subject to con-
16 I. INTRODUCTORY MATERIAL
d it ion (2.11), while b(x, > 0. Then for any t; from [0, 7"] one has the
estimates
estimates
0; min ^mis a(x, 0); mln
0; max-^-j ;
*1
«*. max«(*, 0); ^ max/e*<W>J, (2.21)
in which is equal to maxg (- a (x, t)).
This theorem is proved in the same way as Theorem 2.1, i.e., the function
v(x, t) = u(x, t)e~^ is taken and all possible locations of a positive maximum
and a negative minimum of it are analyzed. All of the possible situations coin¬
cide with those considered in Theorem 2-1- A difference in the estimates obtained
for them exists only in the case when these extremums fall on S( . Thus, if v
takes its greatest value in Qc at some point (x°, t°) 6 S(^, then at this point
bi vx.>0, (since the vector b m (b j, • • • , bn) forms with tbe outer normal to Sj.
at the same point an angle not exceeding tr/2); hence t>! < the~^'i^\
|(*0,«0) J(x0,f0)
by virtue of tbe boundary condition (2-20) - Analogously, if a minimum point
(*°, t°) for v falls on S , then at it b,vr. <0, and hence
x, i 1 * ~
v fl n - ^ie~ /b\ n n by virtue of (2.20).
((* .< ) |<* ,o
Theorem 2.3- Suppose all of the conditions of Theorem 2.2 are fulfilled ex¬
cept ijjj. > 0, which is replaced by the condition > — &q> ^q = const > 0,
and suppose
max|att, a,|< Mi. <*„ = max(— a(x. t))
«/•
and
r A.i
6;COs(n, xt)~ U cos(n, b)>6, 6 » const > 0.
Then
§2. CLASSICAL STATEMENTS OF THE PROBLEMS
17
maxl«|-<«i*c7' max j max|i|>|: max] u(x, 0)1; max | f\I, (2.22)
Qt \ sT a qt I
where c and c j are constants depending only on the numbers fij, oQ , 8
and the boundary of the domain Q, which is assumed to belong to the class O2.
For the proof of (2.22) we introduce in place of u {x, t) the function
w (x, t) = a (*, t)rf>(x), where <p(x) is a function of class 02(fl), that satisfies the
conditions
min <p(x)>±. «p|s==l. =
where m = const > 6q/S. Then - l/<f>bi ^.1^ = nti(.cos (n, x^) > 8m > by. The
function w, as is easily seen, satisfies in ^ an equation of the same type as
that satisfied by u(x, t), and on Sj. it satisfies tbe condition
biWXt + {b — =1>9lsr,
so that tbe conditions of Theorem 2.2 are fulfilled. In view of tbis, estimates of
type (2.21) are valid for w, from which follows the estimate (2-22) for a.
Remark 2.1. As can be seen from the proof of Theorems 2.1— 2-3 and Corollary
2.1, they and their assertions (2-12), (2-21), (2.22) remain in force when the
coefficients o.(*, t) of equation (2.2) have isolated poles in * of less than first
order. Hie latter means that at a singular point * = *° tbe product ».(*, t)(x - *°)
must tend to zero for * —*
From Theorems 2.1—2-3 follow uniqueness theorems for the classical solu¬
tions of the boundary value .problems. Namely:
Theorem 2.4 Problems (2.2), (2.8), (2.9) and (2-2), (2.8), (2.20) under
conditions corresponding to Theorems 2.1—2-3 cannot have more than one classi¬
cal solution.
Indeed, the difference u (*, t) of two possible solutions of any of these
problems satisfies the corresponding homogeneous equation and homogeneous
initial and boundary conditions, so that the corresponding inequality (2.12),
(2.21) or (2.22) with f “ <t>0= t/i = 0 is valid for it, from which it follows that
us 0-
We will now show that the Cauchy problem also cannot have two distinct
classical solutions. For this purpose we will prove
18
1. INTRODUCTORY MATERIAL
Theorem 2.5- Suppose the function u(*, t) is continuous in the strip
nr = i|*|
< °°» 0 < t < T\ and its modulus does not exceed some number M-
Suppose, further, that it satisfies equation (2.2) for 0 < t < T, that the moduli of
the coefficients a.., a. of (2.2) do not exceed c, and that a (%, *)>-Og, where
c and ag are non negative constants. Then the estimate
max|a(x, f)| <(max| a(x, 0)| +/max\f\)t*J (2.23:
r e„ «r
is valid if condition (2.11) is fulfilled.
For the proof of (2.23) we consider the function
w(x, t) = u(x, t) e~(a°1 ~ c, — c2t—^ (x5 -f c30.
where Cj |u(x, 0)|» c^ =. maxjj^ |/(x, t)|, and R, < and Cj are arbitrary
positive numbers. We calculate (£ + oQ + «)t»:
(^-j-fl0 + e)® = — c2 [1 -(-(« +flo +
— -JF — 2 + 2aixi + (a + «o+8) • (JC2 + «aOj
~(a+a0+e)cu
Let us select the number c ^ for fixed e > 0 so large tbat the expression multi¬
plied by M/R2 is nonnegative. Then (£ + ag + t)«J < fe~(a0^ * - c2 < 0. In
addition, on the lateral surface and lower base of the cylinder Qj.(R) =
11*| < R, 0<t<T\ the function w is nonpositive. Therefore, by virtue of Corol¬
lary 2.1, w is nonpositive through QT(R). Take an arbitrary point (*°, t°) of
the strip 0 < t < T. Then according to what has been proved w (*®, i®) < 0 for
|*°| < R. Letting R tend to infinity in this inequality, and then letting t tend to
zero, we obtain the desired estimate for u (x, t) from above:
a(x°. f°) < (c, +c2t°)e<*'.
For the estimate of u (*, t) from below it is necessary to take tbe function
«(*. t) = a(x. + c, + c2< +
with the same values for the parameters as above. For it (£ + 4- c)w > 0, and
on the lower base and lateral surface of Qj-(R) it is nonnegative. Therefore
w (*°, t°) > 0 for |*0! < R, from which, letting R tend to <», and to zero, we
obtain the estimate (2.23) for u (x, t) from below.
§2. CLASSICAL STATEMENTS OF THE PROBLEMS
19
Remark 2.2. Theorem 2.5 may be strengthened both in regard to the conditions
on the coefficients (for example, one can admit a specified rate of growth of them
for |x| —> o° or the existence of singularities of the type mentioned in Remark
2-1) and in regard to a replacement of the condition maxjj^. |u| < oo by tbe con¬
dition |u(x, t)| <c«CI* with c and (^arbitrary. We will not cite here the dif¬
ferent generalizations and strengthenings of Theorems 2-1—2-5 but refer the reader
to the relevant literature [ 10,25,3 lb, 34,46,47b, 63,90,9 3*107,109,110,123c,d,l30,etc.]-
From Theorem 2-5 follows the uniqueness theorem for the Cauchy problem in
its classical statement.
Theorem 2.6. Under the conditions of Theorem 2-5 the Cauchy problem has no
more than one classical solution in the class of bounded functions.
Below, a result is established that is analogous to Theorem 2.3 in an un¬
bounded domain fi. It concerns an estimate for tbe maximum of the modulus of a
bounded solution of the problem with oblique derivative (2.2), (2-20), (2.8).
Ibis result is also true for the first boundary value problem in an unbounded
domain (see [62]).
Theorem 2.7. Suppose that all of the conditions of Theorem 2.3 are ful¬
filled, with the domain SI being such that there exists a function qtGO in it
having the following properties;
q>(*)60*(Q), «nln<ji(x)>^. <p|s=l.
where m = const > tQ/S. Then the estimate
la sT
-f Sup|/(JC. Ol'V*' (2-24)
qt *
is valid% where c and A are constants depending on the b.t b, the coefficients
of the operator £, and $(#)•
Proof. The function v (*, t) = it {x, t) U) is a solution of the boundary value
problem
20
I. INTRODUCTORY MATERIAL
JSPv am Jg'v — 2ya,j (r*1)
+ <P [«*/ (.<p-\ - *„ «P~%lX/J « = /<P.
H=.o —(x)fp(x),
Mx,+HSr=>h>lsr-
where
P = * = * ^ > 0.
We also introduce the (unction
w(x, t) — v(x, *) e-'«.+«> < — Cl~c^~~(xi+ c3t), (2.25:
in which the constants c^, c^, Af, 3^ are defined as follows:
c, = 6 4-sup I$0(x)I <p(*) 4- sup (2.26)
where § is an arbitrary positive constant,
cj — sup|/(at. 0|qp(-*),
Or
/M = sup|«(jc, t)|,
Qt
% =tnax 10; inf (x, 0 + <f> f(T1)^ — a,/ j.
The constant for fixed e > 0 is chosen in such a way that (£ + Oq + f)w(x, <)< 0
as was established in investigating the function (2.25) in the proof of Theorem
2-5, such a choice of c^ is possible.
Let Qr be that part of the domain 0 lying in the ball : |*| < R. Its
boundary S(^) is composed of two parts: 5(^) - R) + ^5?)’ w*lere ^Ir) ~ Kj
and ) is that part of the boundary of the ball K^ lying in O. We will show
that if R is sufficiently great, then to (x, t) < 0 for * € Indeed, from the
definition of tv and tbe constants c j and M, and also from the inequality
(£ + a0 + f)w < 0, it follows that the function u/(x, t) cannot achieve a positive
maximum in (1% x [0, T] and for * 6 On S^’j the function ui [x, t) satisfies
the boundary condition
§2. CLASSICAL STATEMENTS OF THE PROBLEMS
21
bfWxj + pu> = i|)q)e-(®»+e) ‘ — t>iPX/ — P F,
where
F(x, t)*=e, -+-«s*-f Cj/).
Therefore at a maximum point on
pw < i|xj>e ' ,fl*+e)' — btPXl — pf. (2.27)
But by virtue of (2.26)
lb,rXij<maXlbtlll*LjJL.
Thus for large R, at a maximum point on Sfa
P®<0.
This proves that the function w (%, t) does not in general have a positive maximum
point in Slg x [ 0, T). Thus
w(x, /)< 0.
Letting /?—►«> and f —* 0, we see that everywhere in Qj.
v(x, t)<(c, +c2t)e*t.
Since 8 is arbitrary, this gives the desired estimate (2.24) from above for
u be, t). The estimate from below is obtained in an analogous manner. Theorem
2.7 is proved.
We will apply this theorem in §6 of Chapter IV to a problem with directional
derivative in a half-space, where the function whose existence is in the
conditions of the theorem, is constructed very simply.
Theorems 2-4.and 2.6 permit one to draw certain conclusions concerning the
uniqueness theorems for the boundary value problems and for the Cauchy problem
both in their classical statement and for nonlinear parabolic equations. Thus, for
example, suppose u '{x,‘ t) and u "(*, I} are two solutions of equation (2.4) in
which the functions a.. (x, t, u, p), a (x, t, u, p) have partial derivatives with re¬
spect to u and p. We subtract (he equation for u " from the equation for u ' and
22
I. INTRODUCTORY MATERIAL
write the result first as follows:
(a'_a")t — al]{x, t, u', u’J(«' — «")^
~~*V/ J £ a‘j[x'*' «'•+-(!—*)«". ra; + (l-T)u;]rfT
0
1
+ J £a\x, t, ta'-f-(l — t)«", t)B']dt = 0.
0
and then as a linear equation for v «* u * - u ,4:
vt — atJ(x, t)vtlKj +*<C*. 0-+- c(jc, t)vs= 0, (2.28)
in which
0 = «/;(■*• 0. <•(*. 0).
o * o *
i 1
r a«i#r.*.] r *«[...l
c(x, o—«;,/*. o J j -4~
0 0
If the coefficients in (2.28) are bounded in Q j (ot at least satisfy the con¬
ditions of Remark 2.1), then from the homogeneity of (2.28) and the vanishing
of v on Fy it follows by virtue of Theorem 2.4 that v(x, t) is identically equal
to zero in Q^., i.e. that u' and u" coincide. One of the sufficient conditions
for this is given by the following theorem.
Theorem 2-8- If the coefficients a;/. satisfy the condition
t, u, p)%,|y>0
and a.. (*, t, a, p) and a (x, t, u, p) and their partial derivatives with respect
to U and p are bounded on any compact set of the values of their arguments, then
the first boundary value problem for (2.4) has no more than one solution in the
class of functions belonging to C(Qj.)f] C2,1 (Qj-) and having derivatives with
respect to x of first and second order that are bounded in Qp
An analogous theorem is also true for the Cauchy problem.
Theorems 2-1— 2.3 and 2-5 are easily extended to quasi-linear equations. We
§2. CLASSICAL STATEMENTS OF THE PROBLEMS 23
cite here a theorem that permits one to estimate the solutions of the first boundary
value problem for equations (2.4).
Theorem 2-9- «(*, f) be a classical solution of equation (2.4) in Qf.
Suppose that the functions ay (*, t, u, p) and aU, t, u, p) take finite values for
any finite a, p, and (*, l) 6 Qj, and that for (*, f) € QJ and arbitrary u
aij(x. t, u, 0)!(£;>0 (2.29)
and
ua(x, t, a, 0)>— £)«* — #}, (2.30)
where h j and b ^ are nonnegative constants. Then
maxl^*. OK inf eKT
Qr * > ».
max
fr
?i*i' /rSr]- <2-J1’
If in place of (2*30) the condition
aa(xt t. a, 0)>—<D(|tf|)|«|—£2 (2.32)
is fulfilled, where > 0 and 0(r) is a nondecreasing positive function of r> 0
satisfying the condition
<30
J = <2’33)
then the estimate
max|tf(x, OK inf<P&),
Qt >•> >
| =» eu max
: j 1: (-{r-*^(g) : *" (mr*X 1*1)} (2.34)
is valid, in which is the inverse of the function </>(£) defined by the
equation
til)
J -_£!-==!„£ (2.35)
o
The proof of the first assertion of the theorem is analogous to the proof of
Theorem 2.1. Essentially, the second assertion is proved in the same way.
24
I. INTRODUCTORY MATERIAL
Namely, we introduce the function v connected with u by the equation u = <f> (u).
From (2-4) we obtain the relation
v. —a..(x, t, u, ur)(vr_ 4- ^t-vwv, )
I UK xf\ xixj <p' x( xf)
4-•£•#(*. t, u. «,) = <) (2.36)
for u. For the function v = ve~A > 0 we have the equation
t. u. +
+ -If-aix, t, u, ux)e~u 4-A,« = 0, (2.37)
from (2-36). If v takes its greatest value in Qj. at some interior point (z°, £°)
of the domain Qj and v Ufll «Q) > then at this point t5(s 0, - aijVx.x. > 0,
v„ - 0, so that u . = <*'(r)t) = <4'(w)u e^* = 0, and hence on the basis of
*i *| *1 •**
(2.37) we have at this point
la (jc. <. u, 0) 4- W (v) ®1 |U(< w < 0. (2.38)
Since by assumption v(xQ, tg) > e~^c0t we have v {x^, tQ) > 1 and hence
«(*0, «Q) > 0- Multiplying (2-38) by u{xQ, tQ) and using (2-32), we obtain
— 024-|—<£>(«) +Xq)'(t»)vl«<0. (2.39)
As is easily seen, the function (p(v) satisfies the differential equation
<f>'(v)v = 4>(<£) and the condition <ji(l) = 0.
Under a variation of v from 0 to « the function increases raonotonically
(by virtue of condition (2.33) ) boa - « to + “. Fot such a <f> inequality (2-39)
takes the form
(X~ 1)«<U («)<&,.
For A > 1 this inequality gives an estimate for b(*q, JQ) from above, namely
§3. ON ADMISSIBLE EXTENSIONS OF THE CONCEPT OF A SOLUTION
“(*<>• to)<-(JZr^W(0)'- (2.40)
By virtue of the relations between it, v and t? and the fact that maxQ^v (x, t) =
v (*0, £0), we will have at any point (x, t) 6 Qy the inequality
v(x, i) = v(x. t) eu < v(xg, tg)eK‘
— v (xq, t0) ek{,-‘d = <p_1 (a (xQ, /0))e*i<-<.>
<r' (srrrhrj < *-■
This inequality in combination with tbe two other possibilities for maxg^v gives
the desired estimate for u(x, t) from above. Analogous arguments with the func¬
tion - u ix, t) give the estimate for u {x, t) from below. Theorem 2.9 is proved.
In §7 of Chapter V an additional theorem will be cited, which permits one to
estimate the maximum of the modulus of classical solutions of equations (2-3) that
satisfy other boundary conditions.
§3. ON ADMISSIBLE EXTENSIONS OF THE CONCEPT OF A SOLUTION
1. Ob uniqueness theorems. If the coefficients and free terms of equations
(2-1) and (2.2) are not smooth functions, then these equations do not in general
have classical solutions. But the possibility is not excluded that they possess
solutions belonging to a class of functions 91 that is wider than is permitted by
the classical statement and yet sufficiently narrow to preserve one of the basic
properties of boundary value problems for equations of parabolic type, namely,
their determinateness (i.e. the uniqueness theorems). It is accordingly natural
here to extend the classical statements of the problems and to replace them with
investigations of solutions belonging to a class 91. We will consider an extension
(and the class 88 of generalized solutions corresponding to it) to be admissible
if a uniqueness theorem is retained for it.
The choice of the class 3! is dictated by the smoothness properties of the
coefficients of the equation (i.e. by the $th-power summability of them and their
derivatives of a particular order for various values of q) and of the other func¬
tions that are known in the problem; but above all it must be made so that the
conditions of the problem can be put in a form that has a meaning for any element
of 3JL1* These requirements do not in general determine the class IS uniquely,
but once a choice of 38 is made, a transformation of the conditions of the problem
is carried out according to a particular criterion. Let us illustrate this with an
1) Hete we are interested only in those cases in which 18 does not depend on the
actual form of these known functions, more precisely, on the location of their singular points.
26
I. INTRODUCTORY MATERIAL
example of the problem
a,—
a\taG — %(■*)• (3-l2)
in which the a>. are nondifferentiable functions satisfying condition (2.5),
f{x, t) 6 LjiQ’f), and 0qGc) € L^CQ). Clearly, for such a- equation (3*lj.)must
be put in a form that does not contain derivatives of the a.j. This is done as
follows: equation (3-1 j) is multiplied by an arbitrary smooth function tj(x> t),
then both sides are integrated over Qy* and in the terms containing the a.^ a
single integration by parts is carried out. As a result we arrive at the identity
J j-i%rndsdt= J/ndxdt. (3.2l)
Qt sr qt
As is well known, it is equivalent to equation (3-lj) if all of the functions
entering into it are sufficiently smooth. Yet in the case of nondifferentiable o-y
(3-2j) has a meaning (for well-behaved functions u and 77) while (3.1|) does
not. We first take as B all functions from having derivatives u% that
are summable on any smooth manifold of dimension n. Then problem (3-1 j),
(3-lj) is reformulated as the problem of determining a function u from 39 satis¬
fying conditions (3.12) and the identity (3-2 j) for any smooth t\. Such a func¬
tion a is called a generalized solution of problem (3-1 j), (3-l2) from class 8L
If, however, there does not exist a solution in 0 (or we are unable to prove
that there exists one), then it is natural to extend St For example, one takes
5SI= IF1 iQj)- But the integral f$j(du/dN)T)dsdt in (3-2j) does not have a
meaning for arbitrary u from IT*’1 (Qj) and so, in order to get rid of it, one must
require that 1ij vanish on Sj.. For such n the identity (3.2j) takes the form
/ (",*1 + au%\) dxdt= J frfdx dt. (3.22)
Qt Dt
which is meaningful for any element u from IT'’’1 (Qy). This permits problem
(3>1) )> (3*12) 10 ke understood now as the problem of finding a function u from
t'J (^7*) £hat satisfies conditions (3-l2) anc* the identity C3-22) for any smooth
§3. ON ADMISSIBLE EXTENSIONS OF THE CONCEPT OF A SOLUTION 27
71 equal to zero on 5j.. Such solutions will be called generalized solutions of
problem (3.1 j) » (3-12) from class 1
If the class IF^’ also turns out to be insufficiently wide, then one can
waive the existence of the derivative u(. But this implies the necessity of chang¬
ing the form of the identity (3-2j), since u( enters into it. And what is more, if
one takes as 91 the space Vfor example, then it becomes necessary to find
another form for the initial ccndition (3-l2)> since the elements of (Qj) do not have
any kind of continuity (even integral) in t. Both of these problems are dealt with
upon integrating the first term in (3-22) by parts and then substituting tfig(x) for
u(x, 0). But the resulting identity contains the term (qu (*, T) rj(x, T)ix, which
does not have a meaning for an arbitrary element u from W}£®(Qy). In order to
remove it one must require that rj(x, t) be equal to zero for t = 2". For such
ijbc, l) the identity (3.2j) takes the form
J (-n + VA)^"- J VK*. 0)dx
Qt a
= ffr\dxdt. (3.2)
Qt
which has a meaning for any function it 6 A generalized solution of
problem (3-1 j), (3-l2) from class W^°{Qr) can now be defined as an element
of 2*°(*?7") satisfying the identity (3-2j) for all smooch i)(x, t) that vanish on
Sj. and for t - T. At this point the identity (3-2j) has absorbed both the equation
and the initial condition.
If we wanted to extend the class further, by waiving the existence
of the derivatives ux of a generalized solution u, we would have to transfer the
derivatives d/dx- from u to a-.riXi, which is impossible for nondifferentiable
a~. Consequently the coefficients impose a limit on the possible extensions of a
class of generalized solutions which was also noted in the Preface. One might
extend the class ^(Qf) by lowering die summability power of the derivatives
ux, i.e. by replacing it with IF*’0(()j), q <2, but we will not do this.
Thus, as the widest class of generalized solutions for equations (3-1j) with
nondifferentiable bounded a~ we take the class the elements of
which do not have any smoothness in f. On the other hand, for (3-lj) >' is
impossible to further reduce the class consisting of those elements
28
I. INTRODUCTORY MATERIAL
of * (Qj) having a bounded generalized derivative with respect to t, since for
such equations there exist solutions u(x, 0 of class (Q j) with a dis¬
continuous derivative u( (for example, the equation u( - a(t)Au « 0 has solutions
<f>U) ^(x)> whete (x) is an eigenfunction of the Laplace operator under a zero
boundary condition with eigenvalue and cf>(t) is a solution of the equation
<(,’ - aKk <j> = o).
The classes of generalized solutions of problem (3-1 j)> (3-1 j) mentioned
above are embedded in each other (the preceding ones in the ones following),
which produces a seeming lack of uniqueness in their definition. For example, a
generalized solution of problem (3-lj), (3-12) bom {Qj) could be defined
as a function from Jj’ *{Qj) satisfying all of the conditions of the problem in the
form mentioned for generalized solutions from of the same problem. But,
as is easily seen, two such definitions are equivalent, with the first proving to be
more convenient them the second. Such a nonessential ambiguity also exists in
designating the class of functions i)ix, t) entering into the identities (3.2j) -
(3.2j) . For example, in defining a generalized solution from VJ’1 [Q jO we re¬
quired that the identity (3-22) he valid for any smooth function ij(x, t) that
vanishes on Sj. Instead of this we could consider jjOc, t) to be an arbitrary func¬
tion from W\,0(Qt). The information which (3.2,) gives turns out to be the same
1 fl
in both cases since, in the space (Q j), the first set of tj's is dense in the
second.
Thus we have shown how the definition of a generalized solution of a problem
is dictated by the class 31 to which it belongs, and therefore in the sequel we
will sometimes limit ourselves to merely indicating the class ffl without citing
the complete definition of a generalized solution from this class.
We will now explain what restrictions must be imposed on the coefficients of
the non principal terms of equation (2.1) to make some of the classes between
^2°(Qj) and Iadmissible for these equations. Providing a first ori¬
entation here are examples constructed by us for tbe equations
(X) ttx, +• a‘ (JC> “) + bi (X) %
+«w«=af-/w (3-3)
with the of/. (x) satisfying only the condition of uniform ellipticity
v > 0.
§3- ON ADMISSIBLE EXTENSIONS OF THE CONCEPT OF A SOLUTION
29
Tbe solutions u(*) of these equations can be regarded as the solutions of the
corresponding parabolic equations u( + 3Ru = df/dx. - f ol form (2.1) that do not
depend on t. By means of these examples (described in detail in the book [*>5q])
it has been established chat a natural class 91 of admissible solutions for (3-3)
when a- a i>(- s a m 0 is the class ^(O), and that in order for it to be admissible
in the general case the nonprincipal coefficients of % must satisfy the restrictions
< oo, q > -2..
]£*?(-*). a{x)
lli-i i-1 In, a
with the case q « n/2 being already singular. Analogous to these conditions are
the conditions
II n /* ||
2 «?(■*.*). 2 **(■*. a(-x' <oo, q>j (3.4)
‘ ' 1-1 ?. m,Qt
for equations (2.1). The necessity of such restrictions is further indicated by the
following examples (given for the Cauchy problem, which facilitates their
construction )-
Example 1. It is eas ily seen that the bounded function u (*, t), which equals
0 for t < 1 and for t > 1, satisfies at all points except (z = 0, t = l)
the equation
n
ut — + yj^a^ = 0. (3.5)
<~i
The function u^^(x, t)= fgu(x, r)dr has continuous derivatives with respect to
* at all points, a derivative with respect to I everywhere except the point
(x = 0, t = l), and derivatives of second order with respect to x everywhere
except on the ray (* *» 0, t > l)» with the derivatives u, . and M /,
V1 i * (1/ * i*t
being bounded in tbe half-space $ > 0, and the derivatives u,.. . having the
“ U) xixi
estimate jitj ^ (*, t)| > c|l»l*l| near the tay (* = 0, « > l). In addition, the
function it^ as an integral with respect to t of the solution it of equation (3-5),
the coefficients of which do not depend on t, itself satisfies equation (3-5) every¬
where except for the points (* = 0, t > l). Yet such properties are also possessed
by the functions “^pe, <)“ fou(k-l) r^r* ^ = 2, 3, • • •. All of them satisfy
30
i. INTRODUCTORY MATERIAL
the homogeneous equation (3-5X are equal to zero for t ■■■ 0, and rapidly decrease
for |*| —> oo. Thus we encounter a violation of the uniqueness theorem for the
Cauchy problem for equations (3.J) in the class of "almost classical solutions.”
From this it follows that equations (3-5), in which the coefficients (*, t) =
of the u have first-order singularities in x, must be excluded. This
also gives condition (3-4) (the case q = n will not be analyzed here). If one re¬
writes equation (3-5) in the form
(3-6)
and identifies the coefficients - re*f/|se|2 and - n(n - 2)/\x\2 with the coefficients
o.(*, t) and a(*, t) respectively of equation (2.1), then the same functions
«(£)(*, t) also indicate the necessity for restrictions of type (3-4) on die coef¬
ficients ai and a.
Example 2. The same function a ix, t) as in Example 1 also satisfies the
homogoneous equation
a,-Att -2(/l |j-“ = 0. (3.7)
One can easily show that the averagings of it in x
u0(x,t)=. j ttpfljc — y|)a(y, t)dy
| jr-y| <p
with infinitely differentiable kernels together with their derivatives with
respect to * of any order, are continuous in (*, f) in the half-space t > 0, while
the derivatives du^{x, l)/dt ace continuous in (x, t) for n > 2, and for n -- 2
have a singularity in t only at the point (x = 0, I = l), that is weaker than any
power {t - l)-a, a> 0. Thus the functions u^{x, t), p > 0 give another example
of a violation of the uniqueness theorem for the Cauchy problem for equations
(2.1), and this time even in the class of classical solutions (if n > 2). By the
same token equation (3.7) indicates another case of inadmissible singularities
in the coefficients. These singularities can be avoided by the requirement
ll°IL.i.<jr<0°- (3.8)
Conditions (3-4) and (3-8) represent extreme cases that serve to indicate the
reasonableness of the following restrictions on the coefficients (x, t), 6f(x, t)
§3- ON ADMISSIBLE EXTENSIONS OF THE CONCEPT OF A SOLUTION 31
and a(x, *) of equations (2.1) :
2K 2*?..
H-l /-I
< CO
with
q, r, qt
(3-9)
2. On the possible smoothness of generalized solutions of linear equations.
In Chaper III it is demonstrated that the restrictions (2*5) and (3*9) on the coef¬
ficients of equations (2-1) permit one to construct for these equations a reasonable
theory for generalized solutions of the boundary value problems and of the Cauchy
problem. As an admissible class SI of these solutions we take the space
^2* which consists, roughly speaking, of functions having derivatives with
respect to x of the first order and a derivative with respect to t of order In
K2,1/3^M ^ere ex*st uniqueness theorems. The restrictions on the f and
under which these generalized solutions are obtained, are also essential and can¬
not be weakened in terms of the spaces L r(Qj) selected-by us. Thus the func¬
tion if/Q must be an element of L^{Q)y while the f ■ and / must satisfy the
conditions
2/?
1.1, <3r
ll/ll
. < oo
(3.10)
with lAj + n/2q ^ < 1 + n/4 and € [l, <»]. We also assume that the boundary
values are reduced to being homogeneous. Later on in Chapter III it is shown
that the differential properties of generalized solutions from (Q are
improved by increasing the smoothness of the coefficients and free terms of the
equations, with this improvement, as in the case of classical solutions, having a
local character (i.e. the smoothness of the solutions in any subdomain Q* of
Q>P is determined only by the smoothness of the known functions of the problem
in an (-neighborhood of this subdomain) * The dependences established in it are
exact, as indicated by the following examples.
Example 3* Consider for t > 1 the functions
i) It is more convenient to regard the case r = oo, q =z n/2+ t > 0 as a special
case of <3*5>) with r » J + nf 2e and q w n/2 + «, or even with r * 1 + n/f, q = ft/2 +
for which l/r + rt/2g < 1.
32
I. INTRODUCTORY MATERIAL
v(x, 0 = lxluPm(y). (3.11)
where the P m (y) = ym e~y and y - x2/(t - l).1' For y > 0 the (unctions P m (y),
when m > 0 do not exceed mme~m respectively, while Pq(r) < l. Jc is easy to
show that
vt ——I * («—y) 7~r •
vX{ = 2Pm_| |j;|a(m -f-X—y) j—j -.
<V,= n^Pm_,{\(rn +l~y?-{nL + X)] —
+ (m + ^ — y) Ai-iy j,
A® = 2jAr|2X/>m_, [2 (m -f X - y)’ +
~f BOt + a. — y) — 2(m+ %.)] yz-j
and that vix, t) satisfies the equation
*/ —A v = f(x,t) (J 12)
with fix, t) having the form - l»Pm_j (y) (c J t «2y + c 5y2). In view of
the boundedness of tbe (y) for m > 0 the function fix, t) has in general a
singularity at the point (0, 1) that is not higher than
c ^ =7 (. (\
with n > 1 - m for arbitrary A and m, and with fi >-m for A = - m. The function
/M(*, «) is an element of (I |* j <1, 1 < I < 2i) with any q and r satisfying the
inequalities
2(1* — K)q<n and (1 — n)/-< 1, (3.13)
from which it follows that
7'+^>, — (3-14)
If A < 0, then equation (3-12% as we see, has an unbounded solution
v w (y); but if A > 0 then this solution is bounded and is even Holder
1) By means of these functions A. V, Ivanov analyzed the accuracy of a number of
the results in [65n] on linear equations. Concerning his examples and examples con¬
structed earlier by the authors of [65n], see [ 65l»n>q}.
§3- ON ADMISSIBLE EXTENSIONS OF THE CONCEPT OF A SOLUTION
33
continuous. It turns out that the latter is nonrandom and applies to all (admissible)
generalized solutions of equations (2.1). In fact, we prove in Chapter III that if
in equation (2.1) /(*, t) 6 r(Qf) (we take fi s 0 for the present) with q and
r satisfying the inequality
-r+-£-<». «.i5)
and if in (3-9) l/r + re/2? < 1, then any generalized solution of (2.1) is Holder con¬
tinuous, and the Holder exponent grows with an increase in the difference
1 - (l/r + n/2q)- Moreover, it is proved in the same place that if f is worse, i.e.
belongs to L r (Q y) with q and r connected by the relation
7- + -!§- = l —(3-16)
in which A < 0, then any generalized solution of (2.1) belongs to the space
^12 r2 connecte<J by the relation l/r2 +-
n/2?2 = - A. Example (3-12) shows that the connections between q^, an<* ?’
r established in Chapter III are exact. In fact, from the membership of f in
Ly r(Q>]') with any r and q satisfying condition (3-14) it follows that (2.1) has a
solution v not belonging to L r^(() *) with q^ ~ B/~2A. Analogously,
by means of the same functions v(x, t), it is verified that the connections proved
in Chapter III between the membership in L r(Qj) of the functions fi (x, t) in the
free term of equation (2.1) and the membership in either L r^(9') or Hl(Q‘) of
the solutions of these equations are exact.
In Chapters HI and IV an investigation is also made of how the differential
properties of generalized solutions subsequently improve as the differential prop¬
erties of the known functions in the equation improve; in particular, an explana¬
tion is given erf when they have bounded or Holder-continuous derivatives with
respect to x and t. The dependences proved are exact in the same sense as the
dependences just adduced by example (3*12) between the summability powers q,
r for f{xt t) and the summability powers r2 ^or an^ genera^ze<^ solution.
This is outlined, by examples that are analogous to those just given.
In Chapter IV we prove die unique solvability of the boundary value problems
and of the Cauchy problem in the spaces ^ > 2 and W^a,m {Qj),
m > 1. The necessity of the restrictions imposed on the coefficients and free terms
of the equations is seen immediately for the case of the spaces They con¬
sist only in the requirement that all terms of the equation Lu * f prove to be
34
I. INTRODUCTORY MATERIAL
elements of the space (Q,f,) when any element of ft'^\Qj) is substi¬
tuted for u(x, f) in them. The analogous conditions for the spaces W^rn,m {Qp)
are that the coefficients a - in (2.2) be bounded functions, that f be an element
of and that the ai and a belong to f,{Qj.) with certain q j
and r.. However, as we will see by Example 4, the requirement that tbe a.. be
bounded is not sufficient for preserving the uniqueness theorems in the classes
Wq’1 (.Qj-), q <n, and it must accordingly be supplemented, for example, by the
requirement that the a - be continuous.
Example 4> It is easily verified that the equation
' (a6' ~
/ xlxi
has for t > 1 a solution in the form of (3*11) as long as the numbers a, m
and n are connected either by the relations
K = —m, a =
or by the relations
2m — 1 a n — 2m
4(n — 1) ’ P~ 4 (/i — 1) ’ (3.18)
« , 3 —2m n 4-tm — 4
o = -jjjrrf)'’ P- 4-d -■ <5-»9)
in which m is arbitrary. Let us choose the parameter m so that a is positive.
Then equation (3.17)will be uniformly parabolic since a+ /3 = 1/4 and the
solution (3-11) will have a singularity of the order of |*|^ with A = - m or
A = - 1. Take the first of these: v^mHx, t) = i*|-2m P m (y). It is not difficult to
show that for m = % + t. 0 < e « 1, the functions t;^m * (*, t) m f (*, r)dr,
are elem ents of W * (0), where II is the strip i \x | < 00, 0 < * < 2i, and
q = n - (j, 0 < fj « 1, which satisfy at all points of U except the points
\x =0, t > lj, the uniformly parabolic equation (3-17) with bounded coefficients
a -. The functions give an example of a violation of the uniqueness theorem
for the Cauchy problem for equations of nondivetgence form (2-2) in the classes
<n. Ch the other hand, as will be proved in Chapter IV, for continuous a-t-
one has unique solvability of the Cauchy problem in the classes If'2’1 (n) with
any q > 1 for any right side from (II) and * 'be o; and a satisfying certain
(necessary and sufficient) conditions. Thus Example 4 indicates that in order to
construct a reasonable theory for the solvability of the Cauchy problem for equa¬
tions of form (2.2) in the classes q > 1, it is necessary to impose on
the Oy, besides the conditions (2-5). some supplementary restrictions. This we
§3- ON ADMISSIBLE EXTENSIONS OF THE CONCEPT OF A SOLUTION
35
do by assuming the a.. to be continuous.
3. On necessary conditions for boundedness in the large of Holder norms of
solutions of quasi-linear equations. We proceed now to an analysis of the quasi*
linear equations (2.3) » (2.4) • For them we shall for the most part occupy ourselves
with searches for necessary and sufficient conditions for die unique solvability in
the large of boundary value problems in the classes HQp)* It is
already known, through an example for ordinary differential equations, that the
solvability in the large (i.e. solvability in domains of arbitrary size and shape,
without assumptions on the smallness of the various functions or their derivatives
forming the equations) of the Cauchy, problem and of the boundary value problems
is not merely a consequence of sufficient smoothness of the known functions form*
ing these equations but also depends on the character of the nonlinearities of these
functions. This is also true for equations in partial derivatives. But whereas in
the case of ordinary differential equations in the form du/dt - f{t, u) with a smooth func¬
tion f(tf u) the basic criterion for solvability in the large of the Cauchy problem is that
the solution u ~ u(e) does not go to infinity in a finite interval of time, for equations with
partial derivatives this requirement is necessary but, generally speaking, not suf¬
ficient, since the solution u(x, t) can "disappear” in a finite interval of time not
only by going to infinity but also as the result of some partial derivative of it
with respect to x having infinite growth.
Owing to the results on linear problems for equations (2*1), (2*2) and to the
Leray-Schauder theorem on fixed points of completely continuous transformations,
the question of solvability in the large of the boundary value problems and of the
Cauchy problem for the quasi-linear equations (2.3) » (2.4) reduces to die question
of a priori boundedness of Holder norms in +a^{Qj<) with some cl> 0 for
all possible solutions of these problems. K the boundedness exists then so does
solvability in the large. One of the principal results for equations (2*3) and (2*4)
is that the norms in the space //1 + a»V2+ a/2^^) for ajj possible solutions of
these equations can be estimated in terms of max^^luj, max^ |e^| and the
known functions in the problems, as long as the latter have a certain not large
smoothness. The only condition that is imposed on the equation (besides the con¬
ditions on the smoothness of ai (x, t, u, p), a{x, t, u, p), a.. {x, t, a, p) as func¬
tions of their arguments ie, t, u, p)) is that it be parabolic with respect to the
solution u{x, t) being investigated, i.e. the fulfilment of inequalities (2.6) or
(2.7) for (*, t) £ Q-j- when u (*, t) and its gradient are substituted for u and p
36
I. INTRODUCTORY MATERIAL
respectively. Thus this result guarantees for the entire class of quasi-linear para¬
bolic equations the "indestructibility” of a solution u (x, l) if maxQj. |u be, t)|
and ma*Qj, \ux (x, <)| do not have unlimited growth, and by the same token the
question of solvability in the large of the initial-boundary value problems reduces
to the question of a proof of the boundedness of maxg^, j«| and mai^ |. As
the examples given below show, for the entire class of equations (2-3) and (2.4)
there is no a priori boundedness of maxg^, lu^j or even of max^ j )u I f°T the solu¬
tions u (x, t) of the boundary value problems. In order for there to be one it is
necessary to impose certain restrictions on the character of the nonlinear occur¬
rences of u and p in the functions a;(x, t, u, p), a(x, t, a, p) and a.-{x, t, u, p).
Let us first explain them for max q ^ |it|. If u does not depend on x, then equation
(2-4) acquires the form
It is known (see the criterion of Osgood in [ 117] and in [ 50]) that for the bounded¬
ness of all solutions u = u.(t) of equations (3-20) in some interval f 0, 7] it is
necessary and sufficient that
A violation of (3-21), (3-22), as is easily seen, leads to the appearance of solu¬
tions of the ordinary differential equation (3-20) that are not bounded in [0, T]. It
was proved above in §2 that the assumption (3.21), (3-22) for all *60
(together with a weakened condition of parabolicity) is sufficient to ensure the
boundedness of any classical solution u{x, t) of equation (2.4) (this was done in
§2 for the first boundary value problem; it is done analogously for the other
boundary value problems). The necessity of the condition (3-21), (3-22) is veri¬
fied by the just-mentioned reference to equations of form (3-20) . k is true that
against this appeal to ordinary differential equations one can raise the objection:
if a solution u (*, t) = u (t) becomes unbounded for t —>! j, then it is also
u,4-a(x0, t, u, 0) = 0.
(3-20)
a(x0, t, u, 0)»> — <|>(|«|)|b| — b.
(3-21)
where 6 * const > 0 and \fi (r) is a nondecreasing positive function of r> 0 satis¬
fying the condition
CO
0 (3-22)
§3- ON ADMISSIBLE EXTENSIONS OF THE CONCEPT OF A SOLUTION
37
unbounded on the lateral surface S;f; is it therefore possible for u ~ u(x, t) to
have unbounded growth when it is known that the boundary (and initial) values of
uix, t) (prescribed in the first boundary value problem) are bounded functions?
One might suspect that "the matter is straightened out” by the boundary condition
and the elliptic part. The following example constructed by A. Friedman shows
that this is not the case.
Example $. Assume that the problem
«J = 0. «|,_0 = 1>0(*). «Uo = 4>iW.
has a classical solution u be, <} in Qj- = 10 < * < 1, 0 < t < T\ for smooth func¬
tions i/iq, and 02 greater than a certain constant
cS=K 0f
It is easily verified that the function z = Cj/Cj- tx(l - x) satisfies the inequality
0.
for t < 4c2 c i - 1/4 + 8^2, and the condition *lpj-= c i/c2‘ Tbe function
v = (z - u)e is nonpositive on T j and for 4 € [ 0, 4c satisfies the inequality
vt — vxx + (*■ — z — «) t» •< 0.
From this it follows that for sufficiently large A > 0 the function v cannot have a
positive maximum in Qic2S^Ac2' ^*>ere^ofe *n @4 c2 t*"e ^uoct*on v ‘s non"
positive, i.e. it > z. But z —> ~ for * = % and i —» 4c 2, and hence u cannot be
bounded in Q^c ^. Thus problem (3-23) does not have a classical solution in Qj
for T > 4^2- Condition (3-21), (3-22) is not fulfilled.
We will now occupy ourselves with determining the conditions that are neces¬
sary in order that |ux| remain bounded for bounded solutions of equations (2-4) -
Mote precisely, we wish to find conditions on the functions (*, I, u, p),
a (x, t, u, p), a..(*, t, u, p) (below we will see that they concern only the behavior
of these functions and their partial derivatives for |p| —* “)> under which for
classical solutions of equations (2-3) (or (2.4)) it is possible to estimate
maXQ1 \uxi (i) in any domain Q' 6 QJ, separated from PJ by a positive
distance d, in terms of max^y, |a| and d, and (ii) for domains max^ , \ux\*
adjacent to some part F' of the boundary Tj., in terms of ma*qj. IuI« some
(3-23)
38
I. INTRODUCTORY MATERIAL
stronger norms of u on and the distance from Q* to r^Xr*. For linear
equations, as is shown in Chapters III and IV, they are merely conditions for a
certain smoothness of the functions a-, a, a;; on T'. But this is not true for non
*
linear equations. We will show that one of the necessary conditions is the condition
\a(x, t, u, p)\ <Ji(|«|)(|/>| + 1)«, (3.24)
where (i(r) is a monotonic increasing continuous function of r i 0- This is proved
by means of the following example, which was constructed and analyzed in §2 of
Chapter 1 of [®5q]
in connection with resolving the same questions for elliptic
equations.
Example 6. The functions u5(x)= (x + d)\ 0 < o <14, satisfy on the closed
interval Q = l0.< * < % I the equations
uxx~\~c \ux\ * +e> = 0 (3-25)
with tm A/2 (l - A) and c = A-1-2f (l - AX they are infinitely differentiable, and
for 0 < A < 1 their moduli do not exceed 1. The coefficient c and the boundary
values of u j (and even max^jj |oj|) are uniformly bounded for all 5 6 (0, %)
and fixed A from (0, 1). Nevertheless the derivatives du g (*)/</* have unbounded
growth at the point x = 0 for S —» 0- The reason is that condition (3.24) is not
fulfilled: the exponent of lu^l in (3.25) is greater than 2.
Let us consider the functions u j (x) as solutions of the boundary value problems
ut~uxx — c |«Jt|2(,+‘'=,0,
it |(-,o = (-c —f-»|,t-o = d\ u 1^, ss*(l -|-6)\ (3-26)
The functions 0j (() = and 02 ^ = (l + 8)^ in the boundary conditions
are infinitely differentiable, while the function (at + d)^ in the initial condition is
bounded. The function u ? , the coefficient e, the functions ijji («), and the
derivatives of the ipi (t) of all orders are uniformly bounded for all S from (0, %),
yet the derivatives u gx are not uniformly bounded in the domains
f 0 < * < 1, tj<*< T I, t j > 0, and hence the desired property does not exist for
equations (3-26) with < > o. Condition (3.24) excludes similar equations.
In connection with the example just given one might raise the question: is
the reason for the unbounded growth of max |u jxl for S —* 0 then that it has
unbounded growth at t = 0? The example which we give below (constructed by
A. F. Filippov) shows that this is not the case.
§3- ON ADMISSIBLE EXTENSIONS OF THE CONCEPT OF A SOLUTION 39
Example 7. Consider in the domain ()j= io < * < 1, 0 < t < li the initial-
boundary value problem
2
“t “xjr ~ * I**!1"* — 0,
«Uo = (l +~)*. «U = 0,
(3.27)
for an arbitrary fixed value of a from (0, I). Suppose that there exists in Q^ a
continuous solution u (x, t) having a bounded derivative ux in QJ and continuous
derivatives u(, ux, uxx in Qy. Consider in the function
I-r^y for 0<*<1 —
i(x + /-l)°+l for
It is easy to see that u > z on and at points of Qj not lying on tbe straight
line x = 1 - t
2
Therefore
where v = z - u, and
is bounded in Qy Since maxQ ^ \ux I by supposition does not exceed a certain
constant it follows from (3-28) that for sufficiently large \ the function v m ve~^‘
cannot have a positive maximum at an interior point of Qj not lying on the diagonal
x + t = I (see §2). Also, at points of this straight line vix, t) cannot have a
maximum by virtue of the fact that
VJC — e~l/ (2i «*) Lr=l-f+0 = + °°-
On the boundary Tj, as we know, 6 < 0, and hence everywhere in the func¬
tion v < 0, i.e. u > z. But this inequality contradicts the fact that u(0, l)= 0,
Uo^ + 0*(*, l)= 1 and u(*, i) is continuous at the point (0, 1).
Thus Examples 6 and 7 show that one cannot replace the exponent 2 in
40
1. INTRODUCTORY MATERIAL
condition (3-24) by a larger one.** If equation (2.3) >s written in the form (2.4),
i.e.
(3.29)
where
j4(x, t, a, ax)
Jxi
+ t. If, ur),
then, since each of the terms making up A may be missing, condition (3.24), when
applied to equation (3-29), assumes the form
In Chapter V we show that the assumptions (3.30) together with the condition
(2.6) on uniform parabolicity are sufficient for the existence of the desired esti¬
mates of max lit^l in terms of max |u| for solutions of equations (2-3) . For the
quasi-linear equations of general form (2.4) we require, besides the fulfilment of
conditions (2.7) and (3-24), the fulfilment of corresponding conditions for the
partial derivatives of the functions «. • (*, t, u, p) and a (*, t, u, p) with respect
to x, u and p, namely
which are consequences of conditions (2.7) and (3.24) in the case of polynomial
dependence of the a -{x, t, u, p) and a(x, t, u, p) on x, u and p for large |p|.
More precisely, for an estimate of max ju^l in terms of with some a> 0
(or, more generally, in terms of max |u | and the modulus of continuity of u (*, t))
we impose the restrictions (3-31) 5“,^/dpj. > da^/du, da/dp^ and da/du,
but actually weaken the assumptions (3-31) concerning da^/dx^ and da/dx^
1) As is seen from the estimates for max \i±xI in Chapters V and VI, it is possible
to insert in the right side of condition (5-24) a slowly increasing factor <M|p|)> namely,
a factor such chat dr/r»/-(]r*{) co.
a
£ + -[(IpI+D2
+i^|(|pl + 1)2+|^|(|p| + 1)+l^l+|'3^|]
(3-31)
§3- ON ADMISSIBLE EXTENSIONS OF THE CONCEPT OF A SOLUTION
41
replacing them fay the inequality
|§|(|Pl +irI+|^-|(|/»i + l)-3<(i(|«|). (3-32)
In estimating max |»xj in terms of max |u| we impose in comparison to the just-
enumerated conditions the additional conditions
I dan (x, t, u, p) I , ,
— a <e(M)+P(|,p|, |«|),
A , t I 1 {3-33>
-^^^■>-[e(|«|)+/>(|p|,|a|)](|p|4-l)=.
in which c(() is a continuous, monoconically increasing nonnegatire function of
t > 0 whose value ((saaxQ^. |u|) is sufficiently small, and Pit, t) is a function
that is continuous on the set ir > 0, £ > Ol, is monotonically increasing in t, and
tends to zero for t—> «> uniformly with respect to t in [0, fj], where tj is an
arbitrary positive number.
Thus we have determined that, in the general case, for classical solvability
in the large of boundary value problems for uniformly parabolic equations (2-3),
(2.4) with smooth functions a;-, a and it is necessary to impose the restric¬
tions (3.21), (3-22), (3-24) , (3-30) and (3.31). The basic aim of Chapters V and
VI is a proof of the fact that these restrictions are on the whole also sufficient
for the solvability of boundary value problems in the class of smooth functions.
In Chapter V an investigation is made of generalized solutions for the quasi-
linear equations (2-3). In connection with this we note that assumption (3.24) is
necessary in order to speak meaningfully about bounded generalized solutions
u(x, c) from for equations (2-3) with smooth functions o£(*, t, u, p),
a be, t, a, p). Such solutions are defined as the elements of W^,<3(Qj,) with
vraimax ^^ |«| < that satisfy the identity
J [ — ttti, -j- at(x, t, u, ri -f- a (jc, t, a, a)n] dx dt = 0 (3-34)
Qr J
for any smooth function ij(*, t) vanishing on the boundary of Qj.. In order that
the integral on the left side of (3-34) be finite with respect to such solutions one
must require the fulfilment of conditions (2-6) and (3-24). For a correct theory of
generalized solutions that are worse than bounded generalized solutions from
W^’° one must impose on the functions a. (x, t, a, p) and a (x, t, u, p) even more
stringent restrictions, which would guarantee the finiteness of all integrals
appearing in (3-34) with respect to any such solution. We formulate these
42
I. INTRODUCTORY MATERIAL
restrictions in the form of inequalities
|a^x, t, u, p)|<c(|p|B,-+ |«|ft)4-<Pi(*. 0.
|a(*. t, u. /?)| -< Ci(|/?|B*-4-1 alp*)-hqv(jr. t),
(3-35)
where the <f>i are elements of certain j,ri thereby permitting singularities of
tbe functions ai and o in * and t.
In §2 of Chapter V it is proved (Theorem 2.1) that if the indices fii, gj1,
rj* are not greater than certain numbers, then any generalized solution of equation
(2.3) from ^2^ (Q f) H Ly T (Q j-) will be bounded. Examples show that the con¬
ditions of Theorem 2.1 are sharp (one cannot increase the indices f}t, <?~l, rj1).
One of these examples is example (2-31) at the end of Chapter I of the book t65<*]
which was constructed by us for die case of elliptic equations.
Finally, we note that for equations (2-4) the generalized solutions t»(*, t),
concerning which it is known only that they are elements of * (Qr), are in
general inadmissible [see in this connection example (2.23) i“ Chapter I of
£65q], and also Example 4 of the present section, in which the function v can
also be considered as a solution of the quasi-linear equation
We will briefly describe the basic results stated in the present book.
The various propositions of classical and functional analysis used in the book
are gathered together in Chapter II. Some of these are well known and are pre¬
sented without proof, while for others an explanation or proof is given. In the
main (§§6- 9)Chapter II is devoted to a study of functions satisfying certain
integro-differential inequalities. A particularly important role in the investigation
of solutions of parabolic equations is played by the function classes 8, and
B2 *, to which, as it has turned out, belong the solutions of these equations and
theit derivatives. It is proved that the elements of 8 are bounded^ and that the
elements of 82 are bounded, and that the elements of B-, and S2 1 are Holder
continuous. The definitions of these classes differ somewhat from those given by
us earlier in the papers [65n,o]t but their investigation is carried out in almost
the same way as the investigation of those originally introduced (see in
with the same values for the parameters a and B as in (3-17)].
§4. BASIC RESULTS AND THEIR POSSIBLE DEVELOPMENT
§4. BASIC RESULTS
43
connection with this §§5—8 of Chapter II of [651] and the paper [51f]).
1. Linear equations. Chapters III and IV are devoted to Linear equations of
second order.
In Chapter IV linear equations with smooth coefficients in the spaces
I > 2, and are studied. Exact dependences of the dif¬
ferential properties of the solutions on the differential properties of the free terms
and of the functions determining the initial and boundary conditions in the various
boundary value problems and in tbe Cauchy problem are established for these equa¬
tions, and the exact estimates corresponding to them are deduced. The epithet
"exact” refers to the differential orders and means the following: if, for example,
the free term fix, t) of an equation is an element of (^y,)f / > 2, then
it is established that the solution is an element of and by the same
token that each term of the equation belongs to the same class as their sum
/(x, t). Thus, from the membership in
fjt-2,1/2 *(£Jy.) 0f ti,e sum is deduced the
membership in the same space of each of the summands. A result concerning a
dependence of the smoothness of the solutions on the smoothness of the initial
and boundary conditions is also exact. In the example just mentioned the member*
ship of die solution uipc, t) in is proved in the case of the first
boundary value problem under the condition that the values of u{x, t) on Fj coin¬
cide with the values of some function xpix, t) from (Qp). Analogous de¬
pendences established in the spaces {Q^), m > 1, as well as a dependence
of the smoothness of the solutions on the smoothness of the coefficients of the
differential operators, are also exact (see in this connection Example 4 in §3
Chapter I).
The first exact estimates for parabolic operators, under various homogeneous
boundary conditions, were established by O. A. Lady^enskaja in 1954 [65d»e,f]
(see also [ 33c]) |n ^ Spaces W^m>m They were obtained by means of a
simple device from the exact estimates given by her earlier for operators of elliptic
type. The analogous estimates in for q 4 2 required a different analytic
technique. These were proved by V. A. Solonnikov in [H^*] (see also t*1^;
111b]) for homogeneous and nonhomogeneous boundary conditions, where the
dropping of the homogeneity of the boundary conditions required the introduction
of new function spaces with derivatives of fractional order and different differ¬
ential properties with respect to the variables x and t (in connection with die
latter see [9; 11 Id; 116b]).
44
I INTRODUCTORY MATERIAL
Estimates in //2+a, l + a/2(^j }
for the first boundary value problem for (2.2) were
proved by C Ciliberto [13] in the case of one space variable and by Barrar p ] and
A. Friedman [31] in the case of any number of space variables. For die second boundary
value problem and for the problem with directional derivative this was done by
V. A. Solonmkov [116c,d] (jn which this was proved at once for general boundary problems for
arbitrary systems of parabolic type) and by L L Kamynin and V. N. Maslennikova [56b,c].
Friedman [31b] has also given estimates of the Holder norm of ux for the solutions u of
linear equations, the leading coefficients of which are continuous, while the coefficients ir
the minor terms and free term are measurable and bounded.
In Chapter IV two distinct methods for the proof of the unique solvability of the boundary
value problems and of the Cauchy problem are stated. Ctae of them, involving exact esti¬
mates, is based on the construction of a regularizor, namely an operator that inverts the
principal part with frozen coefficients and permits a reduction of the problem for I C [0,
with r0 small to an equation of the form (/ + K)u = f, where 1 is the identity operator and
K is an operator with small ncrm. By making use of estimates catching the exact charac¬
ter of the dependence of the smoothness of the solution on the smoothness of all of the
known functions, one is then able to prove the solvability of the original problem when 0 <
* < T for any finite T. This method permitted V. A. Solonnikov [life] to establish the
unique solvability of very general initial-boundary value problems for systems that are
parabolic according to Petrovskii and Shirota, and also for a wider class of sys¬
tems introduced by him. Here this method is stated (with various admissible
simplifications and concrete definitions) in reference to equations of second orde
It should be noted that in this method the restrictions on the coefficients of
the equation and on the smoothness of the boundary and of the known functions
must be exactly coordinated in dependence on the particular class of functions
containing the solution. In Chapter IV the boundary value problems for equation
(2-2) are considered in the classes Hl,l^2(QT) and Wqm,m (Qj), while the
Cauchy problem is considered in the classes 2 and (D^} )
/T\ 7 r _ . n q n + l
(Dn+J “ n x
The second method for investigating boundary value problems is the classical
method of the theory of heat (more precisely, parabolic) potentials, investigated
in detail in the case of one space variable in papers by E. Levi and Gevrey
[69;35], Many papers [21;25;74;79;5>4;llla,c-,etc] are devoted to it, and it has found
expression in the books [25as31;112;116c;l29]. It has been used in proving the
unique solvability of the Cauchy problem and of the basic boundary value problems
for equations (2.2) in their classical statement.
§4. BASIC RESULTS 45
The potentials in terms of which the solutions of these equations are expressed
are formed by means of the fundamental solution of equation (2.2) with variable
coefficients. For equations (2.2) of general form the fundamental solution was .
first constructed in the papers [Z1], in which it was required that the coefficients
a - in (2.2) belong to #2+ + a/Z. In [94a,c;25».b] the restrictions on the
smoothness of the coefficients are lowered to the condition that they he Holder
continuous, and in a recently published note [79] it is required that they satisfy
the more general (than the Holder condition) Dini condition. By the same tokeo the
existence of the fundamental solution and of the classical solution of the Cauchy
problem for equation (2.2) is obtained under restrictions that are somewhat weaker
than those in the first method. But the results obtained by means of potential
theory on the solvability of boundary value problems are not exact, due to signifi¬
cant analytic difficulties arising in the estimate of even the first derivatives of
single-and double-layer potentials near the boundary.
In the case of a half-space it is possible with die use of heat potentials to
obtain exact estimates, both in the norms of HU/2 and in the norms of r2...
for solutions of boundary value problems for equations with constant coefficients.
Such estimates (for the heat equation), which underlie the first method of investi¬
gation for boundary value problems, are stated in detail in §§ 3 and 2 of Chapter
IV.. These estimates could be derived from general theorems on estimates of
integral operators proved in [38c] ancj [116e] (jn [life] they are applied to para¬
bolic systems). But these theorems rest on certain subtle and so far not yet very
widely known (at least among specialists in differential equations) propositions
from the theory of functions. Thus for purely practical reasons we have preferred
the more customary but somewhat lengthy method of estimating integral operators
arrising in the solution of boundary value problems in a half-space, that is stated
in §§ 2 and 3 of Chapter IV.
Chapter III is devoted to linear equations with discontinuous and, in general,
unbounded coefficients. In the main, equations of form (2.1) are investigated in it
under assumption (2.5) on the uniform parabolicity of (2.1). For them it has been
necessary not only to create new methods for investigating and obtaining solutions
but also to develop a new attitude on the concept of a solution itself, having re¬
jected the rigid classical framework (see in this connection §2 of [65c]). This
rejection, which contributed much to progress in the study of boundary value
problems, required justification, primarily from the point of view of preserving the
uniqueness theorems. For equations of elliptic type there has been available a
46
I. I INTRODUCTORY MATERIAL
class of generalized solutions, namely the class of generalized solutions with
order of derivatives half die order of the equation, for which such a justification
was clear; but for nonstationary equations the question has required studying.
Before the fifties there were two methods for proving uniqueness theorems.
The first, proposed by Holmgren, consists in proving the absence of an orthogonal
complement to the range of values of the adjoint of the operator for the problem
being investigated; (he second consists in the deduction of an a priori estimate,
namely the energy inequality. Neither is suitable for the stated purpose. The first,
because in order to carry it out it is necessary to know the solvability of the ad¬
joint problem (and it is a problem of the same type as the one being investigated)
in the class of sufficiently smooth functions, i.e. we must know that which we
wish to prove as one of the last steps in the whole investigation; the second,
because it assumes that solutions of the problem have the derivatives appearing in
the equations, which is not true with generalized solutions in die most interesting
cases.
Proofs of uniqueness theorems for generalized solutions of nonstationary
problems, that do not make use of any information on the solvability of the problem
(itself or its adjoint) in classes of functions having the derivatives appearing in
the equations, were given in the fifties. The first of them [65c] applied to the so-
called "generalized solutions with finite energy integral” (for parabolic equations
these are the generalized solutions from W^iQ■[•)), while subsequent ones
[65d—f; 73] dealt with this and certain other classes of generalized solu¬
tions (concerning them see §§ 3 and 14 of Chapter III).
The existence of generalized solutions was first proved by means of the
method of finite differences [65c]i 1) ansj then by means of functional methods and
the method of Galerkin [65d-f; l24b,c,f;7J], The passage from the classical state¬
ments to generalized ones and, connected with this, the replacement of spaces of
continuously differentiable functions by the Hilbert spaces and also
die methods developed for working with such solutions, which depend very little
on the explicit form of the elliptic part of a nonstationary operator, have permitted
us to proceed beyond the proper limits of boundary value problems for differential
equations and to study a more general object, namely, the problems
I) The book [65c] is Je voted on the whole co a mote difficult subject, chat of
hyperbolic equations, but die methods and ideas stated in it are also directly applicable
to equations of parabolic type. Strictly parabolic equations are discussed in [65d—” ] and [^24t],
§4. BASIC RESULTS
47
~- + S, (/)«»/(/). «(0) = «o. (4.1)
and
* I <42)
«(0)=%. -5r|(_o=«i.
in which it(<) and fit) ate functions of t with values in a Hilbett space H, Uq
and u ( ate given elements of H, and the Si it) are given unbounded operators in
H depending on t as on a parameter. This was done in the papers [65d-f;124b,c]
of M. I. Visik and O. A. Ladyzenskaja (see also their survey [i 24f]), The theorems
on the unique solvability of problems (4.1), (4.2) established in them presuppose
the fulfilment of only such conditions as are satisfied by the first and second
boundary value problems for parabolic and hyperbolic equations and for strongly
parabolic and strongly hyperbolic systems, by Schrddinger’s equations and by a
number of other nonstationary equations and systems of mathematical physics.
This (abstract) approach later received its development in papers [60;61;114;11;
85;etc.] 0f several mathematicians, and the monograph [75] is devoted to it.
Among the papers on the Cauchy problem for abstract equations (problem (4.1)) a
paper by T. Kato [5?*], having a different origin and different applications,
occupies a special place. Its author started from a desire to generalize the results
of the theory of semigroups (see [ISb;120;etc]) to the case of infinitesimal opera¬
tors 5 j depending oo t, and he had before him, apparently, as litmus paper the
Cauchy problem for the one-dimensional parabolic equation in its classical state¬
ment. Extensions of [57a] have been given in [57b,e;l 15a,b;etc.],
But let us return to the parabolic equations (2.1) with discontinuous coef¬
ficients. Essentially two results have been established for them by different
methods: 1) unique solvability of the basic boundary value problems in the func¬
tion classes W^iQj) and V^ {Q y) under the condition that the coefficients of
equations (2.1) be bounded, that (x) belong to £.-,(fl) , and that fix, t) belong
to L^iQj.) [65c—f; 73; 124*—f], and 2) unique solvability in the
class $2' * (t*f) under the additional condition that the a.-x and ai-t be bounded
and that i)/g belong to IF^ (0) [65d—f] (under some other assumptions concerning
the aan analogous result was proved in [33c] for equation (2.2)). In addition,
48
I. INTRODUCTORY MATERIAL
methods have been given [65c;124f] permitting one to prove, roughly speaking,
that an increase in the smoothness of all functions enteting into the problem by
unity in * and by H in t increases the smoothness of the solution of the problem
by the same amount.
Yet everything that had been done on equations with discontinuous coefficients
left a large gap in regard to revealing the true dependences of the differential
properties of the solutions on the known functions in the problem. Thus it was
clear that solutions from can be obtained under somewhat weaker as¬
sumptions than those concerning the boundedness of the minor coefficients. But
will they be necessary? On the other hand, how will the smoothness of solutions
from increase under an increase in the summability power of the free
term? When do they become bounded or Holder continuous? And so on. The first,
and unexpected, result of this kind was a theorem of Nash [8s] on the possibility
of estimating the Holder norm (everywhere except in a neighborhood of a singular
point) for a basic singular solution of the equation
ui--§Fi (a" <*• 0 u*i) ~ 0 (4-3)
in terms of only v and ft from condition (2.1). This theorem and an analogous
theorem of De Giorgi [17] on the elliptic equations (d/dxjia-, (x)u .) = 0 laid the
f
foundation for detecting new connections between the properties of the coefficients
and the free terms of the equations and the properties of their solutions. From
Nash’s result followed the possibility of estimating the Hdlder norm |u|^ for
smooth solutions u(s, t) of equations (2-1) under the assumption that a; a /\ = 0,
and the A(-, a and f are bounded [88;62a,«;0Ic]. But the complexity and opacity
of his proof did not permit one to essentially sharpen his results and find exact
conditions on the coefficients and free terms of (2.1) under which there would
exist some smoothness of all solutions of (2.1). This was done, using another
approach, by O. A. Ladyzenskaja and N. N. Ural 'ceva [651,"]. They consider the
complete equation (2.1) and establish when all of its solutions (including the non-
classical but generalized ones) belong to some function space, in particular, the
Hdlder space Ha’a^. By means of specially constructed examples [65l,n.q] it
was shown that the conditions imposed in this connection on the known functions
and the free terms of the equation are not only sufficient but also necessary.
These conditions are expressed in terms of the membership of the functions in
question to the spaces L m (Q j.) with different q. [65c,d,n] form t(,e basis of
§4. BASIC RESULTS
49
Chapter III. In comparison with them the following generalization is introduced:
instead of the spaces LW(Qy.) the full scale of spaces L r(Qy) is considered
and in terms of it the properties of the known, and in one case also the desired,
functions are formulated (see in connection with this the papers [ 51d,e,f;62c;41]).
The basic contents of Chapter III are as follows. First the unique solvability
of the first and second boundary value problems for (2.1) in the class
is established under the widest possible conditions on all of the data of the
problem. Then we outline step by step how the differential properties of all the
solutions of equation (2.1) improve as the properties of the known functions improve,
in particular, when they become classical. The proof of existence is carried out
by Galerkin's method. This method is suitable both for proving the solvability of
the boundary value problems and for actually determining their solutions. As a
calculating method it was applied long ago to stationary and nonstationary problems.
A proof of its convergence for nonstationary problems commenced with the papers
[ 39; 46b] (see aiso £ 124f]). Other possible methods for proving existence
theorems are described, with references to their origins in the literature, in §§14—18. We
note that in carrying out all of these methods restrictions were imposed on the coefficients
of the equation that are stronger than the restrictions made in §§1—5 of Chapter HL
In Chapter III there is one substantial omission, namely the absence of a proof of
Harnack’s inequality (a formulation of it is given in §10 of Chapter HI). This is explained
in part by the fact that a proof of this beautiful inequality is for tbe present long and com*
plicated, and in part by the fact that it is not used anywhere in the book. For solutions of
equations (4.3)** is established by J. Nfoser [87b]. Its generalization to the full equation
(2.1) is given by A. V. Ivanov [51e], Aronson and Serrin [*32d]) Kurihara [133] and
Trudinger [136], In papers [131**b] of Aronson the existence of the Green’s function is
proved for the equations (2-1) with discontinuous coefficients and its behavior is investi¬
gated in a neighborhood of the singular point.
2. Quasi-linear equations and systems. Chapter V is devoted to quasi-linear parabolic
equations with principal part in divergence form, i.e. to equations of form (2.3> Its main
results are proofs of the unique solvability in the large, in the class of smooth functions,
of the first boundary value problem, the Cauchy problem and a number of problems with
nonlinear boundary conditions, and only under such restrictions as are essential. Preced¬
ing diem we obtain a priori estimates of various norms of the solutions in terms of known
quantities, including both interior (local) estimates, valid for all solutions of the equations
considered, and total estimates in the whole domain Qj>, that take into account die initial
and boundary conditions. Part of tbe estimates are carried out for bad
50
I. INTRODUCTORY MATERIAL
(generalized) solutions, and the others under the assumption that the solutions
ate classical. All estimates could be carried out under not large (necessary) as¬
sumptions only on the smoothness of the solutions being investigated (as was don<
in [ 65q] for elliptic equations). But we did not do this, partly because we did no«
want to make more tedious the calculations that are difficult as it is, partly be¬
cause we would have had to impose additional restrictions on the functions making
up the equation, but mainly because our path of investigation of quasi-linear equa¬
tions proceeds from good solutions to bad solutions and by the same token is not
required in estimates of such a type. We first obtain classical solutions of the
problems, using a priori estimates of the classical solutions and the Leray-
Schauder principle, and then, by reducing the assumptions on the smoothness of
the data of the problem, we obtain worse solutions as limits of the good solutions.
One could also go in the opposite direction: first, under not large assumptions on
the data of the problem, one establishes the existence of bad solutions, and then
one proves an improvement in their properties under an improvement in the proper¬
ties of the data of the problem.
In Chapter VI quasi-linear parabolic equations of general form (2.4) are in¬
vestigated. The following cardinal fact is proved for the entire class of these
equations: if a solution u(x, t) has ut and the uxx from while the ux
are bounded and continuously depend on t in the norm of (fi) (the class of sucl
solutions is denoted by SI), and if the equation (2.4) is parabolic with respect
to this solution, i.e. inequality (2.7) holds under substitution of u(x, t) and
u% (*, i) for a and p respectively, then tbe ux C*» t) are Holder continuous (belonf
to Ha'a^2{Qj.)) and their norms 1^1^ for any Q' CQy are estimated only in
terms of mas^ |ui|, the known parameters characterizing the functions
at- (x, t, u, p), 3a((x, t, u, p)/dpk, p)/da, da.(x, t, u, p)/dx-,
a(x, t, u, p) for u «• u(x, t), p -- ux (x, t), and the distance from Q ‘ co A
corresponding result is also established in a closed domain. This proposition
together with the results of Chapter IV on linear equations provides the possibility
of obtaining estimates of all norms ^ > 1> *n Eerms of ma*Qy lul an^
maxpj, 1*1^| and known quantities.
Later in §3 of Chapter Via priori estimates of max^ » !«*! and maxgr|ux|
are established in terms of quantities characterizing known functions and in terms
of weaker norms of u, namely, lul^ with some a> 0 or max^y |u|. As the ex¬
amples in §3 of Chapter I show, this is not possible for the whole class of equa¬
tions (2.4) but in general only under definite restrictions on the behavior of tbe
§4 BASIC RESULTS
51
equations (2*4) but in geo era I only under definite restrictions on die behavior of
the functions a,.{x, t> u, p) and a (x, t, u, p) for p —» oo. An estimate of max |i/J
in terms of *s carried out under only such assumptions whose necessity
is confirmed by the examples of §3* As to the estimate of max Iu<Qp in terms of
max |«|, there exists one condition (the smallness of t in inequalities (3-33))
whose necessity is not established;^ it is not needed for equations (2-3) but it
is necessary for very close objects, such as the system of Heinz in §2 of Chapter
V1L We note that for individual classes of equations (2*4), in which die functions
a if ($, t, u, p), a U, i, it, p) have a special form, it turns out to be possible to
estimate max |u„j without any restrictions on the orders of growth of the a • and
* */
a with respect to p or else by using only a portion of such restrictions.
The estimates of obtained in §§2 and 3 together with the well-
known estimate of max^^, |u| and the estimates in §5 of Chapter IV with respect
to linear equations with smooth coefficients provide all that is necessary for a
proof of the solvability in the large of the boundary value problems for equations
(2-4). In Chapter VI, unlike Chapter V, we restrict ourselves to only the first
boundary value problem. Its solvability is proved by means of the Leray-Schauder
principle. One could use other theorems on the solvability of abstract nonlinear
equations, for example the theorem of L. V. Kantorovic oo the convergence of
Newton’s method for a solution of such equations, if one observes that the unique¬
ness theorem holds for the problems being studied.
The results on quasi-linear equations stated in Chapters V—VII, except for
the results in §9 of Chapter V and part of §5 of Chapter VI, have been obtained
by the authors of [*>5q] (see [ 65h,n,o,q] aujj also [65PJ51*]), They provide
answers to questions concerning the classical solvability in the large of boundary
value problems and of the Cauchy problem for quasi-linear uniformly parabolic
equations.
An investigation of quasi-linear parabolic equations was begun with papers
of Gevrey C^5]. in them and also in subsequent papers ( 25»;31?80; 114c]
solvability in the small of the boundary value problems and of the Cauchy problem
was proved. Such solvability, as in the case of nonlinear equations of other types,
holds for the whole class of quasi-linear parabolic equations (2*4) as long as the
1) When we speak of the necessity of some such condition for the possibility of
some such estimate, we mean that there exists among all equations of the form (2*4) an
equation, satisfying all conditions except the one in question, for which the estimate is
52
I. INTRODUCTORY MATERIAL
functions entering into the problem are sufficiently smooth and are compatible wit
each other (in connection with conditions of compatibility see §5 of Chapter IV
and the following text). Its proof is based essentially on properties of linear
parabolic operators only and does not make use of any a priori estimates for solu¬
tions of strictly nonlinear equations (the only exception is the well-known estimate
for the maximum of the modulus of a solution, based on the maximum principle) .
Tbe same remarks also hold in the papers of Gagliardo [33d], Friedman [ 31] and
P. E. Sobolevsku ( H4b,c] devoted to different classes of equations of the form
u, — atj(x. t)= a(x, t, a, ux) (4.4!
wi th linear principal part to which the right side is subordinate.
Strictly nonlinear estimates first appeared in the papers of Ciliberto [ 13],
Prodi [ 96 ] and Ventcel' and Oleinik [ 123e,f] devoted to quasi-linear equations
of the form
ut = <*11 (*j. t, a) uXiXt -)-£(*,, t. u)uXl-\-c(xt, t, «) (4-5)
and certain generalizations of them (see [40»I6;*«.]). They permitted one to
prove solvability in tbe large of die basic boundary value problems for these equa¬
tions. But in obtaining these estimates essential use was made of both the special
form of equation (4.5) (namely, the independence of Ojj (*^, /, n) from ux^ and
the less than quadratic growth of a fa ^, f, u, ux) - b fa t, u)ux + c fa j, t, u)
with respect to ux ) and the presence of only one space variable.
The first paper to present nonlinear estimates for manydimensional quasi-
linear equations was [®5hj. The unique solvability in the large of the first
boundary value problem is proved in it for many-dimensional quasi-linear equations
of the form (4.5) under the condition that da., fa, t, u)/du be sufficiently small.
Methods of estimating max |ux|, that are suitable for the general quasi-linear
equations (2.4X are given in it for arbitrarily small subdomains (local estimates)
and for the entire domain of definition of tbe solution u fa, t) (total estimates) .
These estimates have been used in subsequent investigations on the solvability
of nonlinear boundary value problems both for elliptic [ 122*;62e;<S5q] and for
parabolic [ 123a;62a,e;65o,o;91c] equations. They and estimates of the type of
De Giorgi and Nash for linear equations have permitted us to establish solvabilit;
in the large for elliptic equations of the form
atJ(x, u)uXlX/+a(x, a, ax) = 0. (4.6)
§4. BASIC RESULTS 53
under certain natural restrictions on a {%, uy p) [ 122a] analogously [^e] for
parabolic equations of the lotto
ui aij(x, t. u)uXlXj + a(x. t. u. ux)—0 (4-7)
under certain supplementary (nonessential) conditions on a (x, t, tt, p). In § 3 of
Chapter VI we give a derivation ot these estimates, using an approach somewhat
different from our original one, with which it has been possible, as in the elliptic
case [654], to effect a maximum reduction in the number of derivatives required of
the solution.
The general case of quasi-linear equations required the obtainment of still
other nonlinear estimates for solutions u (x, t) of equations (2-3) and (2-4),
namely, estimates ot the Holder constant for a (*, t) and the ux (*, l). These esti¬
mates were given in [ 65n,o]j an<j under conditions that were in a definite sense
necessary. They are stated in Chapters II, V-VII. From the subsequent papers
we mention [29], in which an estimate of maxg^, lu* jl is given for solutions of
general quasi-linear equations in the case of one space variable by making use of
only necessary assumptions on tbe behavior of the known functions
OjjOtj, t, u, pj), a (*, l, a, p) for p —> 00. We present it in §5 of Chapter VI.
In Chapter VII a certain class of linear and quasi-linear parabolic systems of
second order is studied. This class was chosen by us according to the following
principle: if one rejects from a system all minor terms and "freezes” the leading
coefficients (i.e. takes arbitrary values for their arguments), then the maximum
principle must be fulfilled for the resultant linear system, i.e. any norm of a vec¬
tor solution u (3c, l) must take its least and greatest values on ! ’ j-. We have re¬
tained this basic property, which is inherent in all parabolic equations of second
order, for the class of systems investigated in Chapter VH. It has turned out that
the same properties that are valid for one equation of second order, in particular
the classical solvability in the large of the basic boundary value problems, are
also valid for this class of systems.
Tbe last sections of Chapter VII are a survey of what has been done on linear
equations of high orders and systems of parabolic type. Special attention is given
to papers in which exact dependences and exact estimates corresponding to them
are established for their solutions in various function spaces.
3- On some unsolved problems. Let us now indicate several unsolved problems
connected with a further investigation of nonlinear problems of parabolic type.
For quasi-linear uniformly parabolic equations of general form (form (2.4)) it is
54
I. INTRODUCTORY MATERIAL
desirable to understand whether the assumption concerning the sufficient smallness
of t(ju[) in inequalities (3*33) is necessary. It is superfluous for equations of
form (2-3) when m = 2. If it turns out to be necessary for equations (2.4), then it
would be interesting to have an explanation of whether or not it will be needed for
equations of form (2.3) with die a; {*, t, u, p) satisfying the inequalities
v(i«i)(i/>i+ ~-i^ju-p) Uj 4g
<m(|«|)(|pi+i)'*”V
with arbitrary m > 1. As shown in [^’^]} for elliptic equations
~-(a,(x, u, ax))+a(x, a. ux) = 0,
that satisfy condition (4-8), f(|“|) can be arbitrary.
A second series of questions is connected with the replacement of the con¬
ditions (2.6) and (2.7) of uniform parabolicity for (2-3) and (2.4) by the conditions
v(|«l)(W + 1)'"V< —hlj
°Pi y (4.9)
and
v(|«I)(|/»| + I A2 <<*„(.*. t, u.
<4'l0)
with different m j and What are the conditions for solvability in the large of
tbe boundary value problems for such equations?
A third set of questions concerns equations of forms (2-3)®nd (2.4) for which
the a;(*, l, u, p), a(x, C, u, p), t, u, p) have vatious singularities in the
arguments « and p. Such equations are encountered, for example, in Stefan
problems, which can normally be formulated as ordinary boundary value problems
for equations of form (2.3) with elementary nonlinearities in the known functions
but with discontinuities of the first kind in a (see §9 of Chapter V). It is com¬
paratively simple to establish the existence of generalized solutions for Stefan
problems. Yet the classicalness of these solutions, including the smoothness of
the free (unknown) boundary, has been investigated only in the one-dimensional
case. A number of other linear problems with unknown boundaries can be re¬
formulated in the fotm of ordinary boundary value problems for quasi-linear equa¬
tions of form (2.3) with the functions a-fa, l, u, p) and a (x, t, a, p) having dis¬
continuities in u. In problems of underground hydrodynamics equations of form
§4. BASIC RESULTS
55
(2-3) are encountered with die (unctions a(. (x, t, a, p) having unrestricted isolated
singularities in p, and also with a-(*, t, u, p) fot which the function i/(|u|) in
(4.8) vanishes when |it| = 0 (for equations admitting a degeneracy, for example,
filtration equations, see [6;22;54; 103; 130]). It is necessary to construct examples
which would outline the contours of a possible theory of boundary value problems
for such equations.
Further, it is desirable to investigate the general nonlinear parabolic equations
a, = F(x, t. a, B,(. UVj) (4-11)
and above all to prove the following proposition concerning the entire class of
equations (4-11): if u{x9 t) is a sufficiently smooth solution of equation (4*11)
and equation (4-11) is parabolic with respect to this solution, then the Holder
norms |u|^* for u can be estimated in terms of max^ |u, u.%f u%xt uj and certain
characteristics of the known function F {x, t, u, p., p^*), which is assumed to be
smooth. One must then find those conditions on the function F{x, t, u, p., p^.)
under which max^*|uf, uxx\ can be estimated in terms of |it, ux\and
K\($' can be estimated in terms of aa& m&xQj lux I’ an<i so on.
All of these problems are formulated by us for equations of second order. We
believe that the results and methods stated in the present book will prove to be
useful in solving them.
The following set of problems concerns systems and equations of high
orders of parabolic type. It is desirable to carry out investigations for them with
the same degree of thoroughness as was done for equations of second order. At
the present time linear problems ate well investigated in the case of smooth coef¬
ficients. For discontinuuous coefficients the only known results are analogous to
the results in §§1-5 of Chapter III. They concern strongly parabolic systems
and are proved in the same way as for one equation of second order. As to nonlinear
problems, particularly their classical solvability, the results are comparatively
few. Let us point out a number of papers (in addition to those mentioned above)
that are devoted to an investigation of quasi-lineat equations and systems.
They can be conventionally divided into three groups. To one of them we
refer the papers [ Hb—d;22;68»;84»,b; 124d,e;134] devoted to quasi-lineat equations
and systems of any order.- Existence theorems are established in them for gener¬
alized solutions of boundary value problems. For this purpose one makes use of
Galerkin’s method. Beginning with papers of Minty [84] and Browder [11] there
emerged a new feature in the implementation of this method, consisting in the
56
I. INTRODUCTORY MATERIAL
carrying out of weak passages to the limit under die sign of the so-called monotone opera¬
tors. Questions concerning the uniqueness of such solutions under "natural** re¬
strictions and those concerning an increase in their smoothness under an increase
in the smoothness of the known functions in the problem remain unresolved.
To the second group can be referred the papers [60b;57c;l 14b,c;«tc.] £a whicl
there is investigated the Cauchy problem for abstract equations of the form
~-\-S(t)u = f(t. a), a(0) = u0,
where u(t) is the desired function of a parameter e 6 [0, T\ with values in a
Banach (or Hilbert) space H, 5(e) is a linear, in general, unbounded operator in
H depending on ( as on a parameter, and fit, it) is a known function defined on
[ft T\x H and subordinate in some sense to S (t). Tbe results obtained in regard
to its solvability made it possible to prove the solvability of the first and second
boundary value problems for equations of form (4.4) undet the condition that
a (z, t, u, p) have a weak oonlinearity in u and p.
Finally, in the third group can be combined the papers [ J7;65g; 123h-d; 126]
in which are analyzed certain classes of quasi-linear systems encountered in the
mechanics of continuous media and systems simulating them. Of these three groups
of papers we give an account dealing only with the first (see the end of §6 of
Chapter V),
We have also bad to leave to one side many other investigations concerning
the stabilization of solutions uix, t) for t —» oo, the analytic icy of solutions in
x under analyticity of the coefficients of the equations, the behavior of solutions
for |*| —> oo (theorems of Liouville type) and fot the leading coefficients tending
to zero (problems with a small parameter), the analysis of nonclassical problems
for equations of parabolic type (for example, such as in [66]), and others. Also
untouched are questions arising in the theory of probability in the study of con¬
tinuous Markov processes (see [2<‘>27]), including investigations on degenerate
linear parabolic equations.
The extensiveness and diversity of the material, and in certain directions a
definite lack of completeness to it, has not permitted the inclusion of it in this
(even without it, alas!, thick) book.
CHAPTER II
AUXILIARY PROPOSITIONS
In this chapter are collected propositions that concern arbitrary elements of
various function spaces and that for us play an important but auxiliary role.
In §1 are presented the Cauchy and Holder inequalities and various conse¬
quences of them. The second section is devoted to embedding theorems and the
inequalities corresponding to them, including certain multiplicative inequalities
for functions depending in the same way on all of their arguments • xn.
Because of the need to know the character of the dependence of the constants in
these multiplicative inequalities on all of the numerical parameters entering into
them we have had to give a careful derivation of them. In the same section are
formulated a special case of the second basic inequality for elliptic operators and
a theorem on the extension of functions of class V'^(S) from S onto all of Q. In
§3 ate presented embedding theorems and extension theorems, for functions de¬
pending on x and on i, with respect to which the functions have other proper¬
ties.
In §4 are proved several simple assertions on averagings of elements of the
spaces 1,^ r(Qf) and V ^ °{Q j>) and their "cuts" = max In - k, Of.
In §5 are collected the different propositions used in the proofs of Holder continu¬
ity for functions of classes an<* investigation of other properties of
smoothness and summability of generalized solutions with various summability
powers.
The remaining sections (§§6—9) are devoted to an investigation of the func¬
tion classes S and E2 introduced by us. The elements of these classes are
functions satisfying certain systems of integro-differential inequalities. It is
proved that the elements of !I are bounded functions (§6), while the elements of
the classes B2 are Holder continuous in x and t (§§7-9). At the same time
estimates are given of max M for u € SI and of for »€S2 that depend
only on the numerical parameters entering into the definitions of !t and $2-
57
58
II. AUXILIARY PROPOSITIONS
The propositions proved in §§6—9 are used essentially in §§7—12 of Chap¬
ter III and in the Chapters V—VII devoted to nonlinear problems, since in the major¬
ity of cases the solutions u of parabolic equations and their derivatives i*% belong
to SI or B2.
§ 1. SOME ELEMENTARY INEQUALITIES
We will frequently make use of certain well-known algebraic and functional
inequalities. Among the algebraic inequalities we require the following.
1) Cauchy’s inequality
V VwJ-
valid for any nonnegative quadratic form with = aj. and any numbers
*i* V
2) "Cauchy’s inequality with <”
jl*2, u.2)
valid for any < > 0 and arbitrary a and b;
3) Young’s (mote general than (1.2)) inequality
(1.3)
771 T71
valid for any positive a, b, ( and m > 1.
From the functional inequalities we require:
4) the inequalities
K».«)I<M-M.
the first of which is valid for the norm of any Banach space, and the second for
the scalar product and norm of any Hilbert space;
5) the Cauchy-Bunjakovskii-Schwarz inequality
j E <’?**)*• (i-4)
which is a special case of the last inequality;
6) Holder’s inequality
§1. SOME ELEMENTARY INEQUALITIES
him* <n(j
a z-i \a /
in which > 1, 1 = 1;
7) the consequences of Holder’s inequality
r
J #(■*. ow(JC, t)dxdt < J ||aHT| n • IIpII ^ cdt
Qt
<ll“\\,.r,Q 'MU. ' ff>‘- '>»-
‘ ”-l r-t ‘ ”
59
(1.5)
(X.6)
and
^ II 8 !lx/J, ItT, Qj. ■ II * llx1*, n'r, Qr •
X+'F=i* tt~+'7P:"== 1: X' x'’ f1-
We write the latter in a form that is more convenient for us:
(1.7)
\uv
11*. '■ Qt ^ II “ Ilf,, r,. Qr ‘ IIv II W, it, , ft >0, (1.7')
* r,~r # Qr
8) the inequality of form (1.6) for several factors
III "< o
Or (-1
(1.8)
where
?i>i. '■/>>. 2'•f1 —i.
Inequalities (1.4)-(1.8) are valid for arbitrary uXx), v^x) given in fl, and
u(x, t), vlx, t), u.ix, t) in QT, for which the norms in the right side of the corre¬
sponding inequalities are finite.
We note the following useful proposition.
Lemma 1.1. If a summable function it (*, t) in Qj. satisfies for any ij(*> t)
from Lq (Q j) the inequality
60
II. AUXILIARY PROPOSITIONS
atidxdt <c||r,||? r Qt (1.9)
Qt
with the constant c not depending on if, then u(x, t) belongs to the conjugate
space Lq'fr‘(QTX q' =?/(?-1), r'= r/(r-1) and ||a|lq',r',QT^c- Here, as
above, r> 1, q > 1.
§2. THE SPACES »"'(«) AND UH.0). EMBEDDING THEOREMS
In §1 of Chapter I were defined spaces and WHQ) with q > I and in¬
tegral I. These spaces were the subject of special investigations (cf. t11 la;49>
35c,d; 89; 90b; 5;9;38a,b; an{j others]). We present here some of the basic
results of these investigations that are needed for the purposes of the present bode.
They concern the behavior of arbitrary elements of spaces IP* (fl) on manifolds of
dimensions r<s and the convergence relations in these spaces with different
I, q and r.
We recall that the norm in L^(fi) (q > l) is defined by the equality
<«l"dx]" , (2.1)
)
and the norm in IP*(Q) is defined by
I!« |fi?a = 2 {{“))% - 2 2 II D‘xu ||s, q. (2.2)
l-o /-o U)
One of the central formulas, which we will repeatedly make use of, is the
formula of integration by parts
I dx = — J uvx. dx + | av cos (n, x,) ds (2.3)
n q s
for bounded domains Q. In it n is the outward unit normal to the boundary S. It
is valid for any functions u from B'J(fi) and v from IPp(fl) with l/q + 1/p < 1 +■
l/n (more precisely, for specially chosen representatives of them from the func¬
tion classes that are equivalent to them on Q) as long as S has a certain regular¬
ity (for example, if 5 is piecewise smooth). If the function u(x) belongs to
lf*(fl) (or v e then formula (2.3) acquires the form
uxpdx = — J uvXldx. (2.4)
§2. THE SPACES t*(Q) AND 61
We recall that IPj(fl) was defined as the closure in the norm of ^*(0) of
the set of functions that are infinitely differentiable and have a compact support
in SI. Under such a definition of formula (2.4) is clearly valid for any
boundary S of fl. For piecewise-smooth boundaries If^(Q) can also be defined
as the set of all ele meats from IPj(Q) that vanish on S, with the norm of I^^O).
By an embedding operator from the space W^fSl) into a space H'*1 (Sr) (or//“(S,)),
where Sr is a surface of dimension r < n, or a subdomain of fl when r = n, we will
mean the operator that associates with each function from W'p(Sl) Its trace on Sr.
Various sufficient conditions are known under which such operators are bounded
or completely continuous. One of them is the following:
Theorem 2.1. ^ The embedding operator from W^iSl) (I integral) into
£^(Sr), where SI is a bounded domain in euclidean space of dimension n, and
Sr is some plane r-dimensional piece belonging to Q, is bounded if n> Ip, r>n~ Ip
and q < pr/(n — Ip), and is completely continuous for q < pr/(n - Ip). For n -
Ip it is completely continuous for any finite q. For n < Ip the embedding oper¬
ator from IF^iQ) into Ha(Sl) is bounded for a < (lp-n)p; for a < Up - n)/p this
operator is completely continuous. It is assumed here that I > 1, while a < 1.
The basic part of Theorem 2.1 was proved by S. L. Sobolev and V. 1. Kond-
rasov (see [H3a] and [112]). Additions to the propositions proved in U 13a] on
the embeddabiiity of IT^fi) in //“(fl) are given in tbe papers of several authors
(see [33;90b;49a*fet ai.])# hi these papers are described the restrictions to which
the domain fi must be subjected. For domains 0 with piecewise smooth bound¬
aries (their definition is given in §1 of Chapter I) Theorem 2.1 is correct.
This theorem guarantees for any function u(ac) from i^(fl) the validity of
the inequalities
M*.S, S c WI'?Q (2-5)
and
The first is valid for q < pr/{n - pi) when n > pi and for any finite q when
DThis and other analogous theorems concerning an arbitrary element u(x) from
must be understood as the existence of at least one function, equivalent to u(*) on
CL, for which the assertions of the theorems are true.
62
II. AUXILIARY PROPOSITIONS
n - pi, where r > n - pi. The second holds for n < pi with a <(pl - rt)/p, a < 1.
The constants c depend in general on n, p, I, r, q, Q, Sf, but not on u(x).
From these inequalities one easily derives the two following inequalities:
w£B<‘(Kr,.a+iw|a) (2-7)
with q < npAn - p) for p<n and w ith any q for p = n, and
max | a P < e (!) a, |g 2 + |! a f a) <2-*»
for p > n.
We note that for bounded 0 and any function u(x) from £~(U)the inequality
11“ Up, a c 11“ a
with c = (mes which follows from Holder’s inequality, is valid for
'V
p > p.
The following proposition is useful: for a fixed bounded fi the norm IMI^jQ
is equivalent to li*lip JJ + llulli fl> ®»d if the function uix) vanishes on
some part of the (n — l)-dimensional boundary of 0 that has a positive measure,
then IWIp'J) is equivalent to Iuxllpin ■
To die assertions of Theorem 2.1 should be added the following: the norms
of the embedding operators are uniformly bounded for all possible situations of
Sr in 0, and the variation of die operators is small under a small displacement of
Sr and even under a sufficiently smooth small deformation of Sr. Corresponding
assertions follow from Theorem 2.1 for nonplanar sections of 5r as long as they
are sufficiently smooth (for example, if Sr € 0*). Indeed, by straightening them
with the help of £-times boundedly differentiable transformations of the indepen¬
dent variables, we arrive at the case described in Theorem 2.1.
Besides Theorem 2.1 we will need the following mote refined characteristics
of embedding operators, expressed in the form of the so-called multiplicative ine¬
qualities.
Theorem 2.2. For any function u(jc) 6 IT\0), m > 1, and number r > 1 the
inequality
il«lU<Pil“*C,all«lC2 (2-9)
is valid, in which
§2. THE SPACES If''(fl) AND Hl(0) (ft
and-. 1) for m > a - 1 the number q can be any number from [r, oo], while ft =
(l + (m - l)/m r)a; under a variation of q front r to oo the number a varies from
zero to m/(m + Km - l)), including both ends;
2) for n > 1 and m <n the number q can be any number from [r, nm/(n - m)],
if r < nm/{n - m), and any number from bun/in - m), r] if r> nm/in - m), while
/3 = ((«- l)m/(n - m))“; under a variation of q between r and nm/(n - m) the num¬
ber a varies between 0 and 1, including both ends;
3) for m > n > 1 the number q can be any number from [r, <*>) and 0 =
max{?(n - 1 )/n; 1 + r(m - l)/m}“. Under a variation of q from r to “ the number a
varies from zero to nmj[nm + r{m - n)] excluding the right end point. If m > n > 1,
then (2.9) is also valid for q « “ with a = nm/[nm + r(m - «)J for some 0 <
Remark 2.1. The condition in Theorem 2.2 that uix) vanish on 5 can be re¬
placed, for example, by die condition
J u(x)dx — 0. (2*n>
Q
In this case the constant ft in (2.9) will depend on n, m, r, a and on tbe domain
fl, although it does not vary under a similarity transformation of fl. The bound¬
ary 5 must be piecewise smooth.
The first inequalities of the type (2.9) were found independently by E. Gagliar-
do [33d] and O. Ladyzenskaya (see her book Mathematical problems in the dynamics
of a viscous incompressible fluid, Gos. Izdat. Fiz.-Mat. Lit., Moscow, 1961 [English
transl., Gordon and Breach, New Yoik-London, 1963}; see also [90b; 38a, b; 49a, b]).
We shall prove items 1), 2) with the help of a method suggested by Ladyzenskaya and
extended by L. Nirenberg and K. K. Golovkin. We used the same approach in the
Russian edition of this book for the case m > n > 1. Here, for m > n > 1, we demon¬
strate the approach of V. P. Il’in. The expression for 0 obtained by this method is dif¬
ferent from the expression for fi given by the first one (for m> n> 1).
The inequalities (2.9) for m < « are a special case of more general multi¬
plicative inequalities proved with different methods in [33;38a;49a]_ ^ wjjj <je_
rive inequalities (2.9) here for any m > 1 and we will observe on what and how
the constant ft depends.
For the case m > n = 1 we make use of the inequality
which is valid for any s > 1 and q > r. We estimate the integral oo the tight by Holder’s
inequality and then integrate both sides of the resultant inequality over fl. This
64
II. AUXILIARY PROPOSITIONS
gives
f 1 a (x) f dx< f| u (x) \'dxls (| ux \\m a || u (2.j2)
q \ m-1 ’ /
Let us take s « 1 + (m - l)r/m and designate {q - r)/qs * a. Then the parameters qf r,
m and a him out to be connected by the relation a (l/r + On - l)/m) » l/r - \/qt
which coincides with (2.10), and from (2.12) we obtain
«“IU<(‘+ '■)“n«j|am. 0
i.e., inequality (2.9) with /3 = (l + On - 1 )r/m)a . Since under a variation of q
from r to “ the constant ft remains bounded, it is clear that (2.9) also holds for
q z:oo,
Consider now the case n > 1 and m < n. We first show that for any function u(%)
from ff'p(fl) tbe inequality
Ml* a< MM,. a (2.B)
is valid, in which p is any number from [l, «), p = np/(n - p) and c^ m
(n - l)p/(n - p).
We first establish it for p = 1, i.e. we verify that
b<HII-'«#?»• (2-14)
Inequality (2.14) is valid for n ** 2 since
J J x2)dxtdx2
■< f max|a(*i, x2)\dx, f max|u(jr,, xt)\dx2
J * J *i
< J J J
Suppose it holds for dimension n - 1 > 1. Then it also holds for dimension n,
since by Holder’s inequality and our induction assumption
n
J |«1*1 •*„)! dx, ... dxn
< J dxn J |«|rfJt, ... dxn„j)
... dxB.,)T:rr]<
§2. TOE SPACES IP*(Q> AND Hl(Sl) 65
< max-(J \m\dx, ... *e._,)'S=r J jj |||| *^__ d*m
<(J••• dJf»)^rn(J Kld*> ••• **•)"* •
Inequality (2.14) is proved. To prove (2.13) for p > 1 we introduce a new func¬
tion v = u1/*, where 1 /« «= pin - l)/n > 1. It is clear that
Employing inequality (2.14) and Holder's inequality for an estimate of
IMI „/(„-!),n> weeet
from which follows estimate (2.13).
Let us show how to prove (2.9) with it. If m < n, then we estimate ||u||? q
for minlr; m I < g < max I r; in! by Holder’s inequality as follows:
iK.B<ii“irs.aH;«- (2-i5)
where a is a number defined by the equality a/m + (l-* a)/r» l/q, It is obvious
(hat a belongs to the closed interval [0, 13, and when q varies between rand
m the number a varies between 0 and 1, and conversely. From (2.15) and (2.13)
with p - m we obtain inequality (2.9)
66
U. AUXILIARY PROPOSITIONS
The case m > n > 1 is considered by following the methods of V. P. II ’in.
The number q, by assumption, is taken to be an arbitrary number from [r, °°] for
m> n, and from [r, ~) for m = n. We first assume that q is subject to the addi¬
tional condition q > m. It is sufficient to prove inequality (2.9) for functions
uix) that are infinitely differentiable in En and equal to zero outside of fi (since
they are dense in IT * (ft)). For such functions, Poisson’s formula fora Newtonian
* m *
potential
Breaking tbe domain of integration En into two parts: {|z| < cr} and I \z\ > <xj, cr > 0,
and carrying out an integration by parts in the second integral, we obtain
Tt
implies die representation
n
We multiply both sides of this equality by non 1 and integrate with respect to <*
from 0 to an arbitrary k > 0. This gives us
uix + z)dz .
Upon making some simple estimates we obtain
\ \u{x + z)\dz
i<A
We let
§2. THE SPACES AND Hl(Q)
67
vAx, h) = J
&*(* + z)
dx-
-iz.
v-Ax, h) - ~ 5 [uix + z)\dz
hn (z|<A
and estimate and in tbe following manner:
I 1/m~l/ q
UjOc, h) <
[|,LK
.(* + z)\mdz
z|<A * |z|°J U«I<M*I J
where the numbers a and b are connected by the relation
1
+ 4 1
with a < n, b < n. Since
‘4) >“[l~s] '*_1,
it is possible to select such numbers. Thus
oj* "l/m + l/q
!!»>,(*, A)| < i«u/*-i/m+iA>)|
? “ (n - i) /m(n - «) /?
ML <2-l6>
In ao analogous way
v2{x, A) C j„(* + z)\'iz Il/r [ / dz I
* LUllfc i U-l<* j
l/r'
.1 -l/r
«- l*l<*
68
II. AUXILIARY PROPOSITIONS
wl-l/r+l /q
*>1, S <2-17>
Combining (2.16) and (2.17) we obtain
1-l/m + l/g ul-l/r+l/q
w*s
or more briefly
M?<v4A + BA
-5 .
This inequality is valid for any h > 0. Minimizing the right side with respect to
h, as is easily done, we obtain
i.e., inequality (2.9). The constant /3 in it tends to some finite number for q —»<»
if m > n, and to <» if m ■» »•
If in case 3) our addition assumption q > m is not fulfilled, then we take
some q > m and make use of die inequality
IMI < WZ.MrI-T,
H ?
where l/q - y/'q + (l - y)/r, and consequently y = (l/r - 1/q)/{l/r ~ l/q ),
yG [0, l). By estimating II “ 1!^* herefrom above by inequality (2.9) (since q“ > m):
o i, „l-a -i l/r - l/q
Nk</3K»: wir ,a
~_r- », • l/r + 1/a - l/m
we obtain (2.9) for the case q < m.
Theorem 2.2 is proved.
We will make use of a number of consequences of inequality (2.9). One of
them is the inequality
(2’18)
§2. THE SPACES AND 69
valid for any function uix) 6 in which c ~ /S j mes
p > n for B > 3,
p > 2 for n = 2,
p > 2 for n = 1, (2.19)
and Pi = 2U - l)/(n- 29 for » > 3, = 2(p - l)/(p - 2) for n = 2 and /3| = 2 for
» » 1. For uGc) that do not vanish on 5 but satisfy condition (2.11), the constant
depends on n, p and on the boundary S. Inequality (2.18) follows from (2.9)
when in the latter one puts m = 2, q = 2p/(p - 2), r < 9 and estimates j|“llrjQ by
Holder’s inequality
IMIr,a<il«!l =1. . (mes 0)7-V.
P-2
Another inequality that will be made use of is
II« IIjp a< e || «x || Q+ct || a || a, (2.20)
P-2'
where p > n and > 2, < is an arbitrary positive number, ce «=
(p- n)/p(/3j\jn/ptand the constant /Jj is tbe same as in (2.18). Ine¬
quality (2.20) is obtained from inequality (2.9) for m = 2, r = 2, 9 = 2p/(p - 2) and
an estimate of its right side by Young’s inequality (1.3).
For estimates of integrals over the in - 1 )-dimensionai boundary S of an
^-dimensional domain 0 when n > 2, we need in addition to (2.5) tbe following
inequality of type (2.9):
II“II?1s<cII^II?>hII«||2,"q“. a = -f — . (2.21)
which is valid for any function uix) from lf|(£l) with /q uix)dx *> 0. In it q is
any number from [2(n - l)/n, 2(n - 1 )/(n - 2)] for n > 3 and any number from ll, <»)
for n = 2, while the constant c depends only on n, q and the sutface S, which is
assumed to be piecewise smooth.
An important role in the investigation of operators of elliptic type is played
by the inequalities —> ([65b, d])
IIa lik'd ^ c II Ils. a IIa II2, n* (2.22 )
which are valid for an arbitrary function uix) from IP2^) satisfying one of the
homogeneous boundary conditions, ajj * 0 or Su/dl + <ru|j = 0 (l is a given vec¬
tor that forms an acute angle with the direction of the normal to S at the same
70
H. AUXILIARY PROPOSITIONS
point), and an arbitrary uniformly elliptic operator £ of the form
tftt = atJ (jc) « + at (x) uXt -\-a(x)u
with continuous oy, with o(. from L^iCl), q > n, and with a € i^y2(£l), where
5 = maxi9; 41. The constants c and Cj in them are determiaed only by the coef¬
ficients of £, the boundary S (which must be from class 0*) and the known
functions entering into the boundary conditions.
Inequalities (2.22) are also useful in the investigation of parabolic operators.
Besides them, we need a generalization of (2.22)
v(||«02«-2'«. -2>«) + *ilM£a- v —const > 0. (2.23)
or, more precisely, a special case of it in the form
|v|l«„|||0<(^’«. An)-f-c, II «* Il!,a- ».24)
where fu = (<?/dx,) (a.. (x) ux ), while the «,.(*) are continuous and
I / ) 7
vraimaxj-;|5o^/3*^| < oo, k, i, / = 1, •. • , n. Inequality (2.23) is valid for any function u
from P'l®) and any two elliptic operators £ and £ of the same type as in (2.22). De¬
tailed proofs and an analysis of inequalities (2.22), (2.24) are given in Chapter
III of the book [s 5i].
The space IP*(fl) with nonintegral I will be used in Chapters IV and VII.
r /i
It is a Banach space consisting of the elements of with finite norm
II “ ll«?a — (.(“))«? a + II « ili!,&). (2.25)
where
M
MI?.'0 = 2 2II^IU
/=" w
Such spaces were introduced (first for q = 2, and later for any q > 1 in
the study of properties of functions from il^(Q) with integral I on manifolds Sr
of dimensions r < n and in the proof of exact converse propositions concerning
the extension of a function from Sr onto all of fl. A theorem of this type, in
which functions from 5^(0) are studied on the boundary S of Q, is cited below.
§2. THE SPACES W^iQ) AND hHQ) 71
It plays an important role in the study of boundary value problems with nonhomo*
geneous boundary conditions, especially in the proof of exact estimates for
their solutions. For a formulation of this theorem we must introduce classes IT*
with nonintegral I on a curvilinear surface S. This is done by means of a pata-
metrization of the surface. Let Sj, • • • , S^, • • • be a finite or infinite (in the
case of an infinite S) family of subsets of S such that U4S4 - S and for each
point x e S it is possible to find an such that *6 Sj and the distance from
x to s\exceeds a certain fixed number S. Further, it is assumed that each
Sk intersects with only a finite number of other subsets, not exceeding some num¬
ber m, and is mapped onto some canonical domain a of (n - X)-dimensional euclid¬
ean space, for example a cube, by means of a transformation of class R* , I' > I.
Suppose that under a mapping of onto a a function uix) ix 6 5^) goes over
into «(A)(z) (z € a). We put
|l«lfe?S = {S(ll«(»(*)lWp.
The norms corresponding to die various coverings of a surface by sets S^
satisfying the above-mentioned conditions are equivalent.
Theorem 2.3 [3Jc;lllc], Suppose uix) e W^(Q), where I is a positive in¬
teger and q > 1. // S € 0^\ then u(*)|^ G ? (S) and the inequality
Sl/9> - c!l“l 5^0 ‘s valid. Conversely, any function uix) 6 IF* ~*^(S), de¬
fined on S, can be extended into the domain fi so that uix) € If* (fi) and
Tbe constants c in both inequalities are independent of u.
It was subsequently determined that the so-defined spaces W^iQ) fix q £ 2
do not form a ''continuous” scale in I. It "tears” at integral 1. In order to
create such a scale the spaces IP*(ft) for integral I must be replaced by the
spaces Blq(fi). For a convex domain Q the norm in B*( fi) can be defined by for
formula (2.25) if in the right side one takes instead of
(i_i)Va a x
+ D'-‘a(y)f \x~iF*Y' a26)
72
II. AUXILIARY PROPOSITIONS
Let us formulate three mote propositions that are known from the theory of
functions of a real variable.
Lemma 2.1. If a sequence of functions u^ix), 4 = 1,2, • • • , converges
strongly in L^iCl), q > 1, to a function uix), then it is possible to extract a sub¬
sequence which converges to u almost everywhere on Cl. And the convergence of
the u^ to u almost everywhere on fl implies an almost uniform convergence on
fl, i.e. one such that for any <> 0 there exists a set S!(C fl with vaesCf < e such
that the u^ converge to u uniformly on fl - flf.
Lemma 2.2. If ||«4|l9 q < c, q > 1, for a given sequence of functions U/J.x),
k = 1, 2, • - - , then it is possible to extract a subsequence from iu^l that is
weakly convergent in LiXl). If§uk\\qn < c, q> 1, and §fk\\q< q • i n(mes 12’),
q' = ql(q - 1), where n(r), t 6 Ef = [0, °°), is continuous and fi(0) = 0, for all
k — 1,2, ... and arbitrary Cl' C S2, and if the uk(x) converge to u(x) almost every¬
where on Cl and the fk converge to f weakly in Lq,(Cl), then faukfkdx —*■ faufdx.
Lemma 2.3. Let fix, u) be a measurable function on the set I* 6 Q, « e
(— oo( oo)} that is continuous in u for almost all x from Cl. If a sequence of func¬
tions Ujix), 4 = 1,2, ■ ■ • , from Lj(fl) converge& almost everywhere to uix) 6
£j(£2) and ||/ix, u^(ac))||^ jj < c, <? > 1, then the functions fix, u^ix)) converge to
fix, uix)) in the norms of Lq*iCl), q < q, and u/eakly in L^iCl). If, in addition,
it is known that \\fix, u^ix)) ||? q - < fiimes fl '), where /x(r) is a continuous func¬
tion of r > 0 that is equal to zero for r = 0, while fl1 is an arbitrary measurable
subset of Cl, then the fix, u^ix)) converge to fix, uix)) strongly in L^iCl).
Let us cite one more assertion, which is a generalization of one^of the propo¬
sitions of K. Friedrichs on functions from to functions from (see
[15b], Chapter 7).
Lemma 2.4. Suppose the functions 4>kix), k = 1, 2, ■ • • , form an orthogonal
basis in L-^Cl). then for any c > 0 there exists a number N( such that the ine¬
quality
Nt w/
g(U,^)2| + (2.27)
is valid for any function uix) from MO) with m > 2n/in + 2) for n > 2 and with
m > 1 for n = I. The number N( does not depend on u.
§2. THE SPACES AND Hl(Q)
It is sufficient to prove the validity of the inequality
M2.fi + s)
73
(2.27')
with arbitrary 8 > 0, and e > 0, since from it, by virtue of the orthonorrnality of ly^i in
L2(0) and inequality (2.5) for Sf = fi, <j = 2, / ~ 17 p = m, there follows
fNe.$
WI2.fi * [ j (“> ^k>
j, (“> tk)2 j 'A + (c8 + () NI^q ,
i.e. the assertion of Lemma 2.4.
Let us assume that inequality (2.27 ') does not hold. Then there exists an
fp > 0 and a sequence of functions uf, r = 1, 2, • • ■ , from such that
> (1 + 8)
+ tcKH”ta
for any r. Hence for the functions vf •» u /||»r ||2 (j we obtain
i = Ki2ifl > u +«(J(», ^A)2 j54 + <0l«r\\%,
so that the norms liMI m\)’ r - 1,2, • • • , are uniformly bounded. By virtue of the
compactness of the embedding of ^(Q) into L^SSt (Theorem 2.l) there exists a subse¬
quence vr^, p = I, 2, • - • , that is strongly convergent in £>2(fl) to some function v, with
M2 A = Jl^rlU.fi “ *• The functions Pr^ vr-^ s S ^P.j (ur^ , also converges
strongly to i>, since
||t>- Pr vr |2 q » ||Pr (t»- vr ) + (£- Pr )w||2 q
P p *■»** p p p ****
< II*' ~ vrpll2,fi + IK£ - ^ Mh.fi — 0 for p -* “•
In view of this, if in the preceding inequality we pass to the limit with respect to
the subsequence rp, p -* <», we obtain 1 > 1 + 8, which is impossible. Lemma
74
H. AUXILIARY PROPOSITIONS
2.4 is proved.
§3. DIFFERENT SPACES OF FUNCTIONS DEPENDING ON * AND t.
EMBEDDING THEOREMS
We will first prove some properties of tbe functions uix, t) from the
space V2(QT), and consequently also from the space C V2iQT). Sup¬
pose uix, t) is an arbitrary element of ?2^*? 7^’ For almost all t from to, 71 in¬
equality (2.9), which when m » r = 2 can be rewritten in the form
!!«(*. OII,.a<fMK<*. OllTS.
«-■5—(3.1)
holds for it. Here q 6 [2, 2n/(u - 2)1 for n > 3, q 6 [2, “>) fot n - 2 and q €
[2, °°] fot n - 1, while /3 depends only on n and q as indicated in Theorem 2.2.
From (3.1) we have
l
MU
V
J KIEV*)'
vrai max ||a(x, Olll a*
«<<<r
For a = 2/r we obtain the series of inequalities
II“II,.r.Qr<Pvrai #maxu(x. t) • ||«,||^ (3.2)
In them l/r + i*/2q - n/4, with
'€[2,
oo|,
«4S- A]
for
« > 2,
'€( 2.
oo].
?€12. oo)
for
# = 2,
'•€14.
ool.
?6l2. ool
for
ft* 1.
Estimating die right side of (3.2) by Young’s inequality (1.3) with n = r/1
and € = 1, we obtain a second series of inequalities:
SPACES OF FUNCTIONS DEPENDING ON x AND t.
75
II “ llj, r. Qt ^ P TIIU* fti. Qr
+P(1-7)vrai0^rll“IU<Pl“l«r-
(3.4)
In them q and r are subject to the same relations (3.3) as in (3.2). For q = r =
2Oi + 2)/» inequality (3.4) is given in the paper
Let us denote by the set of points x from Q at which |uix, *)| > 0,
and by Qq the set of points (*, t) from Qj> at which |uix, i)| > 0. Take arbitra:
q > 1, r > 1 and q^ € [l, q\ € [l, r] connected only by the condition r^/r -
qx/q.
From Holder’s inequality (1.7 ') applied to uix» t) and the characteristic
function x *of £he set Qq we have
If one of the indices q^ is infinite, then, by analogous arguments, we obtain
r
r-r,
(3.5)
t i
and
(/
r
ii “ il, vrai m2ax !«(■*.<)
i i
Here
fi (0) = mes ft £ [0, T]: mes A0 (t) > 0).
From inequality (3.4) with r and 9 satisfying conditions (3.3) and from ine¬
qualities (3.5) it is not difficult to deduce that for any function uix, l) from
76 11. AUXILIARY PROPOSITIONS
0
V2(<?t'> ***** “bittaty q^ > 1, fj > 1 subject to the conditions
and
tl
e[x.c°],
?>e[f
2n ^
’ M«-2) J
for
ft 3,
n
e(f-°°].
*€[#
. oo|
for
n = 2,
ri
e[x.°°].
*6[f
, ooj
for
I,
the inequality
1
*t
II “ 1WV r,
,er<Pl
“lOr?(0),!
a
xi+T
(3.6)
is valid, in which
T
£ (0) — J mes «> A0 (t) dt.
o
The notation jt*/ri(0) means vrai max 0< ^j.mes^^MpU) fot ~, while
M(0) «= mes it 6 [0, T]: mes -40(t) > 0) fot = 00 .
In the special case = Tj = 2, Kj - 1/2 it follows from (3.6) that
li “Ik «r•< P(mes I«l«r- (3J)
Inequality (3.4), derived above from the multiplicative inequality (2.9),
also holds fot functions uix, t) from V 2^ that do not necessarily van¬
ish on Sy but satisfy for ail t €. to, T] the condition /jj it(x, t)dx = 0. The
constant /3 in (3.4) depends in this case on the surface S, which is as¬
sumed to be piecewise smooth, although /3 does not vary under a similar¬
ity transformation of the domain £1.
Let us now assume that uix, t) is an arbitrary element of P'2(|? j).
We denote by u0(t) the average value of uix, t) in 0. The difference
SPACES OF FUNCTIONS DEPENDING ON x AND t. 77
it(x, t) - Uq(t) satisfies the condition Jq(u - u^)dx = 0 for all t € [0, T], and
hence inequality (3-4) Is valid for it. On the other hand, the following estimates
are obvious:
H“ll«. '.<?r -^11“ — «oB».r, Qr -HM* r. CtT
1
<!!“ — “o!l«. r. 0. + vral maxja0(/)ir7 mes*Q,
r o««r
Ia ~ °o tor I “ kr ■+■ I “o |<?r
— 10 la, 4- vrai max |a0(0|mes2Q,
T 0</«r
1 i
vtal max | (*)J ^ vrai max || a)(, „ mes'7 Q < J a lQ mes ’'? Q.
»<<<r o<<<r ■ r
From what has been said it is clear that for any uix, t) from wi'h arbi¬
trary parameters q and r satisfying relations (3.3) one has the estimate
(3.8)
T"T
I 1 I n V
c = 2p + T~ mes""5* » Q — 2p + \rT mes” 'QJ
The constant /3 in (3-8) is determined only by r, re, q and S and does not de¬
pend on T or mes Q.
Inequality (3.5) holds for any u(x, f) from VLetting rj = q^ 2,
r = q = 2(n + 2)/n in it and using (3.8), we conclude that for an arbitrary function
u(x, t) from V2(QT) an estimate of form (3-7) is valid:
II■“!l2. <tr ^C (nies I“l«r-
= 2P -f- (7'^rn«S-
, (3.9)
where the constant j8 is the same as in (3.8).
An analogous estimate can be derived for the norm ||u|^ r $if instead of
inequality (2.9) one uses inequality (2.21) with a = 2/r. Namely, for arbitrary
uix, t) from V^.Qt) and q and r satisfying the conditions
78
II. AUXILIARY PROPOSITIONS
71 — 1
/■ 612. ool.
2(« — i) 2(« — 1)
it ' n — 2
fot n~^r 3,
(3.10)
'6(2. ool <r€H. oo)
fot rt = 2.
inequality (2.21) implies the estimate
0.11)
with the constant j8 depending only on n, r, q and on local properties of the sur¬
face S, which is assumed to be piecewise smooth.
In the case n = 1 we can derive an estimate of form (3.11) if instead of (2.21)
we make use of the inequality
which holds if u(x, t) vanishes at least one point of fl. Integrating it with re¬
spect to (, we obtain, after some elementary estimates,
for an arbitrary function u(x, t) from
Let us prove the following proposition, which is a special case of more gen¬
eral facts established in the papers [89;38b],
Lemma 3.1. Suppose u(x, t) in Q.p satisfies a HSlder condition in t with
exponent a and Holder constant ft j and has derivatives u%, which for any t from
[0, T] are HSlder continuous in the variables x, more precisely, are such that
b*(s, /)-<a2(s, t)2 J |ux(x, t)u(x, t)\dx, s£S,
a
From this follows the inequality
(3.12)
P
0 <t<T
max osc, [ax(x, t), KP(\Q]
SPACES OF FUNCTIONS DEPENDING ONx AND t.
79
Suppose, further, that the domain Cl satisfies a cone condition, i.e. there exists a fixed
spherical cone K of some altitude d such that, no matter at what point of 0 its
vertex is placed, the cone itself can be swung so that all of it is located in 0.
Then the derivatives ux satisfy in QT a Holder condition in t with exponent S =
o-l3/(1 + 0) and constant p determined only by a, fi, p2> d an^ the solid angle
at the vertex of the cone K.
For arbitrary points x' and x", which can be connected by a rectilinear seg¬
ment [x *“] belonging to the domain fl, one has the relation
n-
K
du (x, t) du (x\ f)
Jl dl
=u(x", t") — a(x'. r) — u{x". t') + u(x', t'). (3.13)
In it 3/dl denotes differentiation along the segment Ix', x * ]. By virtue of the
conditions of the lemma the right side of (3.13) does not exceed - t' |a,
and
It-
du (x, t') du (x1, t')
dl
dl
dl
yiI+P
In addition,
f r du(X, n
dKx'-nij,
= |*"~ x'\
| du <£, O
j -dl -
dl \al
1 ~Si
*»(*', t')
du (x\ t")
du (x\ t')
dl
dl
dl
where x is some point of the segment [*', *"] lying between x' and as'. On
the basis of these estimates there follows from (3.13) tbe inequality
! du (■*', t’) du (x', t')
dl
sr
<2|ij
'Ip
I x" ~h 1 x,r ■ - x’ j
(3.14)
Letting * ' be the vertex of the cone K, we minimize the right side of (3.14)
with respect to *' over a segment of some ray I of length d belonging to K 6 0.
As is easily calculated, for |«" - <' |“< (/Sftj/Pj)^1 this minimum is equal to
2tTTF'+ P'1^)
80
II. AUXILIARY PROPOSITIONS
I ±S£.n - | < n | e -1' I &.
Since the ray I can be arbitrarily directed in K, this implies the validity of lie
lemma.
Let us now formulate some propositions concerning the spaces V^’KQj-)
and H2l-l(QT) used by us in Chapter IV. Their proofs can be found in [38b;49c; 116b]
A definition of the spaces for integral I and of the spaces
for nonintegral I is given in §1 of Chapter I.
Lemma 3.2. Suppose the domain 0 satisfies a cone condition (see Lemma
3.1). Then for any function u the inequality
{“)qt < {u)qt + c,fi-r | u |(“>r.
is valid; here r and S are arbitrary numbers from the intervals (0, /] and
(0, min Irf, \pf\) respectively, where d is the altitude of the cone, and Cj and
c2 are constants depending on r, I, n and on the angle at the vertex of the cone K.
Lemma 3.3. Let fi be the same as in Lemma 3.2. Then for any function
uix, t) from V2l'KQf) with integral I the inequality
is valid under the conditions p > q, 21 - 2t - s - (l/q - l/p) in + 2) > 0. In addi¬
tion, if 21 - 2r - s - {n + i)/q> 0, then for 0<A<2/-2r-s ~ (b + 2)/q
«*>©»
(for nonintegral 2l — 2r — s - (n + %/q this inequality is also valid for A = 2/ -
2r-s —+ 3/?)- The constant 8 here must satisfy the condition
0<6<min(d; yT),
and the constants c j—eg depend on I, r, s, n, q and on the angle at the vertex
of the cone K (cf. Lemma 3.1).
SPACES OF FUNCTIONS DEPENDING ON * AND t.
81
Spaces with nonintegral I will be used in Chapters IV and
VII. Let us first define them for functions given in a cylindtical domain ■
a x (fj, r2), where a is some domain in (» - l)-dimensional euclidean space
En-l and - »< rt< r2<». The space consists of functions with
finite norm
IM|«> = + 2| \\DrDszu\\ „
11 ni.*Ti,T2 (f1-aTi,T2 . (0<2r+*<O 1 * ?"CTTi.r2
where ~
«■»&,.„ - o-™
and for 0 < a < 1
mlUor
J fajax J !•(*. o-*<* 0|*1-_^=1_ry.
'jTj <f o
mt,aT
T 2.
= (f dJt Jd< J|o(*' f)"'0(x* ol* u-n1^) '
\o 0 0 /
Finally, let us define spaces and also with nonintegral
I > 0, in which the elements are functions given on a curvilinear surface Sj. = S x
(0, T). As in the case of V^(S), these spaces are defined with the help of a para-
metrization of the surface S. Let S be covered by sets Sj, • • • , Sj in the man¬
ner described in
lowing notation :
ner described in §2 in the definition of the spaces If^(S). We introduce the fol-
5^=5* X (0. T),
or=ffX(0,n
(a is the canonical domain into which each set 5^ is mapped). Let h^Kz, ()
(z e o) be die function into which die function u(x, t) (x e Sk) goes under the napping of
Sk into cr. We define die space r^2(sr) as the sec of functions with finite norm
82
U. AUXILIARY PROPOSITIONS
l<V “ { ? (I “(fC)^ *>CJf ■ ai8)
and tbe space as (be set of functions with finite norm
|4£==.up|«<*>|£. (3.19)
The norms in the right sides of (3.18) and (3.19) are defined with the help of
(3.16) of the present chapter and (1.10) of Chapter I. Under different coverings
of S by the Sj tbe corresponding norms (3.18) and (3.19) turn out to be equivalent.
Finally, we note that in the definition of Ml,t/2(Sj.) it is sufficient to require
that the S4 be mapped onto o by means of a transformation of class Hl.
The spaces are needed to obtain exact results on the solvability
of boundary value problems for linear parabolic equations in classes W^jn,m{QT).
This is connected with the fact that the differential properties of the boundary
values of functions from classes W2m,m(Q j*) and of certain of their derivatives
can be exactly described in terms of the spaces V^'^2[Sj) with nonintegral I
(namely, with I = s - l/q, where s is an integer). We have
Lemma 3.4. If u € W2m,m(Qj.), then for 2r + s < 2m - 2/q
(Q) Kq
In addition, for 2r +* s < 2m - l/q
1 * J
2m —2r—s— , m—r——•---
o;o;«|s.6^ s 7a,(sr>
and
n«iiif (3-20)
§4. ON AVERAGINGS AND CUTS OF ELEMENTS OF
L?(n>, LqjQT) AND V\’°iQT)
Let «(*) be a function from L^(Q), q > 1. We will denote by u^Xx) the
function u^kHx) = maxiuix) - k, o!, which obviously also belongs to L^(Q) for
any k. It is easily seen that if u and v belong to £3), then for almost all x
from Q we have the inequality
§4. AVERAGINGS AND CUTS OF ELEMENTS
83
| ' (x) —1**> (*) I < I a (JC) — v (x) I. (4.1)
one of the consequences of which is the well known
Lemma 4.1. If a sequence of functions u (x) from L^iQ), p = 1,2, • • • , con¬
verges to a function uix) in the norm of £?(£1), then the sequence u^ix) conver¬
ges to a<«(*) in the norm of L^(Q).
The following assertions are also valid (see, for example, Chapter Q of
Lemma 4.2. If uix) 6 LiQ) and has a derivative u~. € LAO), then u'^ has
H * 9
the derivative 6 L.^ifl); in particular, if uix) 6 then u^Hx) e IP*(Q).
If, in addition, vrai maxj u < k0, then for k > k0 the function u^ix) belongs to
*J(G).
Lemma 4.3. Suppose that the functions uix), Uj(*), u2(x), • • • belong to
LqiQ) and have generalized derivatives ux.upx., and p = 1, 2, ••• , from 0) for
some i. If ||ap - u||?q — 0 and ||up*. - 0 for p -> ■», then the func¬
tions and du^/dxi converge in L^(S1) to and dJ'^/dx. respectively.
Let us now consider a function uix, t) belonging to V2’°iQj.). Tbe function
u^Xx, t) <= max. tuix, t) - k; 0! lor any t from [0, T] will belong to L2iSl). By
virtue of inequality (4.1) we have
|| d»\x, t + A/) — «<*> (x, t) ||2> 0 < || a ix, t -+ AO — a <*. 0 |l2.n.
and therefore the functions u^Xx, t) are continuous in t as elements of L2iQ).
In addition, for almost all t € [0, T] they belong to B^Ul), with lit4A)(*. t)|2 0 <
Uu^Oc, z)|| 2 Q. In view of this we have
Lemma 4.4. If uix, t) belongs to V2’®iQj.), then u^ix, t) also belongs to
v\’Hqt).
From Lemma 4.3 and inequality (4.1) it is easy to deduce
Lemma 4.5. Suppose the sequence u^ix, /} € Vlf^iQy), p •* 1» 2, • • • converges to
uix, t) C V\’°{Qt) in the norm of V2’°iQj). Then i/*)\q^ —> 0 for p-*<*>.
We note also the following simple fact.
Lemma 4.6. If on some part I' * of the surface S j. a function uix, t) from
^oes not exceed k0 (i.e. vraimaxp , u < kQ), then for k > kQ
the functions u{k\x, l)£(*, t) belong to V\>°iQT), where (ix, t) is an
84
II. AUXILIARY PROPOSITIONS
arbitrary smooth function that vanishes on Sj\F*.
On the other hand, if F' is an arbitrary part of a surface T j. and £(*, /)
vanishes on Tt\T', then «<*>£ for k>kg belong s to anrf vanishes
for t = 0.
We will consider the averagings
«„(Jt)= j %(\x — y\)u(y).dy
U-yKp
— J rfy (4.2)
lx~y\ «i>
with a sufficiently smooth nonnegative averaging kernel <a(|f |) that is equal to
zero for j£"| > 1 and is such that /|^j< i <o(£)d£ = 1 (concerning them, see 1113a]
and jf u(x) is defined in a domain fl, then an averaging *>> is clearly
defined in any subdomain fl' that is separated from the boundary $ of fl by a
distance not less chan p. Suppose uix) is an element of some Banach function
space B(fl). It is known that if the elements of 8(0) are continuous with respect
to a translation in the norm of B(fl), then the averagings u Jjc) for p — 0 converge
to uix) in the norm of B(Sl'), where 0 ' is any interior subdomain erf fl, since
II “p
J a>MU)(ii(je — z)— u(x)\dz
< J «pU)|| “ (X — Z) — U (X) ||fl (Q,;
UKp
dz
< max \\u(x — z) — u{x)\\B(£n~*0
ui<t> K }
for p 0.
In the sequel we need averagings not in all arguments but only in part of them.
If, for example, an averaging is carried out only in the variables Xj, • •• *m,
then the smoothed functions corresponding to it are defined in the domains SI'
belonging to SI together with their translation by p to any direction lying in the plane
{xm + j = 0 xn = 0}, and if in the norm of fi(fl) there is continuity with
respect to such a translation, then the averagings will converge in the norm of a
space B(fl ') to u for p —> 0.
Thus we will deal with functions <*(*, t), x = (xj, - • • , xn), defined in a cylin¬
der Qj = fl x CO, T], and we will consider for them averagings in the variables x
of the form
§4. AVERAGINGS AND CUTS OF ELEMENTS
85
a„<*,<)= J top(| JC —y|)u(y. t)dy, (4.3)
|x-y|<p
where is the same averaging kernel as in (4.2), and averagings in t
(Steklov averagings)
t+A
ah(x, 0 = x J “<*• T)rft- <4-4)
t
It is clear chat the first of them is defined for all sufficiently small p > 0 in
cylinders of the form Qy = O' x [0, 71, where fl* is sense interior subdomain of
ft, while the second is defined in cylinders Qj-g - Q x [0, T - S] for h < 8.
From what has been said above, it follows that if uix, t) belongs to some
with q and r > I, then the functions (4.3) and (4.4) converge to it in the norms
of Ly J.Qf) and L^riQ j„g), respectively, for p and h —* 0. Then we may con¬
clude that for certain sequences p^ 0 or hk -* 0 they converge to uix, t) al¬
most everywhere in Q f and respectively.
Let us assume that uix, t) belongs to the space The elements of
dils space are continuous in the norm of V\,0(Qf) with respect to a translation
in. the direction of the t-axis. This follows from the definition of j.) and
the properties of /^(^p). It is therefore possible to assert that for A -» 0 the
functions (4.4) converge to uix, t) in the norm of ^ (we note
that Ufi 6 ^ < 8).
Let us formulate this in the form of a lemma.
Lemma 4.7. If uix, t) belongs to class V^iQf), then the averagings
“/,(*, f) for h <8 belong to class with
\uh ~ for h-*0.
If uix, t) g L riQ j.), then the averagings converge to u for h -* 0 in the
norms of L^Qj-.g).
Let us now consider the averagings up of form (4.3) with infinitely differen¬
tiable kernel. We have
Lemma 4.8. Let uix, i)€. Then for an arbitrary cylinder Q’j. =
0' x [0, 71 and p <d‘, where d‘ > 0 is the distance from 0 ' to A, the aver¬
agings u Jx, t) are continuous in Q j, have continuous derivatives D^ufi of any
order, and
86
II. AUXILIARY PROPOSITIONS
|ap — «» 0 for p—y 0.
On the other hand, if u 6 {.Q ^), then the Up converges to a for p —» 0 in the
norms of (Q j.).
The validity of the last assertion was noted above. Assuming that a 6
let us estimate the expression
\ap(X, 0-«p(x',oi==p-n| J ®(1~1I)“(y. t)*y
I |jr-y|<p
— J W (—|^-1) a (y. t') dy
f<v fixed p, under the condition that the point (x ’, J') € Q!p tends to (*, t) 6 Q j.
It does not exceed the sum
P-"J> ~• (— f11)[l"(y- Wdy
4-P-• J ffl(-'l:'~-yl )|a(y. i')\dy.
a
in which the first term tends to zero for * ' —» * because of the continuity of the
kernel &>(£), while the second tends to zero for t’ —> { by virtue of the contin¬
uity of the function u(y, t) in ( as an element of L2(0), since
p-n j to^l£rilL^|a(y, t)— a(y, t’)\dy
< c (p) II “ O'. 0—«(y. O H2. a-
We have proved that the Up are continuous in Q'j. The continuity in Qf of
the derivatives D^up is verified in exactly the same way, in view of the infinite
differentiability of the kernel u(f)-
Let us show that as p 0 the ujjc, t) converge to u(x, t) in the norm of
for u € By virtue of what has been stated above it is suf¬
ficient to establish that the norm \u(x, t) - uix + f, tends to zero for |f | ~> 0.
This is true for ||«x(*, <) - ujjc + f, <)t2,(>| by virtue of the membership of ux
in L^Qj). We will verify that it is also Hue for maxo^^ylBfe, t)-“(* +f. £)|l2>n'•
§4. AVERAGINGS AND COTS OF ELEMENTS 87
Foe any finite set tj, • • • , t^ 6 [0, T] it is possible to satisfy the inequality
AT11 “(x’ tk) ~ “ (x+ 6. h) 1U, a. < e, (4.5)
where c is an arbitrary positive number, as long as \€\ is taken sufficiently
small. On the other hand, by virtue of the uniform continuity of uix, l) in ( from
[0, T] as an element of k2(0) *s possible to assert that
* faX * ,cnlX ,,]lu(x' *>-“<*• '*>H2.a <e. <4.6)
'*]
if the distances between adjacent points l/(~i and 4*1, , N, are suffi¬
ciently small.
From (4.5) and (4.6) it follows that
max i\u(x, t) — b(* + |, 0||, 0, <3e.
o<kt *
Thus it is proved that |uix, t) ~ uix + (, t)f q j, — 0 fot |£| -♦ 0, from which
follows the validity of Lemma 4.8 on the convergence of the ujx, t) to uix, t) in
the norm of V2iQ^) for p-* 0. Lemma 4.8 is proved.
The following simple proposition is useful.
Lemma 4.9. If uix, t) 6 V\’°(Qf) and vrai maxs^u < kQ, then for k>kQ
the functions u^Xx, t) « max \u^x, i) - k; Ol belong to y_ j), h<8.
Its validity follows from the inequality
vrai max u .(2, t) < vrai max u(*, t)
sr-s st
and Lemmas 4.6 and 4.7.
In several sections, in the course of investigating differential properties
of solutions we will deal with difference ratios. In this connection it is necessary
to bear in mind that fot any functions uix) and u(*)
. A(v(x)tiix))^=^pu(x)-hv(x fA*n)^.
x 4-Ax» = (x, ... x„). (4.7)
and for a composite function fix, uix))
88
H. AUXILIARY PROPOSITIONS
df df\xi 1 - r) + (* + A*ft) t, uix) (l - r) + uix + A*A)r]
\ ‘ i s* *•
and
JU- = f ~ ax
du J du aT’
In the sequel we will use the formula for summation by parts
J ,(jrt dx „ _ | ^,v(x + &Xk)dX' (,.9)
12 a
which is valid for any functions u € £m(0), t> 6 Lm < (£2), 1/m + 1/m' = 1 one of
which has a compact support in SI and for all sufficiently small |A*4|.
Lemma 4.10. If a function uix) from Lm(0) has a generalized derivative
uxk C then for any strictly interior subdomain SI of Q
for |Ajc*|_>0-
Indeed, for almost all x for which x, * + A*j 6 £1, the ratio Aa/A*^ is
equal to (l/A*fc) %k a^(*,, , **_i, (, *4 + 1. * » *„)<*£, i.e. coincides
with a Steklov averaging (cf. (4.4)) of the derivative ux^, and therefore Ab/A*a
fear |A —» 0 converges to ux^ in the norm of Lm(Q ').
Sometimes, instead of proving that a function uix, t) has a generalized deriv
ative itj from L-^iQj), we will verify that the integrals f Qj. (A u/A t)*dx dt
are bounded uniformly in At. In this connection we will have in mind Lemma 4*11
(see for example [65q]);
Lemma 4.11. Suppose that uix, t) € L^Qy) and that for all positive At<hg
the integrals
J | “(■*. *H-AQ — ujx, t) |*rfTrf|» (4.10)
Qt-ai
do not exceed some constant c. Then in Qy there exists a generalized deriva¬
tive u, and
§5. SOME OTHER AUXILIARY PROPOSITIONS
89
II “< H2, qt < c~
From the uniform boundedness of the integrals (4.10) it follows that for some
sequence A ft —> 0, k ■* 1, 2, • • ■ , the difference ratios Au/A ^t converge weakly
in S > 0 to some function w(x, t) C /< jlG* y ~j)- We apply the formula
for summation by parts (cf. (4.9))
I t)d*dt — — f »(*, t+M)~$.$dxdt
0 T Qt
to uix, t) and an arbitrary smooth function v{x, t) that has a compact support
in Qf-5, and then in this equality pass to the limit with respect to the chosen
sequence Ajt -> 0. As a result we obtain an equality from which it will follow
that wix, t) = Bt(x, 1) in Qj-.$ and, in view of the arbitrariness in the choice of
8 > 0, in all of Q j..
It is not difficult to prove the following proposition.
° 1 1
^ Lemma 4.12. In the subspace of the space $2’ (Qf) consisting of elements
of W^’^iQy) that are equal to zero for t - T, functions of the form d^ify/i^ix)
are dense, where the d^it) are arbitrarily smooth functions that are equal to zero
for t - T, and the { $*(*)}, k = 1, 2, . , ., is a fundamental system of functions in
the space ^(Q).
§5. SOME OTHER AUXILIARY PROPOSITIONS
In this section we will cite some important lemmas of various kinds; part of
them we will prove, but for the rest, the proof of which is given in the book [65q],
we will confine ourselves to an exact reference.
Lemma 5.1. Suppose there is defined in a ball Kp= I*: jx| < pi a non nega¬
tive function uix) € W\iKwith uix) - 0 on some set §,0 of positive meas¬
ure. Then for any measurable set £ from Kp the inequality
ft, I •
J u (x) 91 (*) dx < ft, tries j \ux(x)\‘R (x) dx (5.1)
* ° %
is valid, where j3j = (l + nKn)2nn~l, and 51(a) = 5t(|*|) is an arbitrary nonin¬
creasing function of |*| with values from [0, l] that is equal to unity on $0.
Consider the case n £ 2. For almost all * C S and x‘ € we have
90
H. AUXILIARY PROPOSITIONS
u (x) = U (X) — U (X')
= f I«y(y)|rfr,
o 0
where y — x + r«u and <■> = (* * — x)/\x’ — *| is a unit rector issuing from the point
x. We multiply both side^s of this inequality by fl(x) and on the right we insert
3l{x) under the integral sign, and replace it by My). The inequality is not vio¬
lated by doing this. Then we integrate both sides of it over x1 € ao<* * € 5> •
This gives us
mes g’u | a (x) 91 (x) dx
1
Ix'-Xl
< J dx J dx' J \uy(y)\9l(y)dr. (5.2)
« if. o
Setting 9i(y.) equal to zero outside of Kp, we estimate the integral over 60
in the following manner:
f d*' J \Uj(y)\yi{y)dr
U'-jrl
< I S
2p U'-Jrl
< J |x' — Je1 Je' — x| J d® J
(y)l»(y)
Kfi
From this and (5.2) we deduce
inesifo f f | ay(y)| St(y)</j> max f -———(5.3)
4 *rp ,Eitn i*—yl
The integral / • Jg dx/\x - yl"""1 for any point y € K does not exceed
(l + B«(1)mes1^nS. In fact, the part of the integral corresponding to die domain
of integration & f|i*: I* ~ y| < SI does not exceed j ix/\x - y\n 1 =
n«„S. The remaining part obviously does not exceed Sl“BmesS. Let us take
§5. SOME OTHEK AUXILIARY PROPOSITIONS 91
8 = mes S- Then it is clear that j < (l + »*„)mes &, and hence from (5.3) we
get the desired inequality (5.1) for n > 2.
In the case re = 1 inequality (5.1) and the even stronger inequality
J a(*)9?(*)£?.*<mesg’ J |ux(x)| W(x)dx (5.1')
*
are deduced directly from the Newton-Leibnitz formula
X
u(x)=u (x) — u (x') = J By (y) rfy.
X’
Concerning it is sufficient here to assume only that it is not empty.
We will make use of two consequences of Lemma 5.1. The first of them is
the inequality
f a2 (A ) (*) dx < fJ2 p5 f 4 (x) St* (x) dx.
«0 * 4 (5.4)
p2 = 4p^;.
which is valid for an arbitrary function »(*) from that is equal to zero
on a set £0 C Kp tot the same SKx) as in (5.1).
The other consequence of (5.1) is the inequality
(/ - k) mes At „<(? J Mdx.
^ (5.5)
P = P,kJ.
It holds for any uix) C W\(Kp) and arbitrary values of I and k for I > k. Here
and below p is the set of points x from for which uix) > k.
Inequality (5.4) is easily deduced from inequality (5.1) if in (5.1) instead of
uix) and 5K*) one takes the functions u2ix) and ?i2(*), and fra the set € one
takes the whole ball K . For a proof of inequality (5.5) we take in (5.1) as
uix) the function
92
U. AUXILIARY PROPOSITIONS
«(*) =
I— A. for x£Altp,
U{x)~k. for
0. for *€Kp\<4*lP.
and as the sets &q and § the sets ((|J and Kp respectively. The func¬
tion SKx) is set identically equal to 1. This gives
J u(x) dx <pi mes (XpP\ /4t, p) (x»p”) * J I “» !rfy'
Ko K0
From it follows inequality (5.5) if one notes that ff( u(x)dx = }a , u{x)dx >
(2 - ft)ves . But if one takes as € the set ^ then instead of (5.5) we
obtain the inequality
(/ - k) mes p <P, p) mesT Ai.t / Kldx (5.
6)
established by DeGiorgi [173.
For n = 1 the stronger inequality
j*(l — k) < J Kld* (5.6')
‘,4,pNAl.(>
is valid for arbitrary positive and f2* Inequality (5.6 *) follows directly from
the Newton-Leibnitz formula.
Remark 5.1. Inequalities (5.1), (5.4)—(5.6) ate valid not only for the balls
Kp but also for any domains that are star-shaped with respect to die set &q.
The following two lemmas apply to the case n > 2.
Lemma 5.2. Suppose that in a domain 0^ of diameter 2R there is defined
a nonnegative function v(x) suck that for any ball with center in the
estimate
J «(j:)djc<cp*-s+“, a>0
n a«
o t
holds. Then [or any function £(x) from the inequality
J v(x)^2(x)dx-^.CiRtt J&{x)dx (5.7)
efi
§5. SOME OTHER AUXILIARY PROPOSITIONS
93
is valid, with the constant c j depending only on c, a and n.
A proof of this proposition is contained in Lemmas 4.3—4.4 of Chaptet II of
[65q]. Inequality (5.7) also holds for functions C(x) that vanish on only part of
the boundary of Q^. In particular, we will make use of the following variant of
Lemma 5.2.
Lemma 5.3. Suppose that = KR f) 0 m the conditions of Lemma 5-2,
where the boundary S of the domain Q contains a plane part Sj and the ball
does not intersect with S\Sj. Then inequality (5.7) is valid for any function
£(x) from that is equal to zero on the spherical part of the boundary of
Qjj. Here, as above, the constant c l in (5.7) depends only on c, a and n.
Indeed, consider the domain fi^ that is symmetric to fig with respect to Sj
and extend the functions C(x) and vix) onto in an even manner with respect
to Si. Then all the conditions at Lemma 5.2 will be fulfilled in the domain
* 'V
Qjg U Consequently, inequality (5.7) will be true for the domain Qg (J Gjj
and hence also for the domain
For » «* 1, instead of Lemmas 5.2 ami 5.3 we will make use of the following
proposition.
Lemma 5.3 1 • If 0^ is a segment of length 2R and u(x) is a nonnegative
function from then the inequality
J «(*)£-(*) j£(x)dx, c,—2||#||1>u (5.7')
is true for any function (ix) from l^at IS ^quai to zero at at least one point
Xq C QjJ.
Inequality (5.7 * ) follows from the fact that for any x from 0^
x a
£*(■*)= J Z,(y)dy <2/tj^dy.
aR
The following simple proposition is useful.
Lemma 5.4. Suppose uix) is a bounded function from ^25+2^®’ * - an^
£(x) is a smooth function such that the product tt{x)£(x) vanishes on the bound-
ary S of the domain Q. Then
94 II. AUXILIARY PROPOSITIONS
\\uxn*dX
a
< 16osc* {«, Q) J |«JS4d)dx, (5.8)
a
c = t&-f-s2.
In order to prove this lemma (cf. Lemma 4.5 of Chapter II of [^5q]) we trans¬
form the integral Jq |uj2s +2 £2 jx means of an integration by parts and carry
out elementary estimates with the help of inequality (1.2) of Chapter II:
f I Uxfis''%idx= J \ux\lsuxiux.C2dx
a 1)
== — J [U(J.-)-“(^o)l[^“l«r|-’^S4 uxi2s\ux(*~iuxkx.uxip-
U
+ iuXl I ux I2' &,,] dx < J |^e I ux f*+2l?
+ | a, rvuU2 + e |«, r21*
where *0 is chosen so that the product [«(*) - u(x0)] £(x) vanishes on S. Letting
f = 1/4, we arrive at estimate (5.8).
The following lemma is well known.
Lemma 5.5. Suppose that a nonnegative function y(t) is absolutely continu¬
ous on [0, 71, equals zero for t * 0, and satisfies for almost all 1 the inequality
< c U)y(t) + 5(«) (5.9)
dt
with nonnegaiive functions cit) and Sit) that are summable on [0, r]. Then
y(*)<exp| J <r(x)</T 5J &(i)dx, (5.1(>)
(ty (t) ^ ,,,
-iri-<c(0exp
In particular, for c(t) «■ c and nondecreasing 'Sit)
§5. SOME OTHER AUXILIARY PROPOSITIONS
95
y(t) < (e«« _ l),
(5.12)
c
dt
(5.13)
Foi a proof of this lemma we multiply both sides of (5.9) by exp I -
transfer the first term on the right side to die left side, and write the result in the
form of an inequality
From it and (5-9) follow (5.iO)-(5.13).
Ve distinguish in the form of separate lemmas the following two propositions
on numerical sequences connected by recursion inequalities. The first of them
is taken from 1^51] (Lemma 4.7 of Chapter II).
Lemma 5.6. Suppose a sequence yk, h = 0, I, 2, • • • of nonnegative num¬
bers satisfies the recursion relation
■§f y(0exp c(t)df .
Integrating it with respect to t from 0 to we obtain the inequality
yM<ct>X+e. * = o.i
(5.14)
with some positive constants c, c and b > 1. Then
(5.15)
In particular, if
(5.16)
then
h
yh<M~T
(5.17)
and consequently y^ —* 0 for h
This is proved directly by induction.
96
II. AUXILIARY PROPOSITIONS
Lemma 5.7. Suppose nonnegative numbers yh and zh, h = 0, 1, 2, • • • , are
connected by the system of recursion inequalities
(5.18)
z*+i<c*‘(n+*H.
where c, b, c and S are certain fixed positive numbers, with b > 1. Then
-± ( .±\rh
where Vh<M> “. “) . (5.19)
J _ J_ 1_ Ijhe I 1
A.=:min|(2c)-*S"4rf; (2 c)~ ‘ f, (5.20)
as long as
1
and (5.21)
Indeed, inequalities (5.19) are by condition valid for h = 0. Suppose they
hold for yh and zA. Then on the basis of (5.18)
! Ml+i »f, ‘+»\
y*+i < c*"2 U* ~ j = 2«a,1 ^6a ' d \
But, as is easily calculated, the right sides of these inequalities do not exceed
^-ihn)/d an(j (^-<fc+l)/rfjl/(l+f) tespectjyeiy. an(j hence inequalities (5.19)
also hold for + j and */, + [• The lemma is prbved.
The following lemma is very useful in proving the Holder continuity of func¬
tions; it is analogous to Lemma 4.8 in Chapter II of [<>5q].
Lemma 5.8. Suppose a function uix, t) is measurable and bounded in some
cylinder Qp0 = Q(p0, $0Pg)- Consider cylinders Qp=Q(p, <?0p 2) and Qbp that
are coaxial with Q and have a common vertex (center of the top) with Q pQ>
where b > 1 is a fixed constant, and suppose that for any p < b~lp0 there is
fulfilled for uix, t) at least one of the relations,
osc (a; Q„) <Cjp8 (5.22)
§5. SOME OTHER AUXILIARY PROPOSITIONS
97
osc i“; Qp) < n osc {a; Q^) (5.23)
uiiM certain positive constants Cj, S < I and if < 1. Then for p < Pq the esti¬
mate
osc {a; Qp) < cpfV, (5.24)
is valid, where
a — min f — In* rj; 6}; c = i“ max
o0t=osc (a; Qp,}.
Proof. Let us take a sequence of cylinders Qp^, p^ ~ b ^Pq, k « 0, I,
2, • ■ • , of the type described above and denote by the oscillation of i»(*, t)
in Qp^. From the hypotheses of the lemma it follows that
(»* < max {c,p£; A—1.2 ^ 25)
and
<o
From this and (5.25) we have for = 4*a<Uj, 4 = 1,2, • - • , the estimates
j/B < max i*0,nw4_1|
= max b\yk,,} < max y ] (5.26)
and
yotsss®o^c&~a.
From these estimates we see that for all k = 0, 1, 2, • • -
% < cb~ab~ka «<rS-° |a.
Let os now consider a cylinder with arbitrary p < p0. For some k > 1
we have
P*<P<P*-1.
and therefore
osc {a; Qp) <osc (a; <?p# J < cb-%afi_i < cPo"V.
i.e. the lemma is proved.
la obtaining estimates of Hdlder norms for vector functions we will make use
98
II. AUXILIARY PROPOSITIONS
of a more general proposition:
Lemma 5.9. Suppose that in a cylinder Qp - Q(pQ, 0Qpjj) there are given
measurable bounded functions ul(*, (), ••• , uN(x, t) and w *(*,«),•••, wNHx, t)
possessing the following properties: for any pair of cylinders Q p- Q(p, ®gP2)
and Qbp having a common vertex and axis with QPq, where b > I is a fixed con¬
stant and bp < pQ, there exists a function wr(x, t) such that
osc fwr; Qto) >6] max osc («*; QSp), (5.27)
p i-i N
and at least one of the inequalities
and osc \w'; QJ < (5.28)
osc {t»r; <?p| <nosc [wr; Qipj (5.29)
is valid. Here b, 8j, Cj, 8 and rj are fixed contants, and ij < 1. Then the esti-
mates
osc {u‘; < cp0-“p°, / = 1 N, (5.30)
hold for p < p0, where
a = -^min { —lnAri; 6).
c = ^>a<A,'+1,61', max {c,p*; a>u*a/v'}. (5.31)
©0 = max osc j®'; Qp,).
I, .... N,
The proof of this lemma is analogous to the proof of the preceding lemma.
We will not reproduce it here; instead we refer the reader to [65?,Chapter II,§ 4],
Remark 5.2. In the conditions and assertions of Lemmas 5.8 and 5.9 all of
the cylinders can be replaced by their intersections with some fixed domain Q.
Finally, let us prove a proposition that permits us to estimate
/@j, cxp[btS®X%, i)]dxdt for generalized solutions of linear equations.
Lemma 5.10. Suppose a function uix, t) belongs to Iis bounded
from above on Sj, and for all k > k = vraimaxj^, u satisfies the inequalities
! a**’ |yr < Y W + M-7 <*)]. <5'32>
where fiik) = /J mes A^fidt, A^it) is the set of points x from SI at which
§5. SOME OTHER AUXILIARY PROPOSITIONS 99
u(x, t) > k, y and k are positive constants, and the values of q and r are sub¬
ject to conditions (3.3). Then € 2^(jQj) with any q j > 1. Moreover, there
exist positive numbers b and B determined by T, mesfl, y,k,q,r and k such
that
J e*B<V »dx dt < B. (533)
Or
Remark 5.3. In conditions (3.3) r can be infinite for any n, while q can be
infinite for n = 1. In these cases we assume that fj. mes
vrai max y]mes for r « oo and
- (r- -
(k) = I j mes« ,4*(0cif I —mes'' {<£[0,7']:meSi41,(<)>0j
for q «
For a proof of the lemma let us consider a sequence of levels
“ *a-i +* ^ £ A = 1 • 2, ...,
where is a positive number that will be chosen below, and the sequence of
quantities corresponding to them is
1
H —
From (5.32) on the basis of inequality (3.4) we obtain the recursion inequal¬
ities for
(*#+! — **) **+> ^ II r. <?r
<p|^)IQr<pv(yj‘-+^).
from which it follows that
^+,<PV [(A + I) z\+* + k-lzh]. (5.34)
Let us show that for sufficiently large hg
zh<£U-\ A = 0. 1 (5.35)
where K is some constant. Indeed, if (5.35) holds for zthen for Z m we will
have by virtue of (5.34) the estimate
100 II. AUXILIARY PROPOSITIONS
*:AtJ<PY*[(ft-H)VV1‘*4 *0 l]Xe-(44,).
from which will follow the desired estimate (5.35) for + j as long as
Pt*[(h + k£']< L (5-36)
The function (A + l)e few all h > 0 does not exceed 1/kck and hence, in
order to satisfy (5.36), we choose and A from the condition
PY (— ^x*x + j 1;
for example, we take
*>*'*• ,,!7’
For such k0 and A inequality (5.35) will hold for all h «* 0, 1, 2, • • • as long as
it is valid for A = 0, i.e., if Zq < A, But for zg we have the estimates
*o=(j nws« ,**.«)«») <(*o"fer'll«(?lilj.r,«r
<(ft0-*)-,p|o(«|Or
r IfK I 4-X
< (*„— *)_lpvL*ines » QT ' mes^QT’J,
so that tbe inequality z q < A is satisfied if we set
*„= max [2(Jve;
r~ i±2 i±2- i ri
ft 4-PvX"l[ft mes » Q7" r 4- ires^Qr' J). (5.385
Thus (5.35) is proved. Let us now consider the quantities
mes Q^+i = mes |(*. t)£QT- “(■*• 0 >
= J mes A*h+)(t)dt.
0
By virtue of j
(**♦«
< max j| *<**>(*. 0|I,e<!^*>L
o^»<r 1
it follows from (5.32), (5.35) and (5.37) that
§5. SOME OTHER AUXILIARY PROPOSITIONS
101
■nesQ* + ,<fe0-^
< *»V [(*+1) kJiXe-h)x+’,+u-hf t < aV2r«-»<*+,>.
Because of this inequality
J J Od*dZ + e4"*mesQr,
Or
CO
2 «>**+1 mes Qa 4~ «**• mes Qr
CO
< ^ 2* + «**• mes Qr.
h-0
Thus for b < 2&q! the integral e^^^’^dxdt will not exceed the number
B - \2Te2bko/fZ2U - e "*®4*] + e&*0mes()j.. The lemma is proved.
Remark 5.4. The assertion of Lemma 5.10 remains in force if instead of the
fulfilment of inequalities (5.32) one requires that the inequalities
* J15l /v 1
Sn,r/ (A)*6/ + 2 l»;« (A)
i"*l i—I
be fulfilled for ft > k, where fi^k) = /q mes ri^qL A^it) dl (see Remark 5-3). Ki > 0,
5. < 1 + k., and the and r, are subject to conditions (3.3).
The proof of this assertion is carried out in exactly tbe same way as the
proof stated above, except that a system of recursion inequalities for z=
S/'KkJ is obtained instead of inequality (5.34).
Remark 5.5. If (he requirement of boundedness for vninaij^u is dropped
from the conditions of Lemma 5.10, inequality (5.33) is retained, but die constants
b and B in it will also depend on the form of the domain. Namely, this depen¬
dence will occur through the constant c from inequality (3.8), whereas above
under the conditions of Lemma 5.10 we could use inequality (3.4), in which the
constant j3 does not depend on tbe domain.
102 II. AUXILIARY PROPOSITIONS
§6. ON ESTIMATES OF max|u|. THE CLASS 8(0r, y, r, k, k)
In this section the boundedness of functions uix, t) satisfying certain inequal'
ities will be proved. To such inequalities are subject the solutions uix, t) of
linear and quasi-linear equations with principal part in divergence form and their
derivatives u% if it is understood that the functions forming the equations satisfy
certain conditions. These inequalities in general estimate the growth of the
quantities r-oy) in terms of k, the measure of the set where
> 0, the weaker norm ||w<*>j!z>@(plr)> °\P and °2T- We begin with a more
simple proposition, which permits one to give a total estimate for the maximum of
tbe modulus of solutions in the whole domain of definition.
Let uix, t) belong to V2iQf). We denote by AkU) the set of points *£Q
at which uix, t) > k. The following theorem is valid.
Theorem 6.1. Suppose that vrai max 5^ u < k, k > 0, and that the inequalities
I v*l* r (6.1)
hold for k >k with certain positive constants y and k. Here /i(k) =
/q mesr//^ Ak(t)dt, while q and r are arbitrary numbers satisfying conditions
(3.3). D Then
r JL+1 ,. JL l±i i£i 1
vraimaxa(je, f)<2A|J + 2* *’(PY) * T ' mes * £2j. (6.2)
Qt
where f} is the constant in inequality (3.4).
Proof. Let us take the sequence of levels kk = Jf(2 - 2“*), h
assuming U > k > 0). 2) It js clear that
(V. ~ K) ^ < V.) < il «(*a) II,. <,r-
On the other hand, by virtue of inequalities (3.4) and (6.1)
1 +X
II “(*h) L r, 0r < PI U<*A) l0?. < (**)
and therefore
1)For s = <« and r = *> the quantity is defined in the same way as in Remark
5.3.
2) For k - 0 the theorem is obvious.
= 0, 1,
(6.3)
(6.4)
§6. ESTIMATES OF max |u|
103
— l+K H-X
< •* ' <**) < 4P?2*I*~(Aa). (6-5)
From here and from Lemma 5*6 it follows that will tend to zero for A -* “>
if u^r(ftn) is sufficiently small, namely if
t J_ _t_ i
pj (ft0) = p.f (M)< (4pv)~ * 2-^. (6-6)
In order to satisfy inequality (6.6), we take M m mk, m > 1, and substitute It for
and W for &A + 1 in (6.3)* This and (6.4) give
1 „ 14-11 . l-fM l-fll
r (*)<-s!rr:r r mes * a
Hence we see that for
l-m l-t-x i l
mssl+Pv?' ' mes » Q(4f$v)* 2*1
condition (6.6) with U = mk will be fulfilled, and consequently p(2JO will be
equal to zero, i.e.
vrai max a 2/M = 2mk.
<3 r
The latter coincides with estimate (6.2).
Remark 6.1. The assertion of the theorem remains in force if instead of ine¬
qualities (6.1) the function u(x, l) satisfies the inequalities
N H2S.
K’LCySh/' (*)*6', f6-7)
where i_I
T .1
k, > 0, 6, < 1 4- X/, n, = J mes (i) dt,
o
while the parameters y(. and r. are subject to conditions (3<3)> This is.proved
in the same way as Theorem 6.1.
Remark 6.2. In die conditions of Theorem 6.1 it is possible to drop the re¬
quirement of boundedness for vraimax5^, u and to assume that inequalities'(6*1)
are fulfilled for k larger than a certain 4 > 0. Estimate (6.2) will then be valid
with the constant c from (3.8) in place of /9 from (3.4). Indeed, the functions
a * in this’case are in general not equal to zero on Sj., and hence in deducing
(6-4) it is necessary to use inequality (3.6) in place of (3-4).
Let us turn to local estimates for vrai mas u. Suppose we have
a cylinder <?(pQ, rQ) s= i \x - *Q| < pQ, tQ - rQ < t < tQl and a family of
104
II. AUXILIARY PROPOSITIONS
cylinders of die form
Q(p —<J,p. T — a2x)
= [ Ix — *0| < p — a,p, t0—(l - cjx < t < /a}.
that are coaxial with and have a common vertex with <?(pg. fg) and are such that
4p<P^-0|P<P<P». -y- < T - 02t < t < v (6.8)
We denote by Ak J>t) the set of points from the ball f \x - *0| < p) at which
u(x, t) > k, and by fiik, p, r) the integral /^-rmes^Mj i.t)dl. The parameters
g and r, as above, are assumed to be subject to conditions (3.3), and for r «= ~
it is assumed that
i JL
|i7 (*, p, t) = vrai max mes « /4*.(f).
while for q = <*> it is assumed that y}^r(k, p, r) is equal to
mesr {f6(4> — t, f0j: mes/4*,p(0> 0}.
We will say that a function u(x, t) from PjWPo* r0^ belongs to class
ll{Q(p0, fg), y, r, k, k) it for any k > k and all possible pairs of cylinders Q(p, f)
and Q(p - Ojp, r - o^r) of the above-mentioned form the inequalities
I«(*!Iq(p-o,p, t-o,t,<v{f(olP)"2+'(o2T)-1]||tf(*»| Q(p t)
+ ^--^.P.x)}
are satisfied for certain positive constants y and K.
Let us now consider the case when the cylinder Q(p0, r0) intersects with a part F T
of the boundary of the cylinder Qj- We denote by 21 ($ (pg, rQ) V’ r, k, k) the
class of functions from V2(Q(p0, that satisfy, for k>k>0 and arbitrary
cylinders Qip, r), Qip - o^p, r - oy) ol tbe above-mentioned farm, the inequalities
l«W|?
Q(p-<r,p, t-o4t) (I CIT
<Y{[(alP)-2 + (oJT)-1}|l«wI
2. q( p, t>nor
2(l+x)
r (k.p.X)
Here fi(k, p, r) » /m°axlo,to-rt mesis define<* in the same way as
above; it is only necessaty to understand by J^t) the set of points from the
intersection of the ball i|* - *g| < pi with fi at which uix, t) > k. The para-
§6. ESTIMATES OF nu |u| 105
meters y, r, q and k are the same as in (6-9)-
The following theorem holds.
Theorem 6.2. For any functions u(x, t) from U(Q(,p0, rQ), y, r, k, k) one has
the estimate
f _<■+* r _i ]
vraimax u(x, t) < 2c{ p# j{«||2> t>)[l + T0 7puJ
+ *r»+(wJp1}- (6-ll)
in which the constant c is determined only by the quantities y, q, r and k. An
analogous estimate holds for functions of class 0(pQ, fg) f| Qj, y, r, k, k), it
being only necessary to replace in (6.11) the cylinders (Xpg, rQ) and (Xp0/2, rQ/2)
by their intersections with Qj,. In this connection the constant c depends only
on y, q, r and k if vrai maXQ( *) < k, while in the general case it
also depends on S.
Let us verify the validity of the first part of Theorem 6.2. The second part
is proved analogously.
We first of all make a change of variables: * - xfl = £>g?, t - » pgt'- In
the new variables x and the inequalities (6-9), after a cancellation by pjj,
take die form
I “‘ft>Ppcp-o,p. Vilc^. p)_i +(a-^r']ll«(*,IIW
. .„ ^2-'- ~.l (6.12)
+ k^i ' (k, p, t)f,
where p‘ = pp^1 and r" = tPq2, while conditions (6.8) take the form
Y < p — ctip < p < I, -!- <? — 02t<t<9 s= Tjpo *. (6.13)
so that in the new variables the cylinder (Hi, 6) = 1|*'| < 1, - 6 < t“ < oi corte*
sponds to the cylinder Q(pQ. r0), while the cylinders $p, ?") =| |?| < p', -?'<?'< Oi,
correspond to the cylinders Q(p, r).
In the sequel we will omit the wave over x, p, I, etc. in writing tbe formulas.
Let us take a sequence of decreasing cylinders
106
II. AUXILIARY PROPOSITIONS
h = 0, 1, 2, • • • , and a sequence of increasing levels kh » M + Ml - 1/2*),
A » 0, 1, 2, . • • , where M and N are certain positive numbers.
As a preliminary we will establish the following lemma.
Lemma 6.1. If a function u(x, t) satisfies inequalities (6.12) (or all k lying
in tie range [M, M + JV], tAen /or tAe quantities
o
y„ = Ar2 J J (u — kkfdxdt
-xh A*k.phm
md 2 *
= Pa- T*)sfV • * = 0' 1
iw'tA tAe k^ and pj indicated above there is valid the system of recursion ine¬
qualities
*♦. < Y.24* [j-r6+4+*y% (■& +■ O’] ’ <6a4)
**♦. < V,2“ + 4- (£ + l)i • (6-15)
where » 2®j8z[y(2* + 0" *) + 2*], 8 = 2/n + 2, on</ /3 is tAe constant in (3-4).
After we prove this lemma it will be possible, by choosing sufficiently large
and equal numbers U and N, to conclude with die help of relations (6.14)—(6.15)
that the yh and tend to zero for A -> <*>. This gives the desired estimate:
vrai a.aX(?( ^ /2^ /2) u <2M.
Thus, let us turn to the proof of Lemma 6-1- Take the sequence of continu¬
ous cut functions £^(|*|) that are equal to 1 for 1*1 < Pa + j> equal to zero for
1*1 si P* ” H(pfc + Pft+iK an<l linear on the segment |*| € [pj+p P*]» 80 t*>at
|CJ<2**< Let
K = J mes'4*(N...54W<tt-
~t4+l
We estimate the y^ + (, using inequality (3.7), in the following manner:
o
y*+* J J {u — kh+lf&dxdt
"T*+1 %+i.p,('i
§6. ESTIMATES OF max |u| 107
< B'AT2^ I vtai max f («— fc»+i)2£* dx
\ %+1.Pft(0
_j_2 | J Mt?a + (b— ki,+if^hx\dx dt I
~xh+1 *
(6-16)
The right side of (6*16) is estimated by means of the obvious inequality
^■^(A/i+i A*) (6.17)
and inequality (6*12) is applied to the functions i^h + l) the two cylinders
Q(ph, ri + 1) and (Kpk, rh). This gives
yh+, < 2(52[(**+. - A*)"' Nf { v [(4*+4-f e-V**)
X N-J| «<*- •) IP, «(„, <*) + *»+VJ 0+X)] + 4-\)
< 2‘2 Vi [(*fc+1 -**)"1 Wf[2“yi+4 + kl+lN~*zl+yyl\. (6*18)
Taking into account the choice of the k^, it is easy to see that inequalities (6*14)
follow from (6.18). In order to prove inequalities^ (6*15) we note that
(A/j+l — Aa)2 2/,+x = (Aj,+1 A*)2 M’ft+I
<li“(ftA)^lt.0(pA.,4+l) <PJl“(‘A)^l«(p^*+1)- «•»>
Estimating the right side by means of (6.12), in the same way as was done in
(6.16)—(6*18), we obtain
(^+,-^)J^+1<2p24"*Wy/,
+ 2 f$2V {[(P/i *" Pa) 2 + (T;,+i — **) '] || a**** III, Q (pA, th)
I <y.n
' K 2p2vl(4A+4(l +- Y-") + e-‘2*+J)N2y„ +kfcl+% (6.20)
2(l<y-n
From this, by virtue of the choice of the k^, follow inequalities (6.15). Lemma
6.1 is proved.
Let us continue with the proof of Theorem 6.2. Put N ~ M. From the system
of inequalities (6.14)-(6.15) it is possible to conclude on the basis of Lemma 5.7
108 II. AUXILIARY PROPOSITIONS
that the quantities y^ and tend to zero for h —» m as long as the values of
yg and zQ are sufficiently small, namely, if
I * JL i i |
i’o<mlnt(2c)"T*”‘'6; (2c)' * b~ *d J =Y3.
I
*o<Ya,+*. (6.21)
where d = min 15; «/(I + *)|, 6 = 16 and c = 4yj. We achieve the fulfilment of
inequalities (6.21) by choosing a large number M» In fact, for we have the
estimate
o
y0=sjW“* J J (6.22)
*m, i
In order to estimate we note that inequality (6.20) holds for A = — 1 if
one assumes that A_ j = k, p_ j = 1, r_ j = $. This gives
(Z&F 1(4’(1 +Y"‘) + e"V)||«<* „^(li6l
+ S*[|*<*. J. 9)]M^}<
(AI — rt)3
T 2** +** ULtfSil
X [(4\l -J- Y-'j +0"'2:)||«|!jj(j(1,#) + *Se ' " J- (6.23)
From (6.22) anJ (6*23) it follows that for
r 1 ^ l J
max\V;"TIU'ii,>(3(,.e,; k ~t“V3 *‘1t*>PYt
K
r gfn-n) i£I±KT|7t
x[(4a(l -4-Y_1)-+-©",22)H«Il2.<?(I,B> +-ft2e '«,*]! (6-24)
requirements (6.21) will be satisfied. Thus, for example, if one takes
M -c [m=. (» + e"^) H-1 (> + 0±^)1 ■
f _1 1 2 i 2 ±121)
c== inax|Y3“-i-Y~!!ii*K.pY223; 1 + 2(1‘x,|>Y*x„* J. (6.25)
then the y^ and tend to zero for h -* <*> and, consequently,
vrai max u(x, 2A1.
«(M)
Theorem 6.2 is proved.
§6. ESTIMATES OF tnax|l<| l09
Remark 6.3. Tbe last tern in inequality (6.9) can be replaced by the term of
more general form
AT 2(l+,<.)
2 fe2(U0‘VKiV, u (*. p. t).
i-i
where -rmes Ak ^t)dt, 8{ < min{£(l + K.); K; i, 8= 2/(n + 2), and
y, Kt, q. and r. are fixed positive parameters, with die q. and r\ being subject
to conditions (3.3). Also, the restrictions (6.8) can be replaced by die require¬
ment that
Po - a°Po< P - o,P < P < pu,
T0 — oft < T — < T <Tfl
with some fixed positive numbers cr® and <7®. In this connection an estimate
analogous to (6.11) is valid for vrai maxu(x, 1).
The same remark holds for cylinders intersecting with Fy. All of these asser¬
tions are proved by the same method that was used in the proof of Theorem 6.2.
Remark 6.4. It is not difficult to verify that a function u(x, t) from V^Q^)
belongs to class WQipQ, rfl) f, QT, y, r, k, k) for any cylinder Q(p0, rQ) having
its vertex in Qj, if «(*, t) for all k £ k and 0<«0-r^t0^r satis¬
fies the inequalities
vrai max|| «(**(jf, f)||| Q-f- f f I uM\!&dxdl
' ’ J, Q
<V>| J
I U-X Q
20+v)
h * \ 1
+ fc2 J / f u*j dt
'•“* '■Aku> /
(6.26)
in which yj and k are fixed posidve numbers, r and q are subject to conditions
(3.3), and £(*, t) is an arbitrary piecewise-smooth continuous nonnegative func¬
tion such that u^Xx, t)£(.x, r) is equal to zero on the lateral surface and lower
base of the cylinder Q(pQ, fp) and £(*, t) < 1.
For q « 00 it is assumed that
no
II. AUXILIARY PROPOSITIONS
1 for mes At (I) > 0,
0 for mes A&(t) — 0,
while for r = oo the last term in (6.26) is understood to mean
vrai max I ft dx\
§7. THE CLASS 82«?r M, y, r, S, k)
In this and the next sections we consider classes of bounded-in-modulus func¬
tions satisfying inequalities similar to the inequalities of the preceding section.
Concerning the elements u(x, t) of these classes, it is proved that they are Holder
continuous and an estimate is given of the Holder norm |u|( a\ To such classes
belong the solutions and the derivatives of solutions of linear and quasi-linear
equations with principal part in divergence form.
We will say that a function u{x, t) belongs to class 8£Qy, U, Y> r> $, k) if
uix, t) 6 vraimax^ju| £ M and the functions «<*, t) = ± u(x, t)
satisfy the inequalities
max Hw(*>(jr, Oil? x < *)|l? „
r,<l<r,+T ' '"P-O-I/U 2 ,Kp
+v [(°i p)_2Hic(*>ill,0{p_ t, + (i7<1+K)(k, p, t)] (7.d
and
V{[(°iP)-S+ (V)-1]!!®**1 II*,Q(Plt) + il+X>(*. p. t)}, (7.2)
in which uf-k\x, t) = maxM*, t) - k; Ol; <JKp, r) = Kpx («fl, Iq + r) = ||*~ *0| <p,
t^ < I < «0 + r| is an arbitrary cylinder belonging to Qj.; p and r ace arbitrary
positive numbers; Oj and o2 aie arbitrary numbers from the interval (0, 1);
1) In stead of membership in die class j-), in essence It is required only chat
u 6 V^lQi') “”d that the noim Ja(*. Ol jXI *’* ®"*te ^or each '* f*0® to, 7"].
§7. THE CLASS S2(QT, M, y, r, S, k)
111
ykk, p, r) « mesr/l Ak ^t)dt, with Ak £t) being the set of points x from
Kp at which «Ax, t) > A; M, y, q, r, 8. and k are fixed positive numbers, with q
and r satisfying the conditions
? € (2* for *>3-
?£(2,oo). rg( 2.oo) for n — 2.
^g(2,oo], rgl4, co) for *=1;
(7-3)
and 4 is an arbitrary number subject only to the condition
vrai max w(x, t) — ft < 6.
<?(P. *)
(7.4)
In the case q= <*> (for n = 1) by fi{k, p, r) we understand meslt 6 («0, + r);
mes Ak J.i) > 0|.
Remark 7.1. In the definition of class one could in (7.1) and (7.2) con¬
sider the number y to be different in front of all five terms of die right sides, with
the factor in the first summand of (7.1) being necessarily close to unity, and in
this connection observe a greater differentiation in the dependences of the con¬
stants calculated below on the yjt i = 1, • • • , 5> But this does not interest us,
and therefore we have put one and die same y in (7.1) and (7.2). Furthermore,
all of the results do not change essentially if one imposes on p and r in (7.1)
and (7.2) die additional restrictions p < p„, r < rQ, where pQ and rQ are fixed num¬
bers.
The basic aim of the present section is a proof of the embeddability of
82(Qt’ y, r, S, k) in Ha,a^HQj.) with some positive a determined only by
the parameters M, y, r, 8 and k .
Remark 7.2. It is easily verified that inequalities (7.1) and (7.2) follow from
the inequalities
lkw(*. + *)£(*. „
<11 «.<*>(*, /„)£(*, /0)lg.*p
+V,
112
II. AUXILIARY PROPOSITIONS
in which the same notation as above is used and Qx, t) is an arbitrary piece-
wise-smooth continuous (unction, with values between 0 and 1, that is equal to
zero on the lateral surface of the cylinder Q{p, r). For q > ~ the last integral is
understood in the same way as indicated in Remark 6.4.
As will be shown in the following chapters, the solutions of linear and quasi-
linear equations, and also their derivatives with respect to x, belong to classes
$2 or certain generalizations of them that will be defined below. This circum¬
stance permits one to conclude that all of these functions belong to Hdlder spaces
and to obtain the corresponding estimates for them
We will first prove four auxiliary propositions. They will consist of asser¬
tions on the existence of certain constants. It is essential that they all be inde¬
pendent of the element u(x, t) in question in class and that there be defined
only n and the numerical parameters entering into the definition of class
Lemma 7.1. Suppose a function u(x, t) satisfies inequalities (7.1) and
mesA/l'PU()) <(l/p)mesK^= (,Kn/p)pn, p > 1. Then for any ( from (yj 1/p, l)
it is possible to find positive numbers 0(£) and 6(f) such that if
then
vraimax u(x, f)—*>p’ .
for
mes (Kn\Ak+lHp(t))^-b (£) x„p“
(7.6)
By the conditions of the lemma we have
0-0tp W
-f
On the other hand
(a — kf dx.
(7.7)
§7. THE CLASS ®2(<?p X, y, r, 8, k) 113
Therefore
mes A,+Wl p.„lP(t)<-^|fX„p' + y{(o,|)"2V - <0)p-J
+(!«) **.* ('-'o)r P * Kp - (7-8)
For « < + 6(£)p2, taking into account the assumption H > p"*^2, we obtain
from (7.8) the inequality
mes /4*+;//, p_0,p (0
<rJ{p-,+v[<TrJea)+xf<,+*,-,ea)r(,+x>])x,p*:
here we have made use of the fact that q and r are connected by equality (7.3).
For any (> sj 1/p it is obviously possible to select positive numbers a^, 6(£)
and b(£) so that
+rs{ p'1+v [*r2» ©+*70+,#"‘e af(1+x)]}
= 1-6(I)<1
and hence
mes i4*+at, p (0 < mes Ai,+tH, p_01p(O 4- mes K„ \ Kp_0,p
<(l-*(|))mes/fp.
The lemma is proved.
For the subsequent arguments let us fix p, 5(f) and Mf), for example, as
follows: p = 2, f = 3/4, M£) “ 1/18 and Q(() = $ is a positive solution of the
equation
y [(36b)8 0 4~<1+’t!'10r(,'i'’<)} = -^:.
In addition, we introduce "standard cylinders,” namely, cylinders of the form
Qpm II*-*0I sp. *0 <«< <o+ 0p2>
with arbitrarily situated "vertex” (xg, «0 + Bp2).
Remark 7.3. Lemma 7.1 is also valid for the intersections of the cylinders
considered in it with some fixed cylinder Qj,. It is only necessary in all of the
conditions and assertions to take in place of the balls Kp their intersections
with 0.
We introduce one more noration: Q J.k) is the set of points (*, t) from Qp
at which u{x, t) > k. It is easy to prove the following relations between the
measure of QJ.k) and the quantity p(k, p, Op2), relative to one and the same
cylinder Qp:
114
II. AUXILIARY PROPOSITIONS
2. i 1 —
(Map")* ~ ' *ne87 Qp (*) < i1' (*. P. 0P*)
2_
<(0pJ)”"« mes i Qp (&), for r<?. (7>9)
_7 2 2 1
(*„p")* ~~ mesr Qp(k) > Hr (k, p, Op2)
2. 2. JL
>(9pJ)r ? mes" Qp(k), for f >?•
Indeed, the right inequalities of (7.9) follow from HSlder’s inequality, while the
left follow from the fact diat mesk) — ^ mesr/®mes* '^A^^ijdt
is estimated in terms of ji(A, p, 0p^) mesl ~r/,1 fi from above for r < <7 and from
below for r > 9.
In the following two lemmas we will assume that the parameter q entering
into the definition of /dk, p, r) is finite. As is seen from conditions (7.3), q can
be infinite only for « = 1. In this case it is possible to modify Lemmas 7.2 and
7.3 so as to include the entire range q 6 (2, 00]. We will do this at the end of
the section.
Lemma 7.2. Suppose u(x, t) is an arbitrary function subject to inequalities
(7.2). There exists a 0^> 0 such that, whatever the cylinder C Qj. and the
number k^ > vrai max QpQu(x, l) - 8, from the inequality
mesU0) < 0,p£+2 €7-10)
follows
mesQpo + = 0. (7.11)
Tf
as long as
nx
/i = maxa(jt, t)—k0^pt2 (7.12)
%
The cylinder Qp^^z ** (7.11) has the same vertex as
Thus, suppose the conditions of the lemma are fulfilled. We pass from x, t
to new independent variables lc, T by means of the formulas
x~ JC0 = ?()•?. t — *o = pgf~.
Inequalities (7.2) in this connection take the form
§7. THE CLASS M, y, r, 8, k)
115
I »<*> I5
I f<?(p-a,p, t~o,t)
<Y{[(fl'lP)”:! + (02^j'‘‘]ll “(i) II*, Qip, ?, + (^"Hr<I+’<>(ft. P. T)},
where p = p/pQ, ?* = r/p|, while = Q(l, 0). Let us pass from a to the func¬
tion v = u/H. For it, dividing both sides of the latter inequality by H2 and tak¬
ing into account that //-2p*" < 1, by virtue of (7.12), we obtain
I **' lW-a,p. < V { [(°.P)2 + M'1] (I V$> »!
<3<?-o,p. r ILWIW J I* 112. <J(p, T)
1
+ l»r '"""'(ft. p. T)
<1+x)- - (7.13)
where Ts = k/H.
The function v(x, T) satisfies the conditions of Lemma 6.1. Let us take
as the range [Jf, M + Ml of variation of the parameter It the interval
[*0 = 77-. +
Then, according to this lemma, for the quantities
and
yhs=4 f f (v—k^dxdt
-i
J mesT^j- ^(?)rf?y
the system of recursion inequalities (6.14), (6.15) is valid. From this system of
inequalities it follows by Lemma 5.7 that the y^ and z^ will tend to zero for
h -» •> if y0 and zQ are sufficiently small: ]
yv<k (7-14)
(an expression for A is written out in (5.20)).
But by virtue of condition (7.10)
0 0
y0=4 j J (w — k0f dxdt< J mes A^ , (t) dt
-« Xi
116 H. AUXILIARY PROPOSITIONS
while z0 = (/® according to estimates (7.9) and (7.10)
does not exceed
2 £ 2 £ JL 2
*,* '0{ for r >- <7 and 0r « 8,» for /■-<?.
In view of this we will satisfy conditions (7.14) if we take
e,<mln(x; x^'TxiTmo) for r>?.
8i<min(x,; 01 ' A,2<1+*>) for /■<?. (7.15)
For such a choice of $ j the numbers —* 0 for A —* oo. But this means that
vraimax^^/2 « < + H/2, i.e. (7.11) is valid. Lemma 7.2 is proved.
Let us take an arbitrary (standard) cylinder Qjp~ &2px ^0’ £0 + ^ Qf
and the coaxial cylinder Q having a common vertex with Qjp' l-et
(ji, =s= max u, Hj = min a, « = n, — (j-;.
^ap 0ep
We will prove the following lemma.
Lemma 7.3. Suppose u{x, t) is an arbitrary element of 82^-Qt’ Y’ r’ K^’
For any (?j > 0 it is possible to find an s = s(6*j) > 0 such that either
*> = osc U, <>*,!< 2‘p-r (7-16)
mesQp(|i, — -?pr)<QiPa+i (7a7)
or
mes|(j:, <)€ 9p : “(x-0 < M'2+ 2? } <6iP"‘'2- ^ ’18^
Let oj > 2spnK^, where s will be chosen below. Obviously we have at
least one of the inequalities
1 „ „» (7.19)
mes A m (<0 + 36p2) < ~n nnP"
Mi—f. p *
mes | x £ ATp: «(■*, *0 + 38(5^) < Hi 2"} ^ 3‘**P"‘
Assume that (7.19) holds, and therefore that
mes/I « (fo+3epJ)<4-’‘Bp”. r=l. 2 *— I.
§7. the CLASS %2(Qt, M, y, r, 8, k) U7
Then all of the subsequent arguments are carried out with the function u(x, t).
Otherwise we would consider the function - u(x, *). We denote maxq u by
If <,l*i ~ <a/2s, then mes^Qtj - a/2s) = 0 and (7.17) is valid for any 0j.
On the other hand, if /ij > ftj - &>/2s, then H = jij - (jij - <a/20 > <a/2r - o>/25 >
w/2* > pnK^ for any r < s - 1, and therefore Lemma 7.1 is applicable to u(x, t)
in the cylinder Qp for the level k = fij - <o/2r, as long as <y/2r < S. It guaran¬
tees that
mes >jg
for
'ei'o+sep2. fo+40p*i
and all the more so that
mes \Kp\A a (/)]>’ K)1p\
* 6: [<0 + 36p2. ta 4- 40P2], (7.20)
since jUj - <u/2r+ 3H/A < /ij - <u/2r + (3co/4)/2r =Mj ~ <u/2'"1'2. Inequalities
(7.20) hold for r 6 [lgj&>/S, s - 1]. We will take r from the interval
[[2Jf/8] + 1, s - 1] C[lg2<i)/S) s - 1]. (The symbol [a] means, by convention,
the greatest integer not exceeding the number a.) Let us apply inequality (5.5)
to the function u(x, t) and the levels I = nl - <u/2p+t, 4 = /i1 - <u/2p for l C
U0 + 56p1, tQ + 40p2} and p € [[2M/8] + 3, •s + l] and take into account (7.20).
This gives
~z~rrmes<4 . (0< lSf^'p J I«,<*. Ol**.
V “» ^+T’P a>'«)
where
A « (0 \ ^ » (*)•
p »n—pr • «> »>~pTr-p
We integrate both sides of this inequality with respect to t, using the limits
indicated in (7.20); then we square both sides, after which we estimate the right
side by Cauchy’s inequality
118
II. AUXILIARY PROPOSITIONS
(^r)2«n“2Qp(h—
• 2 'o+46p> <„+40p>
•< [lSjtj’p] p2 J J u\dxdt J mes3Sp(t)dt. (7.21)
(0+Mp1 a>pW <0+39o'
For an estimate of the first integral on tbe right side we use inequality (7.2),
choosing for Qip, r) the cylinder Q2p> and for Qip - erjP, r - a^r) the cylinder
Qp. After obvious simplifications it gives
<,+«p>
<v{[p~2+ (Sep5)"1] (£)*x.(2p)*40p*
+-[tH„(2p)")i48p2]r }. <7-22)
By assumption os £ 2spaK^2, so that for p e [[2M/S] + 3, s + ll it follows
from (7.21) and (7.22) that
mes8Qp(n,— —r)<cp"+2 J mes Sp{t) dt. (7.23)
where
c=(i8x;'e)\ {(e + y2"+2x„+(2vle^) }.
Let us sum inequalities (7.23) with respect to p from [2M/S] + 3 to s - 1,
M
(* “ t^1] “ 3) mesS Q» (Ml ~ £)
first replacing the left sides by mes2Q - os/2s). This gives the inequality
<»+48p>
< cpn+2 J mes Kv dt = c0K„p2'1+4,
from which it is seen that for >*+&&
s*
(7.24)
we will have (7.17). The lemma is proved.
Remark to Lemma 7.3. In the case when in y, r, S, k) the para-
§7. THE CLASS M, y, r, S, k) 119
meter S = ~, i.e. the functions u{x, t) satisfy inequalities (7.1) and (7.2) fot all
k, the parameter s = 4 + [c6t<n/dfi by the same token does not depend on M.
Lemmas 7.2 and 7-3 imply the following proposition.
Lemma 7.4. For any element u(x, t) from 82((?p U, y, r, 8, k) and any
standard coaxial cylinders Qp/2 aK& Qip ® QT *at'In§ ° common vertex, there
is valid at least one of two inequalities: either
osc | u; | (7.25)
or
osc | u; Qp j < (I — -p-} osc («; Q2pl. 0-26)
where s is the number s(6^j in Lemma 7.3 while $l is the number 0^ in Lemma 7.2.
We recall that the numbers 0j and s = s(0j) are determined only by the para¬
meters of the class 8j.
Suppose is the number defined in Lemma 7.2, while s = s(0j) is the num¬
ber determined by 0^ in Lemma 7.3- We assume that ca j « osc {a; Qfi/^1 >
2spnK^2, Then all the more so <o « osciu; > 25pnK^2-
According to Lemma 7.3 this guarantees the validity of inequality (7.17) or
(7.18). Let us consider the case when relation (7.17) is fulfilled (the second case
is considered analogously with tbe function u(x, t) replaced by — u(x, /)). By
virtue of Lemma 7.2 applied to the function u(x, t) in Q and to the level =
*<j - <u/2s ~ *, where /ij = maxQ^^u, either H = maxQ^u - (/it - o>/2s~1) < pnK^,
ot mes + fl/2) ** 0. (Note that the condition kQ £ vrai maxQ u(x, t) - 8
of Lemma 7.2 is fulfilled.) The first assertion of the alternative guarantees the
inequality ma xq^^u<[i1 - to/ 2s"1 + <nl - o>/2s, and by the same
token also inequality (7.26). The second assertion guarantees that
^ <t> , H
ma x«<(i, —+ ~
VjP * *■
- m —2^T + ~ [«« ■-(«*.- ~t)] <*-f
and all the more so (7.26). Lemma 7.4 is proved.
From this lemma and Lemma 5.8 follows one of the principal assertions on
120 H. AUXILIARY PROPOSITIONS
the embeddability of 92(@p M, y, r, S, k) in Ha’a^(QT); namely,
Theorem 7.1. Suppose uix, t) is an arbitrary function from 8j(<?p M, y, r, 8, k.
and Qp^ = Qipg, &Pq) *'* a cylinder of standard form belonging to Qy. Then for
any Qp = Qip, $p*), coaxial and having a common vertex with Q, the oscilla¬
tion of uix, t) in is estimated as follows:
osc [«. Qp} < CfPp-*, (7.27)
where
a= min j — ln4^I -gr); -Tp}. e — 4°max{2M. 2'p^ J,
and the numbers 6, s «* «(0j) rotrf are defined in Lemmas 7.J—7.3. 77te nom-
ber a for 8 = ™ does not depend on U, and for 8 < <» is a mono tonic ally decreas¬
ing function of M/S. The number 0 in all cases does not depend on M. Thus
the class M, y, r, S, k) is embedded in Ha,<L^\Q^), with a independent
of M for 8 =* <*>.
Remark to Theorem 7.1. If inequalities (7.1) and (7.2) are valid only for
t < 62P2 S #2?o' t**en inequalities (7.27) will hold for Q with p < pQ, where in¬
stead of 6 in the definition of Qp and in other places it is necessary to take
6' = mintd, Ojl*
Theorem 7.1 is completely proved for dimensions n > 2. In the case n = 1
we have not considered all of the values of the parameter q that are admissible
by conditions (7.3). Namely, in the proof of Lemma 7.2 the assumption that
q < oo was essentially used, although in conditions (7.3) for n =» 1 the parameter
q € (2, oo]. We will show how Lemmas 7.2 and 7.3 must be altered to make the
proof suitable for all n > 1 and all values of q and r from (7.3).
Instead of Lemma 7.2 one must use the following proposition.
Lemma 7.2 ’. If u(x, t) satisfies inequalities (7.2), then there exists a > 0
such that from the inequality
(V Po* 6($)< ®iPff (7*28)
for the quantity p{k0, p0, &Pq) corresponding to the cylinder QPq of Lemma 7.2
there follows (7.11) as long as (7.12) is fulfilled and Aq > vraimax^^ v(x, t) - S.
The proof of this lemma is based on the use of the same recursion inequal¬
ities (6.14), (6.15) that were used in the proof of Lemma 7.2. The difference
consists in the fact that now the smallness of the quantity zQ is guaranteed
§7. THE CLASS ®2(<?r. M, 7, r, S, Kj
directly by condition (7.28):
221
while the smallness ol y_ is deduced from (7*28) with the help of estimates (7-9):
" r r
It is seen that, in contrast to Lemma 7.2, unrestricted growth of q does not
represent a hazard for estimates of Zq and y^.
Lemma 7.3 must be replaced by Lemma 7.3'«
Lemma 7.3*• Suppose u(x, t) 6 8y, r, S, k). For any 0j > 0 it is
possible to find a number s = s(0j) > 0 such that if the inequality opposite to
(7.16) is fulfilled, then either
Here ftj, 41 denote the same quantities as in Lemma 7.3.
Relations (7.9) show that for q < oo the assertions of Lemma 7.3 ' follow
from Lemma 7.3* But q can be unbounded only for n = 1. For this case, i.e.
for n = 1, we prove Lemma 7.3 ’ somewhat differently, omitting estimates (7.17)
and (7.18). Namely, as in Lemma 7.3> we will assume without loss of generality
that inequality (7.19), and hence also (7.20), is fulfilled.
Instead of applying inequality ($.5) to the function u(x, £) and the levels
I = - &)/2p + 1, k = pj - <u/2p we apply the stronger inequality (5.6) with
(j ~ r/1?* £2 For?<°o this is possible and by virtue of assumption (7.20) gives
1 _JL-
xa * r ^
, 1 £
~~7" n n , .. ^
8 T 8j2 for T^q.
i.
r
r /o+40o»
■f1
■ f,+ 38p*
mes i A u
n.--p-.P
r
(t) dt 1 < 0,p\ (7*29)
or
2
t
(7.31)
122
II. AUXILIARY PROPOSITIONS
But this is also true for q = oo if it is agreed that in the case q = °a the quantity
mesr^ Afc J,t) will be regarded as being equal to zero for mes/4^ J.t) ** 0 and equal
to one for mesA^ £t) > 0.
Integrating inequality (7-31) with respect to t from + 3dp2 to iQ + 49p2,
we get
eP2)
, z.+W
<18(2p)''« J J \ax(x, f)\dxdt. (7.32)
'0+30p> ap»)
where for q = » the quantity fi(A, p, 0p2) on the left side is equal to
tnesU 6 [<Q + 3Op1, «„ + 40p2J: mesAfc £t) > 0|, which conforms to the notation
adopted from the very beginning.
Squaring both sides of inequality (7.32) and extending the same arguments
used in Lemma 7.3, we come directly to estimate (7.29), by-passing the estimate
for mesQ - <u/2s + *); in this connection it is necessary to take
s = [^-+0rr Y02r (e + 4 + 2xe"<1+ X,)J + 4.
Lemma 7*3' can be considered proved. This completes the proof of Theorem 7.1.
Corollary 7.1. If a function uix, t) belongs to class %2(Qj., M, y, r, 8, K),
then for any cylinder Q(p, p2) belonging to Qr the estimate
J «£(*. Od*dt<cIp',+a“ (7.33)
0 (p. P*>
is valid where the constant Cj is determined only by the parameters of class 82,
n and the distance from Qip, p2) to V j., while a is the same as in (7.27).
The validity of this assertion follows from inequalities (7.2) and (7.27).
§8. THE FUNCTION CLASSES 82(?r (JT • • • ) AND %2(QTljr', ■■■)
Functions of class S2(@^ M, yf s, k), generally speaking, cannot be smooth near
the boundary I’y of die cylinder Qp Let us distinguish from it two subclasses
Bj^rUr', •■ • ) and B2(@rur', ■ •• ), where T' is a subdomain on the surface
Srur0, consisting of S j (part of Sy) and Fq ) (part of Tq). The elements u(x, l) of
these subclasses, as will be proved in the present section, are continuous (in the sense
of Holder) functions in <?r IJf' as long as Sj and tfx, <) on T' have a certain regu¬
larity. It has turned out that U F1 > • • • ) and &2• " ‘ ^ cootai“ die
§8. THE FUNCTION CLASSES B2{9rUr’', •••) AND B2«?rUr', •-•) 123
solutions of the basic boundary value problems for parabolic equations.
The class %^Qf UI"", M, y, r, 8, k) consists of all elements uix, t) of
class 8£Qft Y, r> 8, k) for which the functions w(x, i) = + u(x, t) sctisfy the
inequalities „
• *0 (»-«,». t-ffjT) p Qt
C Y {[(alP)"2H-ig, Q(p. T)n q,.+M:~(1+,V.p.f)} W-1*
for any cylinders Q(p, r) .tot intersecting Tj.\F '. In them all quantities have
the same meaning as in (7.2), ezzevt that the numbers k are subject to the ad¬
ditional restriction
k >• max w(x, t), *8-2^
9 <p, t) fl r
and nik, p, r) = /5*a xi 0; «q !m e s ^9 where Ak fi(l) is the set of points x
from KpOil at which w(x, t) > k.
Inequalities (8.1) coincide with (7.2) if the function t) is considered to
be extended onto all of the cylinder Qip, r) in such a way that its values in
Qip, r)\Q{p, r) f) Qj. do not exceed the maximum of uix, t) on Q(p, r) f) F*. For
functions of class U ^, — ) the following theorem is valid.
Theorem 8.1. Suppose the projection of onto SQ ^ satisfies condition
(A). Then any element u(x, t) of class $2iQT |J T', y, r, 8, k) satisfying a
HSlder condition on T’:
oscja; Q(p, p2) H T'X cpf, e>0, (8.3)
belongs to Ha''t^(Qj.\jT'), where a > 0 is determined by n, the parameters of
class S2, the numbers c and e from (8-3) and aQ, 0Q from condition (A). For
8- oo the quantity a does not depend on M- A norm |u|^ in any domain Q' C
Qj separated from T j-\T' by a positive distance d is estimated from above by
a constant depending only on d and the quantities affecting a.
This theorem is proved in the same way as Theorem 7.1. Some small modifi¬
cations are necessary only in Lemma 7.3. In this lemma we were choosing the
number s = s(0j) by making use of inequalities (7.2) and the presence, for at least
one of the functions + u{x, t), of a range [fjj - &>/2|r+2, p^], rj> r^, in the vicinity
of its maximum p^ for which the inequalities
l)That is, the projection of Sj onto Sq * \(x, t): j6S, I* o| by means of straight
lines parallel to the t axis.
124
II. AUXILIARY PROPOSITIONS
mes r/Cp \ A a (01 > 6xnpn, [fo + 30p!*, <0-+46p»l. (8.4)
I m y+2’p J ^
are valid with positive constants r^, b and 6 determined by known parameters
(in Lemma 7.3 one has b = 1/18, rg = [2,1f/8] + 1, while 6 is chosen in Lemma 7.1).
For boundary cylinders we achieve the same result in the following manner.
We separate the boundary cylinders Q2p = &2px f*o* *o + /l@P2^ *nto two tyPes:
the first do not intersect with Sj, but intersect with TJ, and the second have an
intersection with S^.. It is not difficult to see that for a proof of the assertions
of Theorem 8.1 it is sufficient to consider from the boundary cylinders Q2p °f
the first type only those for which 4 )0p2 < 0) and of the second type only
those whose axes lie on 5^.. Let us take a boundary cylinder Q2f> with
and <0 + 30p2 < 0- We assume that
®s=osc (u; Q2p) > 2V'. (8-5)
where = minlf , bk/2 !, while the number s remains to be defined. The oscil¬
lation of the functions + uix, 0) on K-2p by virtue of (8.3) be less than <u/4
if we assume p < I and subject the number s to the condition
2s > 22+V (8.6)
In view of this, for at least one of the functions + uix, t) the range [jjj - <u/4, jtj]
in the vicinity of its maximum fij in Q2p will be free from its values on K2p for
( = 0. Let this be true, for example, for the function u(x, t), i.e. let
«(*, 0)<n,-^, jcg/f2p. (8.7)
Then the subsequent arguments will be carried out with the function vkx, t) and on
the level tj — <u/4, assuming that uix, t) is extended onto the domain of
negative values of f in such a way that for them u(x, £)</<,— to/4. All of these
arguments are based on inequalities (8.1), which, as noted above, for u(*, t) and
k>ji^- at/A coincide with (7.2), and on Lemma 7.1 derived from inequality (7.1).
But for u{x, t) and 4 > ft j - <y/4 inequalities (7.1) in Q p = Kpx [tQ + }6p2,
tQ + 40p2] are consequences of inequalities (8.2) (in fact, inequalities (8.2)
applied, for example, to the cylinders and Kp x [tQ + 3dp2 - p2, *0 + 40p2]
give (7.1) for Q , although with y twice its former value), and hence for uix, t)
and A ^ ftj - oi/4 in Q the assertion of Lemma 7.1 guaranteeing inequalities
(8.4) is valid. One can take for $ the value fixed on after Lemma 7.1. All sub¬
sequent arguments and results are the same as in Lemma 7.3* The only difference
§8. THE FUNCTION CLASSES 82«?rUr\ ••• ) AND 82(Cj-Ur'. ) 125
is quantitive: in the value of s = s(#j) itself, which this time depends on c and
i from (8.3) (see condition (8*6) on s), and in the alternate premise (7.16), which
is replaced by
«> = osc(b; <?2p| < 2V\ E, = min|^, e|. (8.7')
In addition, we assume p < 1.
Lemma 7.2 is applied to u(%, 1) and /1: > — (0/4 in Qp without change.
From it and Lemma 7-3 follows Lemma 7-4 with the replacement in condition (7.25)
of 2spnK^2 by 2sp(l. Also Lemma 7.4 together with Lemma 5.8 of Chapter II
guarantees the desired estimate for the oscillations of u(x, t) in boundary
cylinders of the first type. For boundary cylinders of the second type the
arguments are almost the same as for cylinders of the first type. It is true
that fot them we do not in general enlist Lemma 7.1, but we guarantee ine¬
qualities (8.4) because of the membership of the center of Kp to the bound¬
ary S and because of condition (A), which gives
mes jXp \ p(f)J » mes \Kp \ Qpl > e0x„p\
Thus Theorem 8.1 is proved. Let us introduce the following class of func¬
tions, which are used in the investigation of solutions of the second boundary
value problem.
The class $2(QT\jr', M, y, r, 8, «) consists of elements u{x, t) of class
U rM, y, r, s, k) for which inequalities (8.1) are fulfilled for all k
satisfying the conditions
k ^ max w(x, t) — 6 h k max •w(x, <), (8.8)
<?(p, t)fl<?r <3(p. t)flr'
and the inequalities of type (7.1)
max ||®>*>(x. /)||2 <|| •**>(*. /0) ||2 „
'^Wna^" 2, ATpflQ
+ y[(0iP)'2 || «,<*> II’ Q{p x){]Qt + 0 (ft. p. T)] (8.9)
for all k satisfying the conditions
k max w(x, t) — 6.
QU>.
(8.10)
12<S II. AUXILIARY PROPOSITIONS
In regard to T' « Sy U we assume the following. Suppose S' is the
projection of SJ, onto SQ, Qfl = JKpf]Q, and Kp is a ball with center at an arbi¬
trary point *0 € S' and with radius p not exceeding some fixed number p^ and
the distance from xQ to 5\5'. We require that there exist positive constants
/3j and <?2 such 'hat
1) mesOp> $2mesKp
and
2) for any function u(x) from IT'CO^) the inequalities of form (5.5)
(f-k) mes Au„<^er|'55^ J IO** (8.11)
p ^ p
are valid for arbitrary numbers 2 and k, I > k. As was indicated in §5, inequal¬
ities (8.11) hold, for example, for the cases when Qfi is a convex domain. It is
not difficult to show that properties 1) and 2) are fulfilled for domains satisfying
the conditions that are usually imposed in the proof of embedding theorems ( see
[113a]). For functions of class %2(Qy |J ^ • • • ) we have
Theorem 8.2* Suppose a function u{x, t) belongs to U F' ,M, y, r, S, k)
and satisfies on TJ a Holder condition with exponent e > 0. Suppose,
further, that conditions 1) and 2) formulated above are fulfilled for S£. Then
uix, t) is Holder continuous in <? y U F ' and for any domain Q' C Q j, separated
from r j-\r ' by a positive distance d the norm |u|q> is estimated from above by
a constant c determined only by the parameters M, y, r, 8, K of class S2, d, c,
the norm |a(*, 0)|pQ' and the constants p0, /3j and from conditions 1) and
2). The quantity a> 0 depends on the same quantities as c, except for d.
The proof of this theorem is carried out in the same way as the proof of The¬
orems 7.1 and 8*1. For boundary cylinders of the first type the arguments coin¬
cide with those just described in the proof of Theorem 8.1. For boundary cylin¬
ders of the second type the arguments are analogous to those given for interior
cylinders in the proof of Theorem 7.1, it being only necessary to take instead of
balls and cylinders their intersections with Q and Qj., respectively, and in
this connection to replace mesKp= Knpn *n Lemmas 7.1 and 7.3 hy mesfl^. The
analytic inequalities on which this substitution is predicated are inequalities
(3.7) and (5.5). The first of them was applied above in the proof of Lemma 6*1,
and in the proof of Lemma 7.2 following from it, to the functions from
o
PjWp, r)) (see (6.16)). In the present case for boundary cylinders the functions
«<*>£ do not vanish on all of the lateral surface of the domain Qip, t) f| Qf. In
§8. THE FUNCTION CLASSES •••) AND S2(Qj-Ur'. ”• ) 127
view of this, instead of inequality (3-7) one should use the analogous inequality
(3.9), which is valid fot any function from V^iQj). Since by virtue of condition 1)
we have (eP2)n/2 mes-1n^ < for the domains fix (<Q, tQ + Op2)
considered in Lemma 7.2 it follows that for x («Q, + Op2) one can consider
the constant c in (3.9) *o be independent of p.
As to inequality (5.5), it was applied to those levels k for which
mes(K^\*^ > bp", b> 0 (see (7.20)), and was somewhat simplified to the ine¬
quality
(/ —*)mes/l|,p<-|'p J \ux\dx. (8-12)
Ak, »^At.p
In the present case one should use (8.11) instead of (5.5), and one should also
use (8.11) for those levels k for which mes(0^\ A^ ) ;> bmesQfi. This, by
virtue of condition 1), gives an inequality of form (8.12). In other respects the
proof of Theorem 8-2 is carried out in tbe same way as the proof of Theorems 7.1
and 8.1*
Let us make one more remark that will be useful in the sequel.
Remark 8.1. The membership of a function u(x, t) in this or that class
follows from the function uix, t) satisfying inequalities (7.5) with the functions
£(*, l) indicated there for such distributions of cylinders Qip, r) and such levels
k as are required in the definition of the class 2Sj. In this connection, if
Qip, r) intersects with T j, then all of the integrals in (7.5) are considered to be
extended onto the intersections of the domains indicated in (7.5) with Qj, where¬
as the function £(x, t) as before must vanish only on the lateral surface and
lower base of Qip, r) (but not on Qip, r) p] Qj\). It is easy to see that from ine¬
qualities (7-5) follow the inequalities entering into the definition of the different
classes 9^-
Lemma 8-1. Suppose
#(*. vrai max| «| M,
Qt
and the functions w(x, t) = + uix, t) satisfy inequalities (7.5) for any Qip, r) not
intersecting with T'j\V under arbitrary values of k satisfying the inequality
k > vraimaxpf ui - 8. In this connection it is assumed that all of the inte¬
grals in (7.5) are extended only onto the intersections of the domains indicated in
(7*5) with Qj, while the numbers k > 0, and r and q are subject to restrictions (7.3). Sup¬
pose, further, that F ' satisfies the conditions of Theorem 8.2 and for any Kp
128
II. AUXILIARY PROPOSITIONS
J u\{x. 0)djc<cp*_2+2*, e > 0. (8.13)
*pnro
Then for an arbitrary cylinder Qp ■=*■ Kp x + p2) separated from rT\r’
by a positive distance d the estimate
J «2rfjo«<CiP*+*1 (8.14)
<?pn°r
is valid, where the constants Cj and a > 0 are determined by it, M, S, the numbers
yj, v, g, r and k from (7.5), c and e from. (8.13) an-d the properties of T', with
c j depending also on d.
Inequality (8.14) is proved in the same way as tbe analogous inequality (7.33),
it being only necessary, for cylinders intersecting the plane It *» 0(, to use in¬
stead of (7.2) tbe inequalities (7.5) directly with the function £ = £(x) equal to
zero outside of a ball Kp concentric with the ball Kp.
§9. THE FUNCTION CLASSES 8^1
In the investigation of quasi-linear equations of general nondivergence form,
and also in the investigation of systems of equations, we encounter the need for
further extensions of the classes S2.
We will say that a vector function u(x, t) = (u1(x, «),•••, uN(x, t)), defined
and measurable on Qj, belongs to class ;Wj, Sj, S,, Sj, y, r, S, k) if one
can construct lor it /Vj functions 0*(ti*, • • • , a^), • • • , l(u*, • • • , u^) that
are continuous and continuously differentiable in the domain
*)! < vrai max | u (x. /)||
and such that the functions w\x, t) = <f>Kul(x, t), • • ■ , u^(x, t)), ?=!,••• , Nj,
possess the following properties.
1) vT2iimaxQ^\wKx, 0| <M\ (£j»).
2) For any cylinder Q = K2p* [*0» *o + ^ C ao4* my *i € ^0’ *0 +
there exists an index p for which
§9- THE FUNCTION CLASSES B^1
129
mes Ix g K„'. wp(x, *,) < vrai max in” (*, t)
<?
—fijosc (*'<*, t); Q} I >(1 -63)Hnp”. (9.2)
where K p is a ball concentric with the ball Kjp’ an<^ the Sj, i - 1, 2, 3» are
fixed positive numbers of which 3_, and 8^ are less than unity.
3) For each of the functions u> = w\x, t), /=!,•■•, N^, inequalities (7.1)
and (7-2) are fulfilled lot arbitrary Qip, r) C Qf and o>j, o2 € (0, l) and k satis¬
fying inequality (7.4). In the same way as in §7, pik, p, r) = ^ *Tm^/qAkp(t)dt,
with Ak pU) being the set of points x from Kp at which uix, t) > k; M, y, q, r,
8, K are fixed positive numbers, with q and r satisfying conditions (7-3). For
q = oo we -have
H{/j, p, t) = mes (/£[/0. mes A„, „(/) > 0).
In the same way as in §7, one can assume that inequalities (7.1), (7.2) are
fulfilled only for p < pQ, r < rQ, where pQ and rQ are fixed numbers. It is not
difficult to see that the function class SS^Qj, 'W, y, r, S, k) is the class
®2 *(^r uv sv s2' sy y*r’ 5’ ^ for which “ “» 02(“) “ ~ «» *) =
u(x, t), u>2(x, t) = — u(x, t), Nl = 2, ,lf, = M, Sj = 1, S2 = ^3 “ i/2, while y, S,
r and k ate die same as for fi^Qf M, y, r, 8, k).
We will prove that the functions u(*, t) of classes Bj1 belong to Ha,a^‘KQj).
Theorem 9.1. Suppose u(*, t) = (b1, - • • , u^) is an arbitrary (unction of
class Bj HQft ^1’ &x< $2' sy y* r> ** and Qpq ~ ^Po- ** ° cYlinder
belonging to Qr Then for any cylinder Qp, coaxial with Qaad having a com¬
mon vertex with it, the oscillation of u‘(x, t) in Q p is estimated as follows:
whete ost {«'; QpJ <cp~a$P, i=l N, (9-3)
4° | „aa/v,+i „ nx )
—£ max | 2
the number 9 = #((■/+ l)/2) is taken from Lemma 7.1, wAi/e the number s is
defined beloui in Lemma 9.2. The number a for 8 = no does not depend on if j,
and for 8 < » is a decreasing function of M^8~
The proof of this theorem, like that of Theorem 7.1, is based on Lemmas
J30 II. AUXILIARY PROPOSITIONS
7.1-7.4. Lemma 7.J is valid for any function wkx, t), I ■ 1, • • • , Ny since the
wl satisfy inequalities (7.1). Let us fix in Lemma (7.1) the number £ = + l)/2
and the values ot 6 - $(£) and b = 4(f) corresponding to this we will consider
the standard cylinders = ||* - *0| < p, < t < Ig + 0p2\* For each of the
wKx, t), I * 1, • • • , Ny Lemma 7.2 will also hold, since the v} satisfy inequal¬
ities (7.2). The situation is different with Lemma 7.3. Its assertions can be in¬
correct for any w^, since we have not assumed in this case that together with tv1'
the function - wl also satisfies inequalities (7.2), and therefore to cannot assume
that inequality (7.19) (or (7.19')) is fulfilled, which was the starting point for the
derivation of estimate (7.17). Instead of (7.19) inequality (9.2) is known to us,
but not for all w^\ only for k/ with the special index I = p. Namely, for w* with
this index 1 = p we have
Lemma 9.1. Consider an arbitrary function a(x, t) from My Sy
S2, Sy y, r, S, k) and the functions «/(*, t), I = 1, • • • , Ny corresponding to it.
With respect to my 0j > 0 it is possible to find an s = *(#j) > 0 such that for
any cylinder Q2p~ ^2px ^‘o’ *0 + C Oj. and tj = tQ + 36p2 and function
u>P(x, t) corresponding to it by condition 2) at least one of two inequalities holds:
either
(9-5)
W — OSC [wp; QSpj <2V
or
mes Qp (|x, — < 9iPn+2. <9‘6>
where - vraimawhile Qp - K^ x [tQ + 36(?, + 4&p2] is a cylinder
that is coaxial with Q2p< The number s is determined by the parameters of classes
Bj and for 8 - ™ does not depend on My
The proof of this lemma is the same as for Lemma 7.3, it being only necessary
to use instead of inequality (7.19) the inequality following from (9.2)
me® (^u~l- (9*7)
Let us turn to the last of the lemmas, namely Lemma 7.4, which is a direct
consequence of Lemmas 7.1-7-3* It takes the following form.
Lemma 9.2. Suppose u(x, t) is an arbitrary element of 53^((?p My Sj, Sj,
Sy y, r, S, k) and the wH.x, t), I = 1, * • • , Ny are the functions corresponding to
uix, t). For any standard coaxial cylinders Qp/i °nd Q2pC Qj having a com¬
mon vertex, and function wP(x, t) distinguished by condition 2) for the cylinder
Q2p and tj = tg + 30p2, there is valid at least one of two inequalities: either
§9. THE FUNCTION CLASSES 8^1
131
osciiop; Qp/2^ ^ 2spnK/2,
or ,
OSC |l -_~joSc{u>p, Q2pi,
where s is the number s(0j) from Lemma 9.1, wAi/e the number 0^ is taken from
Lemma 7.2. Tie numbers 6^ and s{6j) ore determined only by the parameters of
class Bj1.
From this lemma and Lemma 5-9 follows Theorem 9.1. It will be used in
proving the smoothness of the derivatives ux. of solutions u(x, t) of general
quasi-linear equations and the smoothness of solutions (and their derivatives)
of die systems of linear and quasi-linear equations considered in the last chapter,
at interior points of the domain of definition. For an investigation of the smooth¬
ness of all of these functions in the vicinity of the boundary of Qj. we need sub¬
classes of class 8j *, analogous to the subclasses U T', • • • ) and
s2(<?r Ur••• ) of class S2(@p ••• )• introduced in the preceding section.
Ve will soy that u(a, t) belongs to class B^H^y (J TMy, Sj, ••• , Sg,
y, r> S, k), if uU, t) € 8^K(?jr» My Sj, 82, Sy y, r, S, k) and if the functions
w\x, £j, I** 1, • • • , /Vp constructed with respect to it possess in addition the
following properties.
4) They satisfy inequalities (8.1) for all Q(p, r) and k indicated under (8.1).
5) For any boundary cylinder Q = Kpx («0, tQ + r) with axis on Sj
(Sj. ~ V fj Sj.) there exists an index p such that
osc Q fl Qt) > 64 max osc la'; Q n Qr], (9-8)
and if '-1 "
osc [wp; QflQr) >6S max osc \w': Q fl T'j, (9-9)
then H N'
vrai max wp < vrai max w* — d6osc [w*-, Q flOr). (9.10)
where 8^ > 0, Sj > 1, 8^ G (0, 1).
For functions of this class we have
Theorem 9.2. Suppose the projection of Sj onto SQ satisfies condition (A).
If u{*, t) € (J r ’, Mj, Sj, • • • , Sg, y, r, S, k) and the functions
w\x, t), I - 1, ••• , Nv corresponding to it belong to H(’e^2(T'), then u(», t) G
132
II. AUXILIARY PROPOSITIONS
yr'), for any domain Q' CQj. separated from I"\r' by a positive distance
d the norm |u|q‘ is estimated from above by a constant depending only on the para¬
meters of class *, ( the norm |u|^«, the constants aQ and dQ from condition
(A) and the distance d. The exponent a is determined by the same quantities,
with the exception of d; for 8 = «> it does not depend on My
This theorem is proved according to the same plan as Theorem 9.1. The
necessary changes in the arguments concerning boundary cylinders are analogous
to the changes introduced in the proof of Theorem 8.1 in comparison with Theorem
7.1. Theorems 9.1 and 9.2 are generalizations of Theorems 7.1 and 8.1, in which
were considered elementary variants of classes B^1 with /Vj - 2, <£j = u and
tj>2 = - u. An analogous generalization can be made for Theorem 8.2 by introduc¬
ing classes of type 1(?j.(jr'> §i> S2. Sy r, 8, k).
Let us formulate one more lemma, to be used in Chapters VI and VII.
Lemma 9.3. Suppose u(*, t) = *)»•••» uN(x, t)) is an arbitrary vector
function from wtth min^j ... ^ vrai miag^u', max^j ... y
vrai max Qj,tt‘ < M. Then the functions N
< (n)“<(a’ uN) - ■1 irt+1 (irff)*-
iml
1=1, N,
for X> 4N possess properties 1), 2) and 5) from the definition of classes
%Nzl(QT U r Mv 8V • • ■ , S6, y, r, 8, k) with Ml = A + N, = 2JV and certain
••• , 5^. For A *= 10/V these parameters have the following values: <5j -
8N/(M - m), S2 = 1/4, S3 = 1/2, S4 - SN/W - m), 5, = 3, S6 = 1/24, Mj = UN, /V, =2/V.
A proof of this proposition is available in the book [65q] (see Lemmas 8.1 and
8.4 of Chapter II). The only difference consists in the fact that in [65q] balls
Kip are considered instead of cylinders Q; but it is nonessential.
CHAPTER HI
LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
The present chapter and Chapter IV are devoted to linear equations and dove
all to the question of solvability of their basic boundary value problems. Equations
with discontinuous coefficients are studied in Chapter III. This is explained by
the fact that the existence theorems are most easily established in classes of
generalized solutions having only the derivatives u% of first order, where there
is no need to assume that the coefficients are smooth. Thus the case of discon¬
tinuous coefficients is studied first, with the coefficients being assumed as bad
as permitted by die uniqueness theorem (see §3 of Chapter I). The properties of
the coefficients and free terms are characterized by their membership in die
spaces L y r(Qf) with this or that q and r.
The solvability of boundary value problems in V2(Qf) *s established by
means of Galerkin’s method, and it is then shown that any generalized solution
from V2(Qt) belongs to
Under a somewhat different form of the conditions on part of the coefficients
and free terms of the equations, generalized solutions from V2(Q-p) a*so belong
to Wl^(QT).
The results of §§1—5 are easily carried over to the so-called strongly para¬
bolic systems (defined in [***]), a special case of which is one parabolic equa¬
tion of high order.
After proving the solvability of the boundary value problems in
(§§1—5) we investigate the progressive increase in die smoothness of the solu¬
tions as the properties of the known functions in the problems are improved,
giving interior estimates for generalized solutions of the equations and estimates
right up to the boundary fot generalized solutions of the boundary value problems
(§§6—12). The dependences proved in these sections are exact (see, for example,
§3 of Chapter I). In §12 the results of Chapter III are linked with the results of
Chapter IV, which are derived without any reference to Chapter III. Sections 14—
18 are devoted to a description of the other known methods for proving solvability
133
134
IB. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
of the basic boundary value problems. In §13 it is shown that diffraction problems
(in which there are in die general case several heterogeneous media) are a special
case of the problems on determining generalized solutions from con¬
sidered in the present chapter.
§1. STATEMENT OF THE PROBLEM. GENERALIZED SOLUTIONS
Tbe present chapter is devoted to a study of the linear equations
ssu, — aMu — — /, (1-1)
where
“ ~STt ^ “xi +°' (x- 0 itx~a(x.t)u,
with discontinuous and, generally speaking, nondifferentiable and unbounded
coefficients satisfying the condition of uniform parabolicity, namely,
n a
(x- *)6/6/< f* 2 I?. v, |i = const >0. (1.2)
i-1 J * i-1
In §3 of Chapter I we discussed the question of admissible extensions of die
concept of a solution for these equations and their boundary value problems, and
we determined what restrictions must be imposed on the coefficients and free
terms of the equations in order that one may begin to construct a theory of these
generalized solutions that preserves die basic features iaherent in parabolic
equations with smooth coefficients and smooth solutions. We will show here that
it is not necessary to impose any further restrictions for this purpose.
The most extensive of these restrictions are the following: the coefficients
a(., b., a are measurable functions with finite norms
(1.3)
1) The assertions of certain theorems remain valid under die condition that the
summability powers of the bj(x, t) are equal to die values of the ends of admissible in¬
tervals. Ve indicate these cases in die Remarks. Their proof does not require any new
methods, but it violates the uniformity of the calculations and dependences. This viola*
tion is connected with an essential point: for these extreme cases die dependences of
the properties of a solution on the 6* have a different form.
2-?; 2*?;
§1. STATEMENT OF THE PROBLEM
135
(1.4)
in which q and r are arbitrary numbers satisfying tbe conditions
?€(|. oo]- '•611. o°) for lt> 2,
?€n< °°]> r€i1-2) lot n~i
(the case r = ~, q-= n/2 + 2(, ( > 0, in (1.3) is contained in (1.4) since L ^ n C
^n/2 +f n/2f+1 an<* t*le indices q = n/2 + e, r = n/2f + 1 satisfy (1.4)), while the
free terms have finite norms
*hQM /£/?<**<«) <(** II f H»,, tv Qr = ^2 »-5)
tO- /-I
in which
r,
.JL==l + n
2?,
4 ’
(1.6)
2]‘ r>^11<21 for n-3’
2], r,6U. 2) for n = 2.
?i€U. 2], neu. 4/3] for s= 1.
Remark 1.1. The cases considered in [65n] are contained here since ^
with q > n/2 is embedded for n > 2 in Lq f, ? = n/2 + e, r = n/2( + 1, e = q -
n/2 > 0, and ?, r satisfy conditions (1.4). It can be assumed in conditions (1.3)
that q and r are different for different coefficients. But we will consider them to
be equal. The case of different q and r is treated analogously and the final re¬
sults are the same. In addition, since the functions from r(Qf) belong to
Lj r>(Qf) with q'<q and r'<r, in conditions (1.4) and (1.6) one could write
instead of the equalities
1.
n-f-4
7+£<«-
u-(-4
2
r i ?i
the inequalities
r, q,
and not place any upper bounds on the choice of qj and /j. But for die sake of
convenience and uniformity in the writing of all of the relations it is more con¬
venient for us to assume that one of the parameters (q and r) is arbitrarily chosen
from the indicated interval, while the other is determined from the equalities of
136
ID. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
(1.4) and (1.6).
Under fulfilment of conditions (1.3)—(1.6), as will be proved in §§4 and 5, tbe
basic boundary value problems for (1.1) are uniquely solvable in tbe class P'j(Qj-)-
Furthermore, under these conditions any solution from of equation (1.1)
is found to possess a certain additional (in comparison with arbitrary elements of
V2((?r)) smoothness in t; namely, it belongs to
We begin with definitions of the generalized solutions of equations (1.1) and
of the first boundary value problem for them that belong to V2(Qf), Y\,0(Qt) and
K2’,/j(*?r>. accordance with the basic purpose described in §3 of Chapter I, we
will mean by a generalized solution from V^iQf) of equation (1.1) a function
u(x, t) belonging to anc* satisfying for almost all ty € [0, T\ tbe identity
/(/,; a, rflss J a(x, t{)i\(x, t{)<tx — j J ur\,dxdt
a 'to
4- J («, »1) + (f. H) 1 dt = 0 (1.7)
0
for all rf(x, l) from that are equal to zero for !»0. Here and below
we make use of the abridged notation
-2*1 (a. ti)= J [ (au “*, + 4- (6/u^ au)t]J dx,
Q
JT2(f. r,)= f (fl% + fi\)dx. (1.8)
2
By a generalized solution from V2(Qy) °f the problem
= “ISt**0’ «L-*W U'9)
O
we will mean a function u(x, t) from j-) satisfying for almost all tj from
[0, 71 the identity
/(/,; u. 10= J ♦hWtiCjc, 0)dx d-l°)
a
for all ij from IP|’*(9jp).
Let us verify that if conditions (1.3)—(1.6) are fulfilled and */>q € L^$),
then all of the integrals in these identities are finite for any functions a and if
from the indicated classes. This is so by virtue of the following estimates (see
§1. STATEMENT OF THE PROBLEM 137
inequalities (1.8) and (3-4) of Chapter II):
J | ataru, | dx dt < || n, ||2> II2 o? f l| «||- -
<?r II 1 IW. r. Qt r
d.ii)
JI I dx dt <|| IT II ttjch o, II11 11$. r- Q
Qj. II I Ilf. r, Qr T
<nly!!PII“xll!li<?7.hl<?r; d.i2)
J | am\ | dx dt < H a ||?> n Q || a ||- - Q || rj ||- - Q
<^2|«|„rhlQ?.; (i.i3)
J l/’ll^d«ll/ILrB<)rllnII,;>r;. *<011/11^. 0rhlPr (1.14)
^7 _
In all of these inequalities q and r are connected with q and r by the equalities
q = q/(q - 2), r = F/(F - 2), while gj and are connected with gj and by the
equalities q{ = g1A?1 - l). = <riAr, - l).
The remaining integrals in (1.7) and (1.10) also obviously exist and are finite
for any u and r) from the indicated classes. Thus the definitions given above are
correct. We note in addition that the values r = <*>, q = n/2 are admissible for
n > 3-
It is possible to define the generalized solutions from somewhat
differently. Namely, u(x, t) is a generalized solution from of equation
(1.1) if u(x, t) € P2«?r) and satisfies the identity
r
/0(a, i0sb— J ari,dx^-f- J [.S’, (a, ii) + ^j(7. r01rf/ = O (1.15)
Qt o
©
for all 7}{x, t) from that are equal to zero for f» 0 god *» T, and «{%, f) is a
generalized solution from V^Qf) Pro^^era (1-9) if u € VjiQf) and satisfies
die identity
/0(a. TJ)= J >h)(J£)»l(Jc. Q)dx (1.16)
Q
o
for all r)(x, t) from ^’^(^r) are equal to zero for t = 7".
Let us show that these definitions are equivalent to the ones given above.
The fact that the latter two follow from the first two is obvious, since, taking 17
138
III. UNEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
equal to zero in (1.7), (1.10) for t > T-e, e > 0, and the number tt > T - f, we
obtain (1.15), (1.16) for such T), and hence for all tj required in the latter defini¬
tions. Now suppose a is a generalized solution of equation (1.1) in the sense of
the second definition. In (1.15) we take 7) in the form <!>(*, ij, t), where
rj{x, t) is an arbitrary function from that is equal to zero for t = 0,
while 4>(t, tj, «) is equal to 1 for t < -1, to zero for t > tj, and to (tj - t)/c
for tj - f < t < We substitute this rf into (1.15) and pass to the limit as (->0.
It is easy to see that the second integral in (1.15) gives in the limit the last inte¬
gral in (1.7). And the first integral in (1.15) is representable in the forra
— J a (<t>T)( -4- 0,1]) dx At
Or
i *'
==— J u<J>r\, dx dt+ — j j ux\dxit. (1.17)
Qt, '■-* o
The function i(rU) - ffl uix, t)fj(x, t)dx is summable on [0, 71, so that (see §4
of Chapter II) there exists a sequence —> 0 for which the integrals
(l/fpj/j-f£ </>it)dt converge to for almost all Ij from 10, 71. The limit
of the integral ~fgt u&rjtdxdt will obviously be -uf\tdxdt, and hence in
the limit relation (1.15) goes over into relation (1.7).
Thus die equivalence of the two definitions of generalized solutions from
V2(Qt) of equations (1.1) is proved. The equivalence of the two definitions
given above of generalized solutions from of problem (1,9) is proved in
the same way.
For an investigation of the properties of generalized solutioas of equations
(1.1) it is convenient to introduce two additional classes of solutions: generalized
solutions from the space an<* ^ronl ***' sPace They are de¬
fined in the same way as the generalized solutions from an equation
(or problem), with the only difference that they are elements of or
or Vlj.’^(Cjr) respectively) and satisfy identity (I.7)(or
(1.10)) for all tj from [0, 71. Both of the definitions of generalized solutioas
from I30^2corresponding to the two definitions of generalized
solutions from 6*ven above are equivalent to each other.
It is clear that a generalized solution from equation
(1.1) or problem (1.9) is a generalized solution from ^(Pf) equation (1.1) or
problem (1.9). And, conversely, if u (x, t) is a generalized solution from V2 (Q-p)
$2. THE ENERGY INEQUALITY 139
(of an equation or problem) and if additionally it is known that u(x, I) is an ele¬
ment of P|’°(@7’)(l',2’^(07'))> A*11 i* will also be a generalized solution from
V\'\QTUV\AkQT» (of the equation or problem respectively).
§2. THE ENERGY INEQUALITY
We will show that for any generalized solution u (x, t) from of
problem (1.9) one can estimate l“l()y in terms of the known quantities in this
problem. For this purpose we will prove as a preliminary the following lemma.
O
Lemma 2.1. Suppose u € satisfies for almost all tj and t^
from [0, 71, including tj = 0, the inequalities
I J</*(*, t)dx^ ’ -+ J (a. «) + ^’2(f. «)]<«<0, (2.1)
in which the functions a., bp a, fi and f satisfy conditions (1.2)—(1.6). Then
M«r< «[!•(*. + Qt]. (2.2)
with the constant c depending only on n, v, /i, ftl and q from (1.2) and (1.3).
From (2.1) by virtue of (1.2) we have
h
1 f a2 {x, t) dx I -f“ v f f a\ dx di
2 i ii a
< f f U-«’+(4£a?+42^+i°i>“5
<T “ L ‘ ‘
+ \fflx,\ + \ fa\\dx dt,
from which, using conditions (1.3)—(1.6) and inequalities (1.6) and (3.4) of
Chapter II, we obtain (in a manner analogous to the derivation of (1.11)-(1.14))
die inequalities
*}<**[ t +v J fu*dxdt< 2J J(^«* + |/#«X4|
140
III. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
in which
/ i
From these inequalities it follows that
min (1; v) |«|2Q( ^<||«(x. /,) ||* n
-f*!*(*«• *2)I“lL . + <s?’(*i* *i)|“k .•
x<l« ‘1 x‘l» *3
where
and
<*> — 2 (11 * ll2l 0<i_ 4 + P H /lift. r„ <?,„ ,,)•
If <2^ *s ^ess than ra*n v^> then with the use (2.3) it is possible to
estimate from above in terms of (]«(*, ()|12>q and known quantities. In
order to do this, we divide the interval [0, T] into a finite number of intervals
t®, ‘‘s=n°f such small length that
I* (^-t. h) < -y s J min (1; v), (2.4)
with the division points I^ having been selected so that they can be taken as
limits of integration in (2.1) (we denote the set of such admissible by §). This
is possible for r < e», with the number of divisions not exceeding some known
quantity. In fact,
Jj M' (<*-,. t„) =* (2p=)' || 3||'_ Cr,
and therefore, if a partitioning is selected so that all <*(<£- j, ^), except pos¬
sibly the last, are not less than Vj/4, then
(*- i)(^-)r<(Wl|5!K.r. ?r.
and consequently the number of divisions s will be bounded from above:
»' < 1 + II 3> 1|,, ,, (2.5)
Since it is necessary that no j, tj) be greater than v^/2, it is pos¬
sible, without increasing the number of divisions, to place the ends t^ in S.
§2. THE ENERGY INEQUALITY
141
For each of these divisions we have by virtue of (2.3) <he inequalities
■J * '"20-T-Br K‘k-1‘ 'k> l“lo.
o <||«(JC. #*_i)||2.Q-hy(<*_!. **)|BI0, ,
<II“(*. <*-.)& q+xI“Iq( (**-.. '*)•
from which follows assertion (2.2) of Lemma 2.1.
Remark 2.1. From the given derivation it is seen that if
ty?\\®\\v, r, Qr< min I1’ V)’ (2-6)
then inequality (2.2) also holds for r => <», since in this case it is deduced directly
from (2.3) without a partitioning of the interval [0, T],
Now suppose u(x, t) is a generalized solution from V^(Qj.) of problem
(1.9). ®e take as i/Oe, t) in (1.15) the function
i
r^(x, 0 = 4 T>dx• (2 7)
t-h
o, ,
where rj(x, t) is an arbitrary element of Wj (^-A £**at *s e<P»al to zero for
t>T — h and for t < 0, and we transform the first term in it in the following
manner:
— J arjj, dxdt = — J u^\tdxdt=: J uktr\dxdt. (2.8)
flf Gr Or
Here we have used the relation
T T-h
J u(t)\(t)dt~ J HjWnW dt. (2.9)
0 o
valid for any square-summable functions u(t) and 7}(t) on [-A, 71, one of which
is equal to zeto on die intervals [~h, 0] and [T — h, 71, and the notation
t+h
O-t J »<*• t)*. (2 10)
t
Equality (2.9) is the result of interchanging the order of integration with re¬
spect to t and r. In all other terms of (1.16) we also transfer the averaging ()j
from rj to its coefficient, taking into account the pennutability of this averaging
with differentiation with respect to x. This gives the identity
J [uh!n+(alju^)+atu+ft)hryi+(l>iuJ<l +««+/)* r\\dxdt^Q. (2.11)
Qr-h
0
for
t<0.
kt
for
0 <<<{,
1
for
1
“7
kih-t)
for
0
for
142 IU. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
This identity is actually valid for a class of functions ij(x, t) that is more exten¬
sive than the class just considered; namely, it is valid for any function fj that is
equal to zero for t> -h) and is equal to some function rj(x, t) from P|> °(Q,,) for
o © 1
t e [0, *j]. Indeed, the set is dense m *e space j). Thus for any
*?(*, *) from T* *ere >s a sequence of functions t) from r^*
* °i n *
that is sttongly convergent to it for m —» on in the norm of (@-4 j-)- We de¬
note by XfrU) the continuous piecewise-linear functions
X*« =
Identity (2.11) is established for f)m kix, t) - 7im(x, t) X^U) when tj < T - 4. It
is easy to see that one can pass to the limit in it as m and k —> o° and thereby
prove the identity
J [««n+{atJuXj -f- at a + f,)„nxi
%
+ (V*, + aa + /)»Tl] dxdt = Q (2.12)
O.
for any function t)(x, t) from when tj < T — h.
In (2.12) we take J}(x, t) = u^ix, t) and represent the first term in the form
J aMuhdxdt = ~ J u\(x, /)<**[
Qt, “
after which we let h tend to zero. This gives
yj «Hx, tO + ^tO. “)]dt — 0. (2.13)
Indeed, by virtue of our assumptions (1.2)—(1.6) on die coefficients and free terms
o
and by virtue of the membership of u in V \ follows that the functions
aijiix, + a.a + f. are elements of L2(Qf), the functions au and 6^u are ele¬
ments of ^2q/{q*V) 2r/(r+i)(C*r)i while the function / is by condition an element
of Lq r {Qf). This follows from conditions (1.5), estimates (1.11)—(1.14) and
§2. THE ENERGY INEQUALITY
143
Theorem 1.1 of Chapter II. By Lemma 4.7 of Chapter II the averagings of functions
from some space L^ r(Qf) converge to them in the norm of this space. But for
elements from V2’°(Qf) their averagings converge to them in the norm of
and in any of the norms of with q and r subject to con¬
ditions (3-3) of Chapter II, including die norm of the space ^2j/(v-l),a-/(r-l./@r)>
which is conjugate to the space ^/(9*-X),2r/(r+l/^r^ w‘*k ? r ta^cen fr°m
conditions (1.3), (1.4). Therefore in the integrals
/ (avuxj -h«/*+f,)h ^ dx dt, j {btttXt+«#+/)A% dx dt (2.14)
% %
we can pass to the limit as h —* 0 for any rj from ok***® A*
limit ‘
J {ai iu*j + atu+/() dt.
9,1 (2.15)
J (b1Ux1 -\-au-\- f}i\dxdt.
%
This is true, in particular, for t] = u. Finally, the integrals Jjju^(x, t) dx converge
to JqU^(x, 0dx for all t 6 [0, T — A] since u(x, t) depends continuously on t
in the norm of L^iQ). Thus (2.13) is proved.
From (2.13) and Lemma 2.1 we get
Theorem 2.1. For any generalized solution u from V2’°(Qj.) of problem (1.9)
the inequality
18 Iqj. ^ c [ II 'I’olla, q + II* II2, qj. +• II /1!^, r,, <jr] (2.16)
is valid if assumptions (1.2)-(1.6) are fulfilled with respect to equation (1.1). The
constant c in it depends only on n, v, fi, and q from (1.2) and (1.3).
Remark 2.2. Estimate (2.16) also holds for r = ■» if it is known in addition
that inequality (2.6) is valid.
Let us now assume that u(x, t) is a generalized solutioa from V\'°(Qf) °f
equation (1.1), so that in comparison with tbe case just considered it does not
o
vanish on Sj.. It is obvious that for it identity (2.12) is valid for all rj from V2'°(Qj.).
We put rj(x, t) = u^ix, t) £2(x, t), in it, where £(*> <) is an arbitrary continuous
nonnegative function from '*((^)> and, transforming the first term as above, we
pass to the limit as h —r 0. This gives the relation
144 III. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
4- f (X, t)g(x, 0 dx I
z df
4- J [-2*1 (#. «??) + JSr2 (? . bS2) - (a, flK,)] dt = 0. (2.17)
0
which is similar to (2.13)* From it follows die estimate
“S|Qr<c[ll“(^. mix. 0)Q-+-1|h2 Qr
+ II&ll„. V Qr + 1! <* k Or + II“ V^ETeTTIk Or)• (2.1®
in which die constant c depends only on n, v, 11, Mi and q from (1.2) and (1.3).
It is derived from (2.17) in almost the same way as die energy equality (2.2) was
derived {com (2.1). The terms in which £ is not differentiated differ from the
corresponding teras of (2.1) only by the factor and are estimated ic the same
way as above. The remaining terms (not represented in (2.1)) have the form
A= 11 -f a,a 4- ft) a2gu( — 0%] dx dt
%
and are estimated by means of Cauchy’s inequality as follows:
|/ll<2(li||8rC||s.0(i
The quantity
as is seen from (l.ll)f does not exceed fij $\uC\Qt^* t*lus (2.19) implies
|/,|<e(|i IKt||2,0( + M|a£|0/i+||ft|l2,Q()2
-M"1 || <r Ig, Qfi 4- II * 1, Q(i- (2.20)
This estimate in combination with the estimate of the other (basic) terms
carried out above gives inequality (2.18).
§}. UNIQUENESS THEOREMS
145
§3- UNIQUENESS THEOREMS
Uniqueness theorems for solutions of boundary value problems and of tbe
Cauchy problem for the parabolic equations (1.1) in the space C2,1(Qj) fl CiQ-p)
were proved in §2 of Chapter I. This was done on the basis of the so-called
"maximum principle,” which requires tbe existence of those derivatives of the
solution that enter into the equation. We will prove here that a uniqueness theorem
for problem (1.9) is preserved for solutions not having the derivatives d/dt and
d2 /dXjdXj.
The first theorems of this type were established by one of die authors of the
present book (see [<5c-«j 124*]). The uniqueness of generalized solutions of
boundary value problems was proved in them for second order equations of hyper¬
bolic and parabolic types in the spaces and respectively. We
will give here the original proof for the case of parabolic equations (Theorem 3-2).
In the process of carrying it out an inequality giving an estimate of the norm of the
solution is established that is weaker than the energy inequality (such inequalities,
following Garding, have been called "dual”). But it is necessary in this connec¬
tion to assume that the coefficients a.- and b. have derivatives a.^ and ..
Ladyzenskaja, Lions and Prodi have since given other methods of proof, in which
the only requirement imposed on the coefficients of equation (1.1) is that they be
bounded. Theorem 3-3 is a generalization of these theorems to the case of un¬
bounded coefficients. A desire to include the more general case, when the coef¬
ficients a-, bp a belong to classes to generalized solutions
from a somewhat narrower class of solutions than the solutions from
in which nevertheless, as we will show below, one has unique solv¬
ability of initial-boundary value problems.
Uniqueness in the class ^*s a simple consequence of Theorem 2.1,
which was proved in the preceding section. From Theorems 2.1 and 4.2, the latter
of which will be proved below, there follows a stronger assertion:
Theorem 3.1. // the coefficients of equation (1.1) satisfy conditions (1.3) and
(1.4), then the first boundary value problem for (I. I) cannot have two distinct
generalized solutions from
Indeed , tbe difference u (x, t) of two such solutions would be a generalized
solution from (Qj) of die homogeneous problem (1.9) (i.e. problem (1.9) with
00 = / = f s 0). By virtue of Theorem 4.2 the function u(x, t) will be an element
of V\'\Qr), and hence inequality (2.18) is valid for it, so that »s0.
146
III. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
In the classes of bounded generalized solutions uniqueness is preserved under
broader assumptions concerning the coefficients &[(%, t). One of the variants of
theorems of this type is Theorem 3.4. For its proof we employ the classical method
proposed by Holmgren at the beginning of die century, which makes use of one’s
ability to solve the adjoint problem in a class of sufficiently well-behaved func¬
tions (we note that die adjoint of an initial-boundaryvalue problem is essentially
a problem of the same type). We will also mention uniqueness theorems for gen¬
eralized solutions of boundary value problems from the space Qf) I6*4*fl
l24f]. These theorems are distinguished by die fact that existence theorems are
readily deduced consequences of them, just as in linear algebra uniqueness implies
solvability of a linear algebraic system for any right sides. We will discuss them
through the example of problem (14.2) in §14 of the present chapter. These
theorems are valid if die coefficients of die equations possess a certain smooth¬
ness greater than that used in the proof of unique solvability in the space
But in return solvability is also proved in a class of better-behaved
functions. This conforms to a general tendency: the better the coefficients of an
equation are, die better is any solution of the equation corresponding to suffi¬
ciently smooth free terms and die worse are the solutions admissible for such
equations, the deterioration being effected at the expense of extending tbe class
of free terms and functions determining die initial and boundary conditions.
Let us proceed to a proof of die theorems.
Theorem 3-2. Suppose the coefficients of equation (1.1) satisfy the conditions
(1.2),
max | a, — bt, (3.1)
and
max lltf“‘azrll (3,2)
00xl o
while the free terms satisfy the conditions
*€MQr>. AQt). (3.3)
in which q = n for n> 3, q > n for n = 2 and q - 2 for n •> 1. Then for any gen¬
eralized solution u(x, t) from V\’°(Qr) of problem (1.9) with ^q(x) = i(x)/dxi,
i/r (x) - (i/jj, • • •, <l>n) € L2iO), the estimate (the inequality dual to the energy in¬
equality)
§3. UNIQUENESS THEOREMS
147
max 4|lt(*.0l|a-f-a#C,feQt
<*&'' + J (3‘4)
holds for € [0, v/4Cj], in which £(*, t) => fgit(x, £)d£, Cj = lOnjjj + 4 +
4c2(l + fi2) and the constant c is taken from inequality (2.18) of Chapter n. In
particular, for i(iQ(*) m »L=q = 0 and f = /= 0 the solution u(x, t) s 0, i.e. a
uniqueness theorem holds for generalized solutions from of problem
(1.9).
By a generalized solution from ^7 °f Pr0^^ern (1.9), in accordance with
the general concept of a generalization of tbe notion of a solution described in §3
of Chapter I, it is necessary to understand a function u(x, t) from
satisfying the integral identity (1.16) or the identity
J {-uti( + [«, A, + (ai -*/)“ + //] \
<>t
+(a — 6uj) «TH- /T)} dxdt — [ ito (*,) t)(x, 0)dx — 0 (3.5)
£
for any function rj (x, i) from that is equal to zero for t = T if tft^x) €
LjW). If on the other hand ^g(») is a generalized function of the form i/>ix. with
ifi-(x) 6 then in identity (3-5) we consider die term — 0)<f* to
be equal to 0)<£t and we assume that 7/Oe, f) has the derivatives
r)x.t from L2(Qt). Under tbe conditions of Theorem 3.2 we assume that u sat¬
isfies an integral identity in the form of (3.5).
For a proof of (3.4) we set 17U, t) equal to zero for t > tj and to
~fti u^x’ T^T ^or e ® t0, tj], where is an arbitrary number from (0, T], and
write the resultant equality in die form
J W - [«, ?\tXj+(«, - *,) n« - /1 j s
~~(a — but) ’l/'H + f*i} dx dt J I|>,t| (x. 0) dx = 0. (3.6)
0
We transform the remaining terms of this equality, bearing in mind that
rj(x, fj) = 0, and then pass from the equality to the corresponding inequality:
148
III. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
J J <*,,(•*. 0) T)^ (x, 0)t}x (x.0)dx+ j r\)dxdt
nl da„ Q"
~2ir ■+ (ai - bi) vu, ~ f,\
Qi,
■+■(“ — blti) tpi, — fil] dx dt —
— J °)dx < J [jMlf Tr''"+
1 ■ Qh
+ TP + n'+ — blxyr?}dxdt
+ fll/IU aII*1 \\ja_ ^t + jlhAx. 0)H2,qH“4 (3-7)
0 0+2 * d-2 '
2
Further, we estimate the term containing i? , using Holder’s inequality and the
embedding theorem (2.18) of Chapter II, as follows:
J («- tf rf* < ||«- ig, B ih n\ <^ihj|Q.
a «-s'
and analogously
p
J ll/!ljl alhlU a^<?ll/llV 2,0 + ‘,lliJll0,-
„•» T75-0 «+7•A (l
Substituting these estimates into (3-7), using condition (1.2) and reducing similar
terms, we obtain
v IITU (x. 0) Ilf, Q 4- 21| r), \\l ^
< II n, III Qh + II * III 0„ +"11 / if*.. 2. 0( + 4II ’f III a. (3.8)
where
c, = 4(|n14 »!*?+ I +cV^c*).
The function r)(x, I) depends on the parameter tj, which bad not been men¬
tioned explicitly. We now take advantage of the arbitrariness in the choice of tj.
We introduce in place of 17 another function £(*, t) = fgu(x, f)if. It is clear
that 7?(*, t) = £U, ij) - £(*, /),
§3. UNIQUENESS THEOREMS
149
and
II n. 111. P(i < 211^ l| || u*. Will a.
In view of this we have from (3.8) the inequality
(V-2 cA) !!?,(*• a+2 ||t,|| <2C, lit,|| 0(+<r^). (3.9)
in which
<r(<,)=iifiiU(,+ii/n^_ >2( Q<t+
For a ([ satisfying the condition
0<‘*<-sr’ (310)
we will have from (3-9), on (fee basis of Lemma 5-5 of Chapter 11,
JII tr (*• *l) l| a + 21| £, l| Qh (M•
and therefore also (3.4). If 4>i ~ f - f— ®» then u(x, t) s 0 for 6 [0, i//4cj]»
and therefore identity (3.5) will take the form
r
J J {—a1it + • ■ • Jdx & *■* 0.
v s
o
i.e. u(x, t) will be a generalized solution from of problem (1.9)
in @v/4c\ T w*t*1 zero t"*1*3! condition for t - i//4c[. Applying estimate (3.4)
to it again in the cylinder O x [iV4cj, v/ZcJ, we find that u a 0 in it also, and
so on, until we exhaust the whole cylinder Qy- Theorem 3-2 is proved.
Let us now give another proof of the uniqueness theorem in the space
not using the existence of derivatives of the coefficients of £.
Theorem 3-3- F°r equations (1.1), the coefficients of which satisfy condi¬
tions (1.2) and the conditions
«!*,.?>§ for »>2. (3-Hj)
?. oo. Qr
2)4 Yibl a
/-I Iml
ilo». 26?. all <|i„ for n= 1. (3.112)
i-l <"l 111, 2, Qr
150
in. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
problem (1.9) cannot have more than one generalized solution from 1
This theorem is a consequence of Lemma 4.3 and Theorem 3.1 of the present
section. Lemma 4.3 guarantees the membership of the difference u of two possible
generalized solutions from ^ of problem (1.9) in ®bere 5
is an arbitrary small positive number. It is applied to the function v(x, I) =
u*{x, t)o>U), where u*(ae, t) is constructed with respect to u(x, I) as described
in (4.9). The function v satisfies identity (4.11) with /,• *» /s 0, i.e., an identity
of type (4.8). It is not difficult to verify that the functions F{ and F correspond'
ing to the case considered by us will satisfy all of the requirements of the first
part of Lemma 4.3, and that in (3-lli) the index q can be taken equal to even
n/2.
Thus, u 6 V^Qj-s), S > 0. Further, conditions (3.11;) imply conditions
(1.3) and (1.4), since for n> 2 we have C Ln/2*f,n/2c+l (Qt*’ w*Iere
f = q - n/2 > 0 and the indices § = n/2 + t, F ~ n/2c + 1 satisfy conditions (1.4),
while for n = 1 condition (3.112) simply coincides with (1.3) and (1.4). In view
of this, Theorem 3.1 guarantees the equality of u(x, f) to zero in Qj-%, and
since 5 is an arbitrary positive number, u = 0 in Qj-
Theorem 3-4. Suppose the coefficients of £u satisfy the conditions (1.2),
(«i-*i)*r 11 “-ly, <* (3 n)
1 Hi, Qf ’’ ^
and
a(x. >-lh, Hi>0. (3.13)
Then a uniqueness theorem holds for the generalized solutions u of problem (1.9)
from ^2with finite vraimax^y,|ii|.
1) Condition (3.13) can be replaced by the condition a(x, t) - dbi(xt t)/dxi > -0(t),
where @(t) is a positive function that is sttmmable on [0, T}. It will be seen from the
proof of Theorem 3.4 that uniqueness is preserved if the condition of finiteness for
vraimax^f |u| is weakened to a condition of finiteness for |a|qtr,Qp ?> and in
return the restrictions on the coefficients of £ are strengthened by replacing (3.12) with
the conditions
§3. UNIQUENESS THEOREMS
151
Such generalized solutions must satisfy identity (3.5) with any r)(x, t) from
W\’l(QT) equal to zero fot I = T and having vraimaxQj,|)j| < <». Suppose u is
the difference of two possible solutions. Then
J
where
dx dt = Im («, tft, (3.14)
Qt
-2’mT| — “ ^ — ST, {aU%) + K — bT) \ + (a“ -1^) *1.
'm (a.tl) = / j
+ [(ara—Sr)-(a ~W\aAdxdt'
and the a™., h™, a™ and am are averagings with a smooth kernel of radius l/m
of the corresponding coefficients, which we can assume beforehand to be extended
onto the whole space with preservation of the properties distinguished by condi¬
tions (1.2), (3.12) and (3.13)- Let us take as j)(x, t) the solution of the adjoint
boundary value problem in Qj-'
==/(*, t), n|sr = 0, r)|(_r = 0, (3.15)
where f(x, t) is an arbitrary smooth function that is equal to zero in a neighbor¬
hood of the base lx £ Q, t ~ T\. This problem (it is a problem of the same type
as the original one and is changed into the usual form by replacing the independ¬
ent variable t with r = T — t), as will be proved in §5 of Chapter IV, has a unique
solution ijm(x, f) in H2*a,1*a^2{Q-p). Let m = 1, 2, • • •• There are two norms
that are uniformly bounded for the full set of solutions fi7ml:
and
max | if11 jSce**7, max |/1=3 cl (3.16)
Qr Qt
mur^cr <3l7>
The first of these inequalities is a consequence of estimate (2.12) of Chapter I
and assumptions (1.2) and (3.13), which are preserved for averaged functions (we
assume that the averaging kernel is nonnegative). The second is easily deduced
from the relation
152
III. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
T T
j J /ifdxdt=J | Jg*mifrTdxdt
t 0 t Q
T
o iil.a+ / J
+(am~ST'mHrfjc dt-
if one takes into account assumptions (1.2), (3.12) and the just proved estimate
(3.16).
Let us now pass to the limit in (3.14) as m —» «>. The right side of (3.14)
will tend to zero in this case, since
| {{a?l-aii)Ux^ldXdt
<S|i^lur«(^-%)«^||20r
and the integral fQ’j.(afj - a^u^dxdt s Im{Qj) tends to zero on a set Q'f C Qj
of measure differing arbitrarily little from die measure of all of Qj-, owing to the
almost uniform convergence of the o™ to o(y, while with respect to any set of
small measure Im(Qy) is uniformly small owing to the summability of on Qf
The integral
I J [(am — bTx,)-(a-*,>,)] «rf dx dt
| Qr
<c,max|a| J — b?X{)—(a — b,Xl)\dxdt
Or qt
tends to zero owing to the convergence of the am - 6^, to o - bix, in the norm
of LyiQj*). Finally,
J[«-0-(flr»i)W/^
Qt
<cs max | u | ( j [(of — bT)—(at — *f)P dx dt\n 0
0* \Qt i !
§4. THE FIRST BOUNDARY VALUE PROBLEM 153
because of the convergence of the a” — to a. — b. in the norm of L2(Qj).
Thus we obtain from (3.14) in the limit
J ufdxdt-*= 0,
Qt
from which, in view of the sufficient arbitrariness in the choice of /, it follows
that u s 0. The theorem is proved.
§4. SOLVABILITY AND STABILITY OF THE FIRST BOUNDARY
VALUE PROBLEM IN THE CLASSES V\M(QT) AND W^'A(QT)
Let us first show that problem (1.9) is solvable in the class namely,
let us prove the following proposition:
Theorem 4.1. If conditions (1.2)-(1.6) are fulfilled and 0o(x) € then
problem (1.9) has a solution from V^Qf)-
For the prove we make use of Galerkin’s method. Ve take a fundamental
system of functions tfr^x), k = 1, 2, • • •, in IPj(O) and for the sake of nonessen¬
tial simplifications we will assume it to be orthonormalized in L2(0,), so that
(lAj., = Sj[. We will seek an approximate solution in the form
«"(*. o-2^(0 ♦,<*).
where the c^(I) = (u'\ 0^) are determined from the conditions
d
dt
-(«". >!>*) +-ST.(«". W = 0 W-H
and
<(0)=(v>|>6) (4-2)
fof h - /V. Conditions (4.1) are a system of linear ordinary differential
equations of the form
i^l+^JM(0eW(0 + y?*W = 0, fe=l /V, (*•»
with summable coefficients and free terms on [0, 71. (The summability of the
Akl(t) and Fk(t) on [0, H is seen from estimates (1.11)—(1.14).) In view of this
problems (4.3), (4.2) have a unique solution on [0, T\, and by the same token the
b'V(*, t) are determined uniquely for any N. Let us show that the norms of the
b in V7(Qt) are uniformly bounded, i.e.
154
III. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
<4-4>
for all <V = I, 2, • • •. For this purpose we multiply each of the equations (4.1) by
its own cj(Oi then sum the obtained equalities over all k from 1 to N, and inte¬
grate the result with respect to t from 0 to Ij. This gives
/ P
yll «"(*. 0!!|a[^'+ J \-3°\ (mn. «") +- -2s, (f. «")] dt = o.
from which, in consequence of Lemma 2.1, we obtain estimate (4.4) (in this con¬
nection we took into account that ||u^(», 0)||j 0 = t(c^(0))2 < ||^q1| q). It
follows from (4) that the functions Z(y ^U) = (u^ix, t), ifi^ix)) are uniformly
bounded. Let us show that for fixed k and arbitrary N >k they are equicontin-
uous on [0, 71. Indeed, from (4.1) we have
f+A<
!**.»(/ +AO-/*. »(0l< I [|-S°.
t
For an estimate of the right side we use inequalities (1.11)—(1.14), (4.4), the
inequality
/-t Al
and the fact that all integrals of 0^ and the known functions (the coefficients
and free terms) encountered in this connection are small over a small volume
Qt I+^( (more precisely, they tend to zero for At —* 0). This gives
|Zjy jU + At) - ^(i)| < f (At)|^£ II2 [j with e (At) not depending on /V and
tending to zero for At —* 0, i.e. the equicontinuity of the I-y j(j), N = k, k +
!,•••, in t. By the usual diagonal process we select a subsequence N , m =
1, 2, • • •, with respect to which the functions Iy M converge uniformly on
fft*
[0, T] to some continuous function 1^(1) for each k = 1, 2, •••. The functions
IfJiOt k == 1, 2, ■ • •, determine a function a (#, /) » ^ To this func¬
tion the converge weakly in uniformly with respect to t from
[0, 71. Indeed, for any function from we have
a
(,liNm —U. t|>)= 2 ($. ♦*) (“A’m — t*)
ftsl
+ K«-«. 2 (♦. t*)(4.5)
§4. THE FIRST BOUNDARY VALUE PROBLEM
155
with
(co \ / 00
S (♦. *»)♦») <«2 2 (<i>. M2) ^c2R(s),
where Cj does not depend on N .
We choose $ so large that CjRis) becomes less than a preassigned * > 0.
On the other hand, for fixed s, the first sum in (4.5) will be, as proved above,
less than e for all t from [0, 71. Thus \{u m ~ u, \JA| can be made less than €
N
for all t from [0, T]. By die same token it is shown that the u m, m a 1, 2, • • •,
converge to u weakly in uaif°rCQty wi£b respect to t € [0, 71. Owing to
(4.4) one can select a subsequence from ju mJ that converges to u weakly in
N
Lj{Qt) together with its derivatives u. m . Without loss of generality we will
N
assume that the whole sequence u m converges to u in this way. By virtue of
a well-known property of weak convergence, inequality (4.4) is preserved for the
limit function u and hence u, will be an element of V^Qj). Let us verify that
it satisfies (1.16). For this purpose we multiply each equation of (4.1) by a
smooth function d^{t) that is equal to zero for t = T, then sum over all k from
1 to <V#< N, and integrate die result with respect to t from 0 to T. After an
integration by parts in the first term we obtain
/#(,\ J [_(*". <)4 ^ („* a>*')
0), <Z>N'(x, 0)). (4.6)
where (x, t) =* ^ 1“ *kis equality one can pass to the limit with
respect to the subsequence N « Nm, m * I, 2, * * •, selected above, assuming 4^
fixed, and thereby arrive at (4.6) for a (instead of u^).
Indeed, the functicuis ^ , aif^x- me elements of wbile the func¬
tions bflft and belong (as follows from estimates (1.11)—(1.14) of
the present chapter and Theorem 2.1 of Chapter II) to be the spaces L (£?r),
*2 2
^q2,r2^T> wit** 92 ~ r2 ~ ~' ” ~ ^ ^
Qi~rj~ 2, respectively, and the indices q, F conjugate with q2, r2 satisfy the con¬
ditions under which L- -(Q-p) is embedded in V2(Qy). In view of this and the
above-mentioned weak convergence of the u m to a one can pass n> the limit in
all volume integrals as m —> But the integral (o m(x, 0), (x, 0)) has for
156 III. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
its limit 0//q(x), (*, 0)), since the u^m(x, 0) converge to <ftQ(x) in LjlQ).
Thus we have proved that the limit function u(x, t) satisfies identity (1.16),
i.e.
/„ («, «>"') = (<|>0, (x, 0) ) (4.7)
for all of the form = l <^( t)with the above-mentioned properties of
d^t) and 4>jt(x). But such are dense in the space of all $ required in the
second definition of a generalized solution from V2(Qf) of problem (1.9) (see
Lemma 4.X2 of Chapter II). In view of this and estimates (1.11)—(1.14) for the
remaining terms, the identities (4.7) hold for all from that are equal
to zero for I =* 7"; i.e. u(x, t) is a generalized solution from of problem
(1.9). The theorem is proved.
Let us now carry out some additional investigations concerning the smooth¬
ness of generalized solutions in t. Suppose u is a generalized solution from
P2«?r) of problem (1.9), and suppose conditions (1.2)-(1.6) are fulfilled for
(1.1). The function u satisfies identity (1.16), which we rewrite in the following
manner:
J — at], dxdt — J % (x) ii (jc, 0) dx
Qt o
= f M-®
Qt
Here
Pl=, — alJaX) — aiu~ft, (4.8 j)
F = — biUiCi — ati, (4.82)
O
i//q € LjiQ), while r), as in (1.16), is an arbitrary function from K'2f) that
is equal to zero for I = 7. From estimates (1.11)—(1.14) of the present chapter
and Theorem 1.1 of Chapter II it follows that the F^x, t) belong to L2(Qj),
while F(x, t) € L q r (Qt) with ?2 = 2?A<? + l)> Ti ~ 2r/(r + l) (an analogous
calculation of the summability indices was made in §2 of the present chapter in
carrying out the passage to the limit from (2.12) to (2.13))- The indices q2, r2
satisfy the same relations as die indices q^ r of the space £,1>ri(<?j,) containing f.
o
Let us show that any function u from satisfying identity (4.8) with
the just-mentioned properties of the functions entering into it, is strongly
continuous in
§4. THE FIRST BOUNDARY VALUE PROBLEM 157
t in the norm of L2(Q) and belongs to V(the defini¬
tion of was given in §1 of Chapter I). For this purpose we will express
identity (4.8) in a more convenient form. Namely, we extend the definition of the
functions u, Fy F and / onto the infinite cylinder 9-oo^so as
follows:
«*(*, t)--
F,(X. t):
F* (x, t) = j
u (x. t),
u(x. — t),
0,
<eio. n.
<61- 7*. 0],
\t\>T,
F,(x, t).
■Ft(x. -/).
0,
Fix. t),
-F(x. -t),
0,
fix. 0.
-fix. -t),
0.
<€io. rj,
te\-T, o).
m>r.
(€10. T\,
/ei-r. 0).
I<l>7\
<€10, T],
t£l-T, 0).
W>7\
(4.9)
and until the end of the section, in place of <?_TC M we will write simply <?.
By virtue of (4.8) we have for these functions tbe identity
J — «% dx dt = j [ F\r\x -f (F* — /•) rjj dx dt. (4.10)
Q « ‘
o
with any function rj from •*(<?) that is equal to zero for \t\ > T. We take a
smooth function <a (*) that is equal to 1 for t 6 [— T + S, T — 5], where 8 is
some positive number, and to zero for |*| > T. We put rjix, t) -co(t) $(#, t) in
— o
(4.10), where $ is an arbitrary element of and we rewrite the resultant
equality in the form
J — v<&t dx dt = J [/7i<a®xi 4- {F*(o — /*« -f u©<) <D] dx dtt (4.11)
Q Q
where v(x, t) = u{x> t)<jj{t). Identity (4.11) is a particular case of the identity
I) Concerning <£q(#) here and below, it is sufficient to assume only that the integral
/q^x) rj(x, 0)dx is finite for any function r){x, 0) from SP'^(Q).
158
III. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
J — v®,dxdt = J | /VDXj + 2 p) dx dt’ (4.12)
Q Q \ f-l /
o
in which v, F^ and Fl are given elements of spaces V^Q)t and
L (Q) with q, and r. satisfying conditions (1.6), respectively, that are equal
°i i
to zero for |t| > T, while <J> is an arbitrary element of Wy (Q). Let us prove the
following lemma.
o
Lemma 4.1. If an element v of the space satisfies identity (4.12), in
which the F. are elements of Lji-Q)* the Fl are elements of L f (0 with q>
^ * o. .
and r. subject to conditions (1.6), and $ is an arbitrary element of (Q),
and if v and $ are finite in t (that is, equal to zero for large |t|), then v €
V 2’^ ((?)• The proposition remains valid if the '* o” of V ^ and means the
vanishing of the elements on only a part x (-©*, *»), Sj C S, of the lateral surface ofQ.
We take as $ in (4.12) the function
t
/) = h~1 j r\(x, x)dt, tl£#2,U(Q).
t~h
and in all of the terms of (4.12) we transfer die averaging ( )- from the second
h
factor to the first according to formula (2.9). This gives the relation
J - VI, </**- J Pih\ + J) /’in) dx dt,
Q Q \ 1-1 /
which for ij(*, t) = y(«)<f> Gc), (where y(t) is some smooth function of t, while
0 (%) is an arbitrary element of can be rewritten as follows:
J Xi(vtr y)dt— J X (Fth. <P,()+j£(^. f>)
dt. (4.13)
Hence by the definition of the generalized derivative d/dt it follows that for
almost all l we have
4t (**• *)=(f<h’ \)+2 (Fi- *) (4-14)
i~i
for any <j>{x) € If'Il(fi).
The function vh is strongly continuous in £ in the norm of L2(^D- All the
§4. THE FIRST BOUNDARY VALUE PROBLEM 159
more so the scalar product (v^, </>) is continuous in t. In addition, has the
generalized derivative v/lt from L2(Q), and so for almost all I
4f<Vk. <f>) = (<V <P)
and
(V <P) = (/?iA. ^)+S(/'v *>)• (4-15)
Hence for any positive Aj and h2 we have
*)~(/W *0 fl(/V/v^ (416)
We put <f> ** vh - vh in ** (which, clearly, is permissible), and present the left
nl "2
side in the form of a derivative with respect to t of the continuous function
- "A2ll|.n; na“ely*
* d H . »i2
2 dt W 1 °
m
"=‘(Fikl — FiAt, «!»,*»—'*Vi)+ S(F*!
Integrating this equality with respect to I, we obtain
l ,
Ti
+ 2 (Pi, — Pkv •Vh, — «4.)
(4.17)
which is valid for all tj and t2- AH of the functions here are finite in t. As Aj
and h2 tend to zero we get
+1| vhtX — Vh,x ||2(, + II t»*, — •Oh, Ht / ,/ Q -*■ 0
(see Lemma 4.7 of Chapter II). In view of this, from (4.17) for Jj = -oo, and arbi¬
trary ij follows the strong convergence of the in L^fQ), uniformly in t 6
(-oo, oo). Therefore the limit function v is equivalent in Q to a function that
is strongly continuous in I in the norm of
160
HI. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
Let us now prove that
00
I <+-*) — v(x, t)\f% adt *=o(h).
-CO
For this purpose we take <f>(x) = (x, t) = v(x, t + h) — v(x, t) in (4.15) and
integrate both sides of tbe equality with respect to t from -oo to + Then in
the right side of the resultant equality we transfer the averaging ( )/t from the
first factors to the second, and the difference A^ from tbe second factors to the
first. As a result we obtain
dt. (4.18)
J A’11| Hi adt= J (A„F„ vhXi) + J)vh)
-OS — OO L /•»!
Estimating the right side by Holder’s inequality and making use of the uniform
boundedness of Q 30^ r< Q and the integral continuity of the
elements F. and F‘ of spaces and r (Q), we show that
*-‘11
'A*° fot
This together with the strong continuity of tiQin f in the norm of and the
membership of v in means that v € {Q). From here and from Theorem
4.1 follows the main theorem of the present section.
Theorem 4.2. Under the fulfilment of conditions (1.2)—(1.6) any generalized
solution u of problem (1.9) from K2((?r) belongs to ^’^(p), and problem (1.0)
is uniquely solvable in if 6 LjCO).
Remark 4.2. Strictly speaking, we have proved the membership of u in
V^(Qr_s), where 8 is some positive number, but not in ^>‘s *s
not important, however, since the solution u and the equation itself can be ex¬
tended beforehand onto the larger cylinder Qr+s with preservation of all prop¬
erties of the functions in question.
Let us explain when a generalized solution u of problem (1.9) is an element
of rpd?r).
The space (*?)> where Q = 0 x (- °o, oo), consists of all elements u of
o
^2>O(0 having a finite integral
/ oo \ 1/2
hi«me=y *~sii«(jc. <+*)—u(x, mi'dh] .
§4. THE FIRST BOUNDARY VALUE PROBLEM
161
It will be a Hilbert space if the norm in it is defined by the equality
II “ II** «*«,, = [(!<«+IIIII 9+III«IH’cJ1'*-
It is not difficult to calculate that
oo
III“IIIq = J f I#(X, o)P|a|djcdo.
-oo Q
whete uix, a) is the Fourier transform of uix, t) with respect to t, i.e.,
OO
uix, a)==J uix, t)e,n‘ dt.
The membership of a function^ u in W^’^iQf) can be defined, for example, as
follows: u is an element of if and only if the function Six, t) =
uix, t)tDit), whete
and
u{x, t).
^ € to. n
u’(x, t) — u'(x, —/) =
uix, 2T
— t), t£\T, 2T\
0,
t> IT,
I,
<6io, n.
—/)==
<3 r—a)
T
'e[>. |t-].
0,
t>\T.
is an element of ((?). As the norm of u in W^’^iQf) one can take the number
Ill'll 1 U •
Let us also introduce the Banach spaces L* r, ? > 1- They consist
of measurable functions u having a finite integral
i
<r. Qt '
.(/(/.
(4.19)
The value of ||u||* ^ is taken as the norm of u in L* qiQf)- It is well known
that
''<?• (4.20)
162
1H. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
and
II “II,.Qr<IM£Uor. (4.20!)
(see [42]t p, 169^ problem 245). In view of this and inequality (3.4) of Chapter II,
we get
IK?,Qr<»«IU3r<f!Kr' (4‘2I)
if r < q and if r and q satisfy conditions (3.3) of Chapter II guaranteeing the
validity of the second of the inequalities (4.21).
Let us assume that the coefficients a{. and af and the functions f. satisfy
conditions (1.2)-{1.5), while the coefficients bi and a and the function / satisfy
the conditions
<H,, r><7, (4.22)
and
II',»<?!. <4-23>
in which the indices 9, r, q^mdr^ are subject to restrictions (1.4) and(1.6). By virtue
of (4.21) these requirements imply the previous assumptions (1.2)~(1.6), and
therefore all of the conclusions made in this chapter are valid. In particular, the
function v{x, 0 ~ uix, ;)&>(£)» where u(xt t) is a generalized solution of problem
o
(1.9) from V2(Qt), satisfies an identity of type (4.12) (namely (4.11)) with the
Fi and F‘ having the properties described there. But, as is easily calculated in
the present case, the functions standing for $ will also be elements of
L* ^ (Q) with r. and q. subject to the same restrictions as the indices rj and
for /. Let us show that in view of this the function v will be an element of
W\M(Q).
* o
Lemma 4.2. If an dement v of the space V2(Q) satisfies identity (4..1T) (in which
F. € L2(Q), Fl€. L* -((?) with q. and r. satisfying conditions (1.6) and
i* o
9; < '() for any <t> from ^’*(0, and if V and 4> are finite in t,
then v belongs to W2’,y*(Q). The proposition remains valid if the "°” over V2>
W2’1 and W2,/* means the vanishing of all of the elements on only a part Sj x
(—oo, oo), Sj C S, of the lateral surface of Q, which can even be empty.
For a proof of this theorem we make use of the Fourier transform with respect
§4. TOE FIRST BOUNDARY VALUE PROBLEM
163
to t, denoting, as above, die Fourier transform of a function t/r (x, f) with respect
■V
co t by iff {x, a), so that, in particular,
oo
v(x, a) = -~ J t)e:~:dt.
-CO
It is accessary to establish (be finiteness of die integral
CO
I J|a||»(x, a)Prfxrfa.
—oo Q
It is not difficult to see that the Fourier transform v Ax, a) of the function
°1
vh(x’ 0 is fot each a an element of IPJ ® > while the Fourier transform of the
function v^x, t) is equal to -iavj^x, a). Therefore (4.15) implies the identity
— tafa, <p) = (Flk, q>,() + Jj (F[, <p). (4.24)
Let us take as <j> the function y (a) a), where xequal to i for
a> 0 and to -t for a < 0, and integrate both sides with respect to a from -■»
to +°»:
oo
J a)|2rfjccra.
-co C
oo f ^ m \
“ f J ( X'VL 4- 2 j^da. (4.25)
-ooQ V t-l t
The functions y{a)F^(x, a), y(a,)F^{x, a) are the Fourier transforms of the functions
FjA(x, t), Ffrix, t), which are obtained from the functions F ^(x, 4), F^ix, t) by
means of the Hilbert transform (see the book [120]). jt js known [120^ that the
Hilbert transform is a bounded linear operator from £<r(— ») into Lf(-oo, oo)
for any r > 1, so that
y J I F'k (x- 0 |r dtj c ^ J | Flh (x, t) \r , (4 26)
and analogously for F^. For r = 2, in place of (4.26) we have the equality sign
if c is put equal to 1. In' view of this and Parseval’s equality, from (4.25) we get
164
III. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
co OB / m \
J Jl«II «*(■*. a)\2dxda= | J I F^v^ ]£J F‘hvJ dx dt
_eo Q -00 Q > /-I /
m
< II ?»k»I'-, IU+$11 »>,.«* "* ">«
^ i« 1 ‘
since the in<lices , q{ ate such that estimate (4.22) holds for them. Passing
to the limit in this inequality as h —■* 0, we prove the finiteness of the integral
on the left, and thereby also prove the membership of v in Lemma 4.2
is proved. From it we get
Theorem 4.3. Suppose a is a generalized solution from 92(Qj) of problem
(1.9), and suppose the a-, a{. and f{ satisfy conditions (1.2)—(1.5). while the
bj, a and f satisfy conditions (4.22), (4.23) with the meaning of the indices the
same as indicated there. Then u is an element of (Qj-).
A proof analogous to that of Lemmas 4.1 and 4.2 exists for the following
lemma. q
Lemma 4-3- If an element v of the space satisfies the identity
J — n®, dx dt as J (/=,i®X(+ FQ>) dx dt, (4.28)
Q Q
o
in which ® is an arbitrary element of Wl,’ (0. while the F( and F ore given
functions from L2(Q) and where q - 1 n/(n + 2) for n > 3, 9 > 1 for
n - 2 and q => 1 for n = 1, and if v and $ are finite in t, then
v 6 ®u( If in this connection F G then V 6 ITThe prop¬
osition remains valid if the " of ^2’*' ^2'^ an<^ ^2’ means the
vanishing of the elements on only a part Sj x (-«, «■), Sj C S, of the lateral sur¬
face of Q, which can even be empty.
For its deduction the term JqF$ dx dt is estimated in the first case as fol¬
lows:
I J FVdx dt I < II Sj Q||<t>||?,.2i Q < *11/^ 2.qII«*IKV
10 I
§4. THE FIRST BOUNDARY VALUE PROBLEM
165
while in the second case
00 CO
J J XFhvhdxda = J ^Fhvhdxdt
-oo fi -co C
<ll^l!2.?.QKlk,'.«<cll F*%.qIKIIV
la both cases we have made use of the embeddability of 1^(0) in L^, (0) (see
Theorem 2.1 of Chapter Q).
Let us now assume that u is a generalized solution from K2(0 of equation
(1.1). Then the function »(*, t) = u(x, t) u> (t) satisfies (4.11) for all 6
if <o(t) is a smooth function that is equal to zero for t <8 and t>T -8, where
8 is some positive number. From this identity, as was shown above, follow iden¬
tities (4.15), (4.16) and (4.24). If in (4.16) we set rf>(x) = [u^(x, t) - vh(x, j)l£2(x),
where C (x) is a smooth nonnegative function of x that is equal to zero on S, we
arrive at an equality from which we deduce the strong continuity in i of die func¬
tion v(x, t) £(x) in the norm of From equality (4.15) with <j>(x) =
C2{x) A&»(», t) we see that the integral
OO
Jft"'ll£Vl6.a<«
—oo
tends to zero for h —> 0. Finally, equality (4.24) with <f> (x) = x(«x) ^(x, a)£2(x)
permits us to conclude that v(x, t) f (x) is an element of If it € V2(Qf)
and satisfies identity (1.16), then the function 0){t) in v can be taken equal to
unity for )(| < T — 8, 8 > 0. Thus we have ptoved
Theorem 4.4. If the coefficients and free terms of equation (1.1) satisfy the
conditions of Theorem 4.2, then any generalized solution u of it from VjiQf)
belongs to x [«, Ij]), where O' is an arbitrary strictly interior subdomain
of the domain 0, e > 0, and tj < T. And any generalized solution a of equation
(1.1) from VjiQy), satisfying (1.16) with ij from and equal to zero for
t= T, belongs to [0, tj]). If the coefficients and free terms of equa¬
tion (1.1) satisfy the conditions of Theorem 4-3, then any generalized solution u
of it from V^Qfl belongs to x [f, Ij]), e > 0, t^ < T, and any general¬
ized solution u from V2(Qt), satisfying identity (1.16) with tj from V^’HQj.)
and equal to zero for t = T, belongs to 1T|>^(0'x[0, tj]).
A generalized solution of problem (1.9) is stable in the norm of tbe space
with respect to variations of die coefficients and free terms of the
166 IH. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
equation, and also of the function tj/g(x) determining die initial condition. More
precisely, we have
Theorem 4.5. Suppose that for all operators
J?mu a u, — {aTj“Jtj -f- a?u) bf uxi -(- amu,
m = 1. 2
the conditions of Theorem 4.1 are fulfilled with one and the same constants.
Suppose the af.(x, t), remaining uniformly bounded, converge almost everywhere
to a.., while the functions a™, 6™, am, fP, fm and tf/™ converge to a., b., a,
ff, f and iAq in the norms of the spaces to which they belong according to the
conditions of Theorem A.I. Then the generalized solutions um from °f
the problems
jg~u*=%L—/". u|Sf = 0, «!,_„ = (4.29)
converge strongly in to the generalized solution u of the limit problem
(1.9).
For a proof of this proposition we subtract from the integral identity (1.10)
for um the integral identity (1.10) for u and write the result of this subtraction in
the form of an integral identity for the function v = um — u in tbe following
manner:
I xTrt dx i;;'' + J I- vmn, 4-
0 Qt,
+ b?*'",’1 + amvmvl] dxdt=* f [(a, j — a"/) «xyn»,
Qt,
+ (°< -0 7) mu, + (bi ~b?) V+(a - °m) 8T>
+(/,- f?) nr,+(/ - n t|] dx at.
We express the right side of this identity as fn [FJnr)_. + Fmr) + (f - fm) >j] dxdt.
VH 1
The functions F™ and Fm are composed of terms of the same type as F, and F
in (4.81—2), and hence, for the reasons mentioned there, satisfy the conditions of
Theorem 4.1. Consequently vm can be considered as a generalized solution from
V\ ,0(Qt) of problem (4.16), in which vm|t=0 - - i[i0, /V™ is -Ff, and fm
§5. OTHER BOUNDARY VALUE PROBLEMS
167
is equal to — Fm — f + fm. According to Lemma 2.1 we have the energy estimate
l»X<*[K-*. II,. IU
+t'mKwr+ir'-l\'r4-
from which it is seen that jt'w,|q —> 0 for m —► <*>, And what is more,
sup0<A< jh Ml vm(x, t + h) - vm(x, 2 ,C?y ^ ♦ 0. anc* under fulfilment of the
conditions of Lemma 4.2 also II!r,ml!iq^ —> 0 for m —> =°.
§5. ON THE SOLVABILITY OF OTHER BOUNDARY VALUE PROBLEMS.
THE CAUCHY PROBLEM
Let us consider the second boundary value problem for equations (1.1), i.e.
the problem of determining u(x, t) in Qj from the conditions
^o = — /; -*L_|_0(Si t)
“l,-o=='MJC)- (5.1)
where o(s, <), as well as ^q(x), and the coefficients and free term f of equation
(1.1), are given functions belonging to well-defined classes, du/m cosa.
—*/ */ •
with a. being the angle formed by the outward normal to S with die x- axis,
while the a..(x, t) are the same as in equation (1.1).
Concerning equation (1.1) and the function </tq (x) we make (he same assump¬
tions as in §§1, 2 and 4. The boundary 5 of the domain fl is assumed to be
piecewise smooth, while the function a(s, t) is assumed to satisfy die condition
(5-2)
where and rj are subject to the restrictions
1 , it — 1 1
r, ^ 2?, ~ 2 ’
?,£(«—!, oo], /"if [2, oo) for a > 2,
?,6(I, oo). r,6(2, oo) for n — Z.
(5.2')
In the case n = 1 we assume that l<J,!r Sj. — rl =
1) For n =1 it is necessary that norms of the form | • j|^ f ^ be replaced everywhere
by norms of the form }| * ||r
168 HI. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
Finally, concerning the function t) we assume that
M?2>r2,sr<*i> (5.3)
where ^ and rj are subject to the restrictions
1 n — 1 n 1
r2 + 2q2 4 + 2’
?2 e V^t\ r2 e [l’ 2] for " > 2>
^2 6 (l, oo], r2 € [1, 2) for n => 2.
(5.3')
In the case it = 1 we assume that < jtj.
With the use of inequalities (1.8) and (3-11), (3.12) of Chapter 11 it is not
difficult to verify that under such conditions on 5 and a the integrals /qJsIiWojMs dt and
/<TJshM ds dt will be finite for any u(x, t) and r)(x, t) from V2(Qj.) and admit
the estimates
T
J J | o«ri|ds dt < || o I Sj. || a
o s * r»“l ’
XlhlU_ ,.*1*1* M*-
»,-i ‘ r,—I ' (5.4)
/o/sl^l** S c \Mq2,r2,$T\v\QT,
where the constant c is taken from inequalities (3.11), (3.12) of Chapter II*
By a generalized solution from (from or Kp«?r» */
problem (5.1) we will mean a function u(*, f) from (respectively from
or V2’/^(Qj') satisfying the identity
r
— ju^dxdt + j UST^a, ti)4 </. tftjrf/
Qr «
-f J (ou-^r)jidsdt= j «h>t*)»]C*. 0)dx (5.5)
ir o
for any Tf(x, t) from ^2’^(Qf) that is equal to zero for t ■» 7\ This identity is
obtained from die identity
§5. OTHER BOUNDARY VALUE PROBLEMS
169
as the result of a single integration by parts with respect to t and with respect to
x. in certain terms and a taking into account of the initial and boundary conditions
from (5,1). In the same way as in §1 we verify that all of the integrals in (5.5)
are finite for any function a from and any function t} from
that in place of ( 5.5) one could postulate the fulfilment of the identities
t
J u(x, 0 *!(•*» t)dx-\- J J —atydxdt
a o Q
t
-+J l-S’iO*. »!)+(/. l) + -2*3(a, 11)1 dt
0
=* j %(x)i\(x, Q)dx (5.6)
a
with
£j(u, tj) - f (cm ~xj/)-qds
S
for almost all t (respectively for all £) from [0, 71 and an arbitrary function
ij{xf f) from
For a proof of the solvability of problem (5.1) in Vwe again make use
of Galerkin’s method. In contrast to §4 the functions \jf^{x)t k « 1, 2,*--, must
form a fundamental system in but not in B^Q), Suppose, in addition,
tjtj) = We seek an approximate solution u‘ in the form
«"(*. Q->J}c"(QiM4
where c^U) = (tt'\ 4‘^ are determined from the conditions
■art*". **)+-§*.(«". +*>+-S'*(«w. w+v, <i>k) = o,
(0) — (i|>0, ♦*).
The subsequent arguments and estimates are analogous to those mentioned in §4.
The fact that all of the functions encountered this time do not vanish on S-p is
not important, it being only necessary in estimating r q in terms of
to use» instead of inequality (3-4) of Chapter II, inequality £3.8) of Chapter
II for functions not equal to zero on For a justification of the passage to the
limit in the term dsdt we note that the uniform unboundedness of the
i N i
norms |u \qt implies by virtue of inequalities (3-1)—(3-12) of
170
III. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
N
Chapter H die uniform boundedness and weak compactness of die it (*, t) in die norm of
r (Sr), where n — 1 /2q^ + 1/fj = n/4 with rj € [2, oo], q^ € [2(n — l)/n,
2(n — 1 )f(n — 2)] for n > 3> with ij G (2, oo], qj 6 [l, oo) for n = 2, and in the
norm of t^(Sj.) for n = 1. This is true, in particular, for = ?j = 2<?j/(<7j - l)>
rj = 7j = 2rj/(rj - 1), where and i-j are taken from (5.3)- On the other hand,
Theorem 2.1 of Chapter II and estimate (5.4) imply the membership of the product
N
o4* 1 in the conjugate space F^Sj.) (L^^iS-f)) of (of L4(Sj.)).
All of this makes it possible to pass to the limit in the integral jc;^^ iis dt
as N —< oo with respect to a weakly convergent subsequence in L-^ - (S j.) (in
L^iSj.)), and then to pass to the limit as N ‘ —* 03,
In this way, by means of Galerkin’s method we have proved die existence of
at least one generalized solution u of problem (5*1) in the class The
membership of such solutions in ancl follows from Lemmas
4.1 and 4.2. Thus we have
Theorem 5*1« Suppose the coefficients and free terms in the equation and
boundary condition of (5.1) are subject to restrictions (1. 2)—(1.6) and(5.2)-(5.3’>»
while fi is a domain with piecewise-smooth boundary S. Then for any $q(x)
there exists a unique solution of problem (5.1) in the class Any solu¬
tion from of problem (5.1) belongs to H furthermore the 6|t> a
and f satisfy conditions (4.22) and (4.23) with the meaning of the indices indi¬
cated theret while o{s, t) € L* „ (ST) and tlris, t) € L* „ (ST) with r• and
rvq{ J rvq2 I k
qk *= 1, 2, subject to conditions (5.2*), (5.3*) and q^ < r^, k = 1, 2, then u
will be an element of
The same result is also valid for the boundary value problem
Su = f,
Su
u|s’r =°- -37 + 0-“|sf = 'fr’ “|r=o = (5-7)
where S^ = S’ x [0, T\, Sj = S" x [0, 71 and S' US" = S. Its solution, if the
assumptions of Theorem 5.1 are complied with, will be an element of (Qf)
that is equal to zero on 5 j. This is proved in the same way as
Theorem 5.1, except that the functions k m 1, 2, • • ■, in Galerkin’s method
must form a fundamental system not in IP^(O) but in the subspace of 1^(0) con¬
sisting of all elements of it that are equal to zero on S'.
§5. OTHER BOUNDARY VALUE PROBLEMS
171
The Cauchy problem. Let us consider in the layer R p = En x [0, T} the
Cauchy problem for equation (1.1) with initial condition
“l,-o = %(•*)• *€£«• (5.8)
By a generalized solution from {Rp) of the Cauchy problem (1.1), (5.8) we
will mean a {unction u(x, t) from V^’^iRp) satisfying the integral identity
T r
— J J dx dt -1- J US?! (a, tj) +-^2(f. tj)] dt
» *„ 0
= /%(•*) n(*. 0)d.s (5.9)
Bn
with
-S’j («. iJ) = J [(«</% + ata-) ^ 4- 4- t|j rfjc,
-2»(f. »!) = J f^)dx
En
for any tj{x, t) from V|’*(/?j.) that is equal to zero for t = T.
It is possible to make other correct statements of the Cauchy problem in
classes of functions with weaker restrictions at infinity, for example, in a class
of functions admitting exponential growth ec'x‘ in x for |*| —» We do not
occupy ourselves with an investigation of these questions here (in this connection
see the references given on p. 19) and we restrict ourselves to an examination of
the solutions of class
The following existence and uniqueness theorem holds.
Theorem 5.2. Suppose the coefficients and free terms of equation (I. I)
satisfy conditions (1.2)-(1.6) with Qp = Rp. Then for any function t/rQ(x) from
i2(£n) there exists a unique generalized solution u (x, t) from class Vl,’'^2 (Rp)
of the Cauchy problem (1.1), (5.8).
Under the fulfilment of the conditions of Theorem 4.3 in Rp a solution of
problem (1.1), (5.8) will also be an element of ^’^(/{jO. The proofs given above
for bounded domains fi are carried over directly to unbounded 0, in particular to
£2 = En, since the estimates on which they are based do not depend on the dimen¬
sions of Q.
172
lU. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
§6. ON ESTIMATES IN THE SPACE
AND THEIR CONSEQUENCES
In the paper [6 5 d] (see also [«5f, 124 f] and [3 3 d]) it was shown that operators
£ of parabolic type possess the same properties as elliptic ones: the member¬
ship of £u in L,2(Qj-) implies (under the fulfilment of some homogeneous bound¬
ary conditions) the membership in L2(Qf) of each of the terms making up £u.
Although this fact was later generalized to the spaces and j.)
(the proof of these more general propositions is allotted a central place in Chap¬
ter IV) we will give a proof of it here both because of the simplicity of the proof
itself and because of a number of consequences that arise in this connection and
do not result from die theorems of Chapter IV. It is based on analogous assertions
concerning the elliptic part of the operator £u ([65b, c.k]; see also §2 of Chapter
II and §8 of Chapter III of the book [s5q)) that are formulated below (in a partic¬
ular c£se) in the form of inequality (6.6).
In order not to overburden the book with calculations analogous to those
carried out in previous sections we will restrict ourselves to a ’'shortened”
equation S-qU - f, retaining only the principal terms. The symmetry of the re¬
sultant equation is unimportant, as is also the fact that we take the first boundary
condition.
Thus, let us take tbe differential operator
oMussaix, t)— (x, t)u
with a > 0 and the o,y satisfying inequalities (1.2), and the boundary condition
We assume from the beginning that ix(x, t) satisfies the inequalities 0 < <
a(x, t) < jij and has a derivative a,, with
r
vrai max |a.(jc. /)|■== |ij(<). f H,(t)dt < oo.
We consider the expression Jq £gii • u dx dt and transform it in the following
manner:
I JT*-udxdt** J [4- (I “2) - j dx dt-
Q, Q,
§6. ESTIMATES IN THE SPACE w\'l(.QT) 173
Hence by virtue of our assumptions on a and die a^ we have
Vi J u1 (x. t)dx-f-2v J u2x dx dt < (i, j" u2 (x, 0)dx
a Qt u
t
+ / fl + M01 / «*(*• 0<**<« + J (^aufdxdt. (6.2)
Let
and
1 -j- (ij (0 = ft CO. v, J fij (0 J u5(x. t)dx = y (0.
H, J «J(*. 0)<**-f J (^0ufdxdt = ^r (t).
a <?,
For y(t) there follows from (6.2) the inequality
< Vf V3 (0 y (0+n3 (0 (0.
from which, in turn, by virtue of Lemma 5.5 of Chapter II, we get
y(0<«P jvf1 J n3(T)rft| J |»j(i)tf (t)dx.
Substituting this estimate into (6.2), we obtain the first of the inequalities of
interest to us:
vi |l B (■*• 0 Hi, a -$* II #JT Hz, Qt
< 1 + exp | Vj-1 J n3 (x) di |v, J (i3(T) dx
X [(i, || u (x. 0) B* 2 + II -2> 111 <,']■ (6- 3)
Ao analogous inequality also holds for die complete operator £ in (1.1) as long
as its coefficients satisfy, conditions (1.3) and (1.4).
Suppose now that the coefficients ai-(x, t) have the derivative with
T
vrai max | a, .,(x, 0|<M0- f H4 CO dx < oo.
174 IU. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
while a is a bounded function satisfying the inequalities 0 < i/, < a(x, t) < fi,.
We consider [q £qU • a(dxdt and transform it in the following manner:
J J?”0a • u, dx dt = («ta* -f- atjuxui^ dx dt
“
from which in view of our assumptions on a and the a- it follows that
YII (•*. 0111 a + J a«* dx dt < £ || ux (x. 0) ||* 0
0/
*
+ TII “< 1819, + k; #•*•“ l|«, + f J * w II <*• *> II *"■
and hence
0III.Qv,II«tIII.
t
<H||«„C*. °)IEL«H-vr'll-sv*IBLQ +«J MOIKIII^'- <6-4>
* 0
We introduce (he notation
/
V J t»4 WII (*. -t) 111 a dx = y (0
0
and
HIM*-0>IIU+vrMmi* Q<~^<o.
Inequality (6.4) implies the inequality
&P- < mr'm(0y(0 + MO^- (0. (6.5)
which, as proved in Lemma 5.5 of Chapter II, guarantees the estimate
y(0<exp /»v-> J n4(r) dx J J (x, (t) «T (t) dt.
0 Jo
Substituting it into (6.4), we obtain the second of the inequalities of interest to us:
§6. ESTIMATES UN THE SPACE W\'HQt)
175
vi|«,(*- 0 til. a -+- V1 !i “< Hi <3,
It J
1 4- exp | nv~l j |i4 (t) dt j /tv~* J )i4 (r) dt
I 0 jo
X [H || <*. 0) || B + V || || J. (6.6)
Remade 6-1. An analogous inequality also holds for the complete equation
(1.1) as long as
r
f vrai max | a„ (jr. t) | dt < oo
and die coefficients at-, b. and a, for example, are bounded by an absolute quan¬
tity. This is proved in die same way as inequality (6.6).
Let us assume now that the a,, have the derivatives da-;/dx. , with
*/ */ *
vraimaxl-^p a(x, 0— 1
Or i d**
and S € (ft. We consider the integral fg Squ(-Au) dx dt and transform it in the
following manner:
J .20“(—&u)dx dt= J ^ /Ux -fAA^dxdt. (6.7)
But by virtue of (2.24) of Chapter II, for any function u from 1^(0)
j" aXu A«djf > ~ v|| «xx |g Q — c, || ux|£ a, (6.8)
where is a constant depending only on u7 n, ^ S. Therefore (6.7) implies
that
II«, (*. t) || Q + IIuxx || Qt < II ux (*. 0) ||| a
+ 2c,||«,|£ 0( + B||A«||’ 0( + e-'||^„«||’. v
from which, choosing < = v/2n and using Lemma 5.5 of Chapter II, we obtain the
inequality
This assumption on the could be weakened somewhat.
176
III, UNEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
II «, (*• t) |j* 4- v l| uxx lla. Qi
<«^.'[||«,(x. 0)||; Q + -f|l-^o«llUj- (6'9)
which in combination with the equation u( - 3Ru + S^u gives the third inequality
of interest to us:
|| ux (x, t) ||j_ a +1| ttxx Hj Qi 4- II llj, Qt
< c2e^‘(t 4- 1) [|l«, (x, 0)||® Q +1| |g, flJ (6.10:
with c2 depending only on n, v, (i, n5 and 5.
The derivation of inequalities (6.6), (6.10) is based on two facts: inequality
(6.8) for elliptic operators and the positiveness of the principal part of the inte¬
grals — «i,Au dx dt and $Qt-u$-aijUx)x.dx dt combining the terms u, and either
-A« or -d(a..u )/dx. in a parabolic operator.
/ */ I
If the boundary of the domain is not smooth, then inequality (6.8) cannot hold
In this case it is possible to obtain interior estimates for u, and uxx by con¬
sidering the integrals
where £U) is a smooth function that is equal to zero on S, and assuming
vtaimx&Qy\da.j/dt\ <*>. This leads to the inequality
IK(*. Ot(*)|l*.0-HI“Ml,0( + ll«,,eil2. c,
<c3e'.' [IIa,(X, 0)£(x)II2.R4-II(S’0«)£11^4-||«xSJll eJ. «■»:
where the iotegral IU.jC.JI2 Qt can be estimated from the energy inequality. If
the initial values of » do not belong to ^(Q), then, by considering the integrals
j J2’0u(u, — Au)mx)xHt)dxdt c x(0) = 0.
we obtain the estimate
t! UX (X * 0 £ (■») X (0 II2, Q II H2. Qt K UXX II2, Qf
< cf* [II C3>) 5x llj, Qt + II “Ax 111 Q( 4- II «^x'x ||,. • (6.1;
§6. ESTIMATES IN THE SPACE W2’l(QT) 177
In accordance with the inequalities derived here one can prove that the gen¬
eralized solutions from ’®(Qt) f°un<i in §4 have finite integrals in the left
sides of these inequalities as long as all of the conditions under which these in¬
equalities are derived are fulfilled. This can be done in various ways. For example,
it can be done at once in the process of proving the convergence of Galerkin’s
method. In this regard it is necessary to show that analogous estimates hold for
AT
all of the approximations u with constants not depending on the order of approx¬
imations. Estimate (6.6) can be established without introducing any changes in
the plan of Galerkin’s method described above. Here it is necessary to multiply
each of the equations (4.1) by the corresponding dc^(t)/it, then sum the resul¬
tant equalities from k = I to k = N and integrate with respect to t from zero to
£. This gives the relation
J [(“?T+ (“*/“*• + ai“N) tttxt + {biUx. + a“K) ut ] dx dt
Qt
= - J (/,«& 4-fa^dxdt. (6.13)
Qi
the left side of which coincides with the integral dx dt, from a con¬
sideration of which inequality (6.6) was derived for the case £ = £q. In an ana¬
logous manner one deduces from (6.13) an estimate that is uniform in N:
OIL+IKIIU,
<er[lKC*. °>li:.a + || (6.14)
,v_
from which it follows that the limit function it for tbe u will have the deriva¬
tive itg from LjfPj’)-
For another case (under which inequality (6-10) was derived) we take as the
basis functions I in Galerfcin’s method die eigenfunctions for the Laplace
operator under a zero boundary condition on the boundary S of the domain fl, so
that
Aij>* = — % \s = 0. (6.15)
For a deduction of estimate (6.9) we multiply each of tbe equations (4.4) by the
corresponding A^c^(t), then sum the resultant equalities over k from 1 to N and
integrate with respect to i. In view of (6.15) this leads to the relation
178
HI. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
— J -?VV • Au* dx dt = J (—^7+ f^kaN dxdt, (6.16)
Q, Q, ‘
the left side of which coincides for £ ■= £g with due integral from which inequal¬
ity (6.9) was derived. In the same way one deduces from (6.16) an estimate that
is uniform in N:
,4ErK«*-'>lt. + I*£ie«r
nH£ t + fi '3^—/ < c‘°- (6-171
JV_
It implies that the limit function it for the it will have derivatives u%% from
L2(QT). From this and from the integral identity (1.16), which the function u
satisfies, it follows that it has the derivative it{ and satisfies for almost all
(*, t) from Qj. the equation £« = dfi/dxi - f; the derivative it, in this connec¬
tion turns out to be an element of L2(Qf) (which is seen fcom the equation). Let
us formulate the proven assertions in the form of a theorem.
Theorem 6.1. The problem
ut~Sj(a‘s(x' *)“*,) — f (*• 0.
“lsr=o, «!,.„=%(■«)■
(6.1®
has a unique solution from V ^ ^ ^2(@7^’ ^0 ® °if sat‘sfies
condition (1.2) and
T
|* I d&i j J
vrai max —r— \dt < 00.
j *€«l * I
If the last condition is replaced by the condition vraimaxg < “ and it
is assumed that S € 0^, then the solution will belong to and will sat¬
isfy the equation for almost all (x, t) from Qy.
Remark 6-2- The first part of Theorem 6.1 concerning solvability in y|**((>j.)
also bolds for the equation
3’au=sa(x. f)ut—J~^(aij(x. t)uXj) = (*, t) (6.19)
with an arbitrary measurable function a(x, t) satisfying the condition 0 < t'j <
a(x, t) </t2 < "• This can be proved by various methods based on inequality
(6.6).
§6. ESTIMATES IN THE SPACE Wig1 (Qj.) 179
For example, lec us consider the auxiliary problems
a(M>(x, (a\fux^j = <^~(x, t),
«lsr —0. «I/.0 = 'J>0(^). (6.20)
where t) and a^.jHx, t) are sequences of infinitely differentiable func¬
tions converging almost everywhere to a and a(.y respectively and such that
i^j/2 < < 2h2 /Q^vraimax^gQ |(?a^"V5<| lit < 2^. If SCO2, then we
approximate S by smooth contours Sm from O2. By virtue of the second part of
Theorem 6.1 the problems (6.20) are uniquely solvable in and for
their solutions um we have inequalities (6.6), which guarantee the uniform esti¬
mate
II\\wh
Because of it one can select from |um! a subsequence giving in the limit the
desired solution of equation (6.19)-
Let us note one more particular case for the equations
S’iusssut~a(x, t)\u — <g~(x, t) £6.21)
with a(x, t) satisfying the conditions 0 < a(x, t) < fi2 < °° and with 3"(x, t) 6
2 ’°(*?7')' consider for them the problem
^l«==ar. «lj7 = 0, o|<_0 = $0(JC)
with i/iq 6 If * (fi). From the relations
— | JS’xU Aa dxdt = — J aT Aadxdt= ^ dx dt (6.22)
0, 0/ <?,
one can deduce the inequality
Ollj, Q-f J a(Au.fdxdt
Q,
<e *■* (|| ux (x. 0) |£ Q+1| <TX 111. 0(). (6.23)
in a way similar to that in which (6.9) was deduced from (6.7). From it and (6.21)
follows the inequality
180
UI. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
-Mn^'OKC*. 0)|!£a + ||<r,!l^(). (6.24)
On the basis of these a priori estimates (6.23) and (6.24), in the same way as
above, it is proved that problem (6.22) has a solution u possessing derivatives
ux and ut, as well as uxx when a(x, t) > 0, and satisfying the inequality
J [a* \ a (A«)2] dx dt -f- vrai max || ax ||£ a
< C (?)^ j if I dx dt + J (x, 0) dxj , (6.25)
in which c(T) depends only on T and
Remark 6.3* In a manner analogous to that of Theorem 6.1 the unique solv¬
ability of the first boundary value problem (and the other basic boundary value
problems) is proved for the complete equation (1.1) under ? - dft/dx. - / £
L2(Qt) and the following assumptions regarding its coefficients: solvability in
W\>l(QT) under the conditions
Idaii (x, t) I
bt (x. t). < (i, (/),
l
T
jni(t)dt<oo, a£Ln t (QTy,
0
solvability in under the conditions
d&u ... _ 1 n 1
■gjj • ~ + —"2 • r <°° (6.26)
and a € r(Qf), r < “ and
7 + £ =
1,
for
n> 4,
T + W~
1. ?>2
for
ft =SS 3,
r> 4,
g = 2
for
« = 3,
r> 2.
a
1!
for
n — 2.
We assume that the coefficients a.(x, t) are equal to zero (which is equivalent to
adjoining the terms
§7. AN ESTIMATE OF mixQT\u\
181
to b’U 4 and du respectively). The conditions on are the same as in
1 v
Theorem 6-1. For r —* <» conditions (6.26) and (6.27) must be replaced by the
condition of uniform smallness of the norms ^da-./dx^, and It°lln/2,Xp
for small p and any t from [0, 71-
Estimate (6.10) is relative to the number of exact estimates. In §§14 and 15
we show how it may be used to prove the solvability of problem (1.9) in ^2’*^
without resorting to approximate methods.
§7. AN ESTIMATE OF max£r|it|. THE MAXIMUM PRINCIPLE
Let us assume that the free terms and the coefficients of the minor terms
of equation (1.1) possess somewhat better properties than in §1, namely,
S'# 2# 2A a; f
li-i ;-i i-i
<l*i.
r. Qj
where q and r are arbitrary positive numbers satisfying the condition
(7.1)
7 +
2?
with
^[20^.»]. r€[i—r- °°]’
0 < X, < 1, for b>2, (7.2)
9611. 00]. T=W]’
0 <x, < for «= 1.
We keep the same assumptions concerning the coefficients a..t namely (1.2).
Let us show that in this case any generalized solution u from y\’°(.Qf) of equa¬
tion (1.1). is a bounded function. More precisely, let us prove the following
theorem.
Theorem 7.1. If the coefficients and free terms of equation (1.1) satisfy con¬
ditions (1.2) and (7.1), (7.’2), then, for any generalized solution u (x, t) from
V\’°{Qt) of equation (1.1) not exceeding k on Tj., the quantity vraimaxQj.u(x, t)
is finite and is estimated from above by a constant c determined only by n, k and
182
HI. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
the parameters v, /j j, q, r entering into conditions (1.2), (7.1) and (7.2), with the
j**
dependence on k being linear.
Remade 7-1. The constant c also depends on mes Qj'- it increases with an
increase in mes Qj-. Below (§8) we give local estimates of max |u |, thanks to
which max^|u| can be estimated for domains Q of any measure, including un¬
bounded ones.
Remark 7.2. In conditions (7.1) and (7.2) q and r can be different for dif¬
ferent coefficients and free terms. The necessary generalization of Theorem 6.1
of Chapter II is given by Remark 6.1 to it.
Remark 7.3- The assertion of the theorem concerning the boundedness from
above of vraimax^ u(*, t) remains in force if instead of (7.1), (7.2) the 6(- are
subject to the requirement
2 b]
n, n, Qt
(7.3)
where and are arbitrary numbers satisfying the conditions
n
for fl > 2
for n
1.
(7.4)
<7, 6 [1,€ [1,2]
This generalization is proved in the same way as Theorem 7.1. The neces¬
sary small modifications will be indicated at the end of the section.
In the same place we will also describe the case when rj ■= <*>, ^ n/2 and
n > 3 in condition (7.3).
Remark 7.4. The assertions ot Theorem 7.1 and Remarks 7.1—7.3 remain
valid for functions u(x, t) from satisfying instead of (1. 16) the corre¬
sponding inequality
/0(u, )?) < J<!>0(x)ii(x, 0) dx,
o
o
in which r)(x, t) is an arbitrary nonnegative function from IPj’HPj.) that is equal
to zero for t = T. The proof of this is the same as the proof given below ol
Theorem 7.1.
§7. AN ESTIMATE OF max<Jr|«| 183
Remade 7.5. Since ~u{x, t) satisfies an equation of die same type as
u(x, t), all theorems concerning an estimate of vraimaxu from above give corre¬
sponding assertions concerning an estimate of vraiminu from below.
For a proof of the theorem we use identity (2.12), in which we take
t|(jc, t) = u^(x, t)= max {«*(*, t)—k; OJ,
This is possible since the function uh and consequently the function iij^
(see Lemma 4.9 of Chapter H) belong to and Since
for almost all t
f "*,(*. <’(■*. Odx — j-jf J | *#'(*> t)]2dx,
& a
we can rewrite the resultant equalities in the form
r,
- f t)fdx\lz,a'+ I J [(<*„«*, + aiu
s o Si
++« + /). ■;"]*** = 0. (7-5)
We let A tend to zero in (7.5). Since a €V ^(Qf), it follows that converges to
u in the norm of 8 > 0, and therefore the "undercut” functions
converge to in the same norm (see Lemma 4.5 of Chapter II). As was
shown in §2 in the deduction of (2.13) from (2.12), from here and from the assumed
properties of the known functions the limit relation for (7.5) will be
i J [«w(*. o]5<**i;:h J J f(v« +«,«+/,)
2 0 Q
+ (A.u^ + au -f /) j dx rff = 0, k>k. (7.6)
We denote die set of points x from O at which u(x, I) > h by /4^(f). Since, by
condition, u(x, 0) < A it follows that /q[b^(j:, q)]2dx = 0. We leave in the left
side of (7.6) the first two trivially nonnegative terms and estimate them from be¬
low, using the condition of ellipticity, while the remaining terms are transferred
to the right side and estimated from above by means of Cauchy’s inequality ((1.2)
of Chapter II). Namely,
184
III. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
-g- J (at, *,)]’ dx -j- v J ( dx dt
a o Ak a)
i,
<~J J [(fl/“+//)+« + /)(«-*)] dx
dt
o Ak«)
<1
<J j* \T“i+72(«y+f%+i“i
o Ab it) L i
dx dt.
(7.7)
from which, assuming A > 1, we get
min I < J J 3S(x. t)[{u-k?+tfi\dx dt.
U W
whete
&{X, t)
= 2
4Sfl5+4£/Hl5>?+2M+i/i
/-I (-1 f-l
For an estimate of the tight side we use Holder’s inequalities ((1.6) of Chapter
II). This gives
min {?'
+ ? ' oan (7.8)
‘ fTp ttp v'»w
Here Qt^{k) is the set of points {%, 0 of the cylinder ^ = Qx(0, fj),
at which u(x, 0 > k. We estimate the norm of the function u standing in the
right side of (7.8) by means of inequality (3.5) of Chapter II in the following
manner:
i(a—*)2i , ■
ft- 7=1r> «<,<*)
t-
V. r. e(i
1. '• Of.
1-t
(7.9
§7. AN ESTIMATE OF max<pj.|a| 185
Here MU) = [*l mes^Ak{t)dtP q = 2q/(q - l), r - 2r/(r - l), § -
V(1 + k), r = 7(1 + *), with k = 2Kj/b. It is easy to calculate that, by virtue of
assumptions (7.2), l/r + n/2? » ra/4 + Kj/2, 1/? + re/2? = 1, and for n > 2 we
have
while for n = 1 we have
?6t2(l -+ k). oo], r‘€[4. 41^]-
Therefore is embedded in L'}'?(Qtj) and IIq can be esti¬
mated in terms of |u^|gt , using inequality (3.4) of Chapter II; namely,
2x
— f (t <P2|«Wf0. (*7 W. (7.10)
where n(k) has the same meaning as in (7.9). We write the second summand in
the right side of (7.8) as follows:
I, 7 \ 2? 8 (i+x)
= /s2f f mes« Ab(t)dty = *5n ' (*). (7.11)
vo
since 7/q = r/ q, while r = F(l + k). Taking into account estimates (7.10) and
(7.11), we obtain from (7.8)
min(4-
r 2x 2 (14-x) -j
<S3%.r, «,,<*> LPVWK’Iq,,-*-*^ r (*)]• (7.12)
Let us choose tj so small that
2h 2*
<?,,(*>/l?(®esC9? <Tmin v}. (7.13)
1)^For q - n - I the numbers q and q will be equal to oo and instead of t4,k) »
A*(tydt we will have (here and below) the measure of the set of points t from
the interval (o, for which mes ^0) > 0.
186 III. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
A
Then from (7.12) it follows that for k > maxil, k\ s kl
£7.14)
where
Since k in (7.14) is positive we conclude from (7.14) on the basis of Theorem
tiplied by Aj.
Analogous arguments are valid for the cylinder Qs = fix *J +j], as long
as their altitudes are subject to requirements (7.13)- Thus, after a finite number
of steps we obtain the estimate
with a constant c that could be explicitly expressed in terms of known charac¬
teristics (see inequality (6.2) from Theorem 6.1 of Chapter II). Theorem 7.1 is
proved.
Corollary 7.1. Suppose all of the conditions of Theorem 7.1 are fulfilled
stant determined, like c in Theorem 7.1, by known qualities.
This assertion fallows from Theorem 7.1 if it is applied to the function
v(x, t) = —U(X, t).
As was noted in Remark 7.3, the assumptions (7.1), (7.2) concerning the bj
in Theorem 7.1 can be replaced by the somewhat broader assumptions (7.3), (7.4).
Let us explain what modifications must be made in this connection in the esti¬
mates just given. The term
in (7.7) is estimated by means of inequalities (1.6), (3-4) of Chapter II as follows:
vrai max a (x, f) ^ ckx
Qt
(7.15)
except it|j-j, < k, which is replaced by the condition “|j-j. > k. Then a solution
uix, t) is bounded from below and vrai minq^u is estimated from below by a con-
§7. AN ESTIMATE OF max<2rju!
187
JO. Q,
r„ Qt, <*>
r\M
2“ i“(S)lo,. • (7-16)
I'"1 l?i,r„ «,,(*)
We wish to cancel the expression on the right side of (7.16) by means of the
similar expression on the left side of (7.14). This is possible, if, for example,
<-Imin(l;v). (7.17)
»i_1
For rj < oo we obtain (7.17) at the expense of choosing a sufficiently small tj.
As was shown in §2 in reducing the energy inequality, in this case the whole
cylinder Q j can be partitioned into a finite number of cylinders Qs = Q x
[ts_j, ( ], for each of which (7-17) will hold, with the number of cylinders being
dependent only on n2 and rj from (7.3), (7.4).
But if fj = oo, then such a partitioning gives nothing, since
it o a a /»
2*?J = vrai max J 2 $ (x, t)
'-f.««<,<») h~l Sj.vo
We assume in this case that we know the quantity n^ = ||«|j M qj, and the
fact that
A (k) == vrai max
o <<«sr
(7-18)
for vrai max^ ?<. y- mes 0. By virtue of the inequality k mes A^it) <
||u(x, f)|j q the quantity vrai max0<i<y mes <4^(l) < k~ —* 0 for k —> <*>, and
therefore inequality (7.17) will be fulfilled for all k> where &2 is such that
^-A(*j)<i- min jl; vj.
Thus, suppose (7.17) is valid. Then, substituting the resultant estimate of
|/| and the estimates of the remaining terms obtained earlier into (7.7), we arrive
at an inequality differing from (7.14) only by the coefficient in the left side. And
188 «I. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
this inequality, by virtue of Theorem 6.1 of Chapter II, implies an estimate from
above for vraimax^u. For r, = o« it will also depend on 1^2•
Let us show that for generalized solutions u(%, t) of equation (1.1) from
V\'°(QT) the maximum principle is valid in the following form.
Theorem 7.2. Suppose u(x, t) is a generalized solution from ) of
equation (1.1) whose coefficients a^, 6. and a satisfy the conditions (1.2)—(1.4),
a(x, t) > 0, and a. a f. s /s 0. Then for almost all (x, t) from Qj
min j 0; vrai rain u (x, /) (x, t)
I rr
<max
\ rT
-<max j 0; vrai max a (*, t) p (7.20)
k ~ max {0; vraimaxj-^ul. From it, by virtue of our assumptions and the choice
For a proof of the right inequality of (7.20) we take inequality (7.7) for
max {0; x
of k, we get
"2
We estimate the right side in the same way as die term b.u u was estimated in
1 Xl
?K>(*• '.)£a + vK>£«(,<- J «■ v-™
e right side in die same way as
§2 in deducing the energy inequality; namely,
f b,uxJ»4xdt < J (-£ u^±wAdxdt
h, «<, <-> J
'■“t*. JL „
I'-'
l'-1
From this and from (7.21) it follows that
min (1, v} 1““%, • (7.22)
1'-* h.r.Q,,
For (j satisfying the condition
§7. AN ESTIMATE OF maxQT\u]
189
•HE- <min{l,v},
, r. Oi
r, Qi
(7.23)
we obtain from (7.22)
i.e. for almost all (x, t) from Qt^ the function u(.x, /) does not exceed k =
and so on, we find that the right inequality of (7.20) is valid. Analogous arguments
with the function -»(*, t) lead to a proof of tbe left inequality of (7.20). If a = 0,
then the zero can be removed from (7.20).
Moser proposed a different method of estimating vrai max\u(x, f)| for solu¬
tions of linear equations of elliptic and parabolic type [*7a, b]i which consists in
obtaining recursion relations between unboundedly increasing powers of u that
turn out to yield a conclusion on the boundedness of vrai max \u(x, t)|. He carried
out this idea for equations (1.1) with «. a b. = a a f-* /( = 0. And with the use of
tbe estimates of minor terms given above it can be applied to the complete equa¬
tion in the following way.
For the sake of simplicity we will consider the solution u(x, t) to be hounded
and equal to zero on Sj.. In identity (2.12) let us put
where uh is the averaging of u with respect to t defined in (2.10), t) is an
arbitrary smooth function, and ~ |r|s with s > 1. It is easy to see that such
a function is admissible for (2.12). We let <f>(uh(x, t)) = v^(x, t) and represent
the first term of (2.12) in the form
After this we pass to the limit as h —» 0 in ail of the terms of the equality. This
is possible, and the limit equality can be written in the form
J “**'(#*)«(*) P**<«==-g- f V\hp<lx I'- f V^dxdt.
J J vV <** [' 4- J [ — -f- a,jUx^jVffP
-+ at /vxtvxjg 4- 2 auvXji&Xt + (ata 4- ft) (tp"a^<5
+ <Pfvxp 4- 2<p't»K^) 4 {btuXl -\-au + /) <p'®£2] dx dt = 0, (7.24)
190 III. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
where v = We put £ s 1 in (7.24) and transfer to the right side all of the
terms except the trivially nonnegative ones: the first integral and the second and
third summands in the second integral. Then we estimate the terms transferred to
the tight side in the same way as was done above with the use of assumptions
(7.1) and (7.2). This leads to the inequality
in which the constant y is determined by the quantities v, ii^, q and r from con¬
ditions (1.2), (7. l)-(7.2), T and mesD, while F « 2r/(r - 1), q = 2q/(q r l).
As above, we let $ = q( 1 + k), 7 «= r(l + k), k = 2«j/b. The parameters Q
and r are subject to restrictions (3.3) of Chapter II, and therefore by virtue of
inequality (3-4) of Chapter II we get
(7.25)
Hence (7.25) implies the estimate
We will take as s the sequence of numbers (1 + «)* with k = 1, 2, ■ • •. If
the norm
is denoted by then inequalities (7.26) for v = +K^ are written as
are written as
©*<(I-+-*)*PY(<Dlt3-H), *—1. 2
(7.27)
Hence we conclude (see Lemma 5.6 of Chapter II) that
®*<I2(1+H)max|l. pv)l *
xa+xf-55 *(®0+1) *’
®*<I2(1+H)maxll, pv)l
X(l+*)“
<®o+ if**’*.
and, consequently,
vraimax|«|i=i|MI == Km
o. r *-*«
vralinaxja|5||a||(
Or
> I 1 + K
<2* maxfl, Pvl’r(«+K) (®o+l). (7.28)
§8. LOCAL ESTIMATES OF max |u|
191
The quantity 4»q = Qj, in the tight side of the inequality does not exceed
(3 |o|qj,, and therefore by means of the energy inequality it can be estimated
in terms of the known constants v, ftj, q and r from conditions (1.2), (7.1) and
(7.2). In this way the desired estimate of vrai max^ is obtained.
boundedness on F^. Suppose conditions (1.2) and (7.1)—(7.2) are fulfilled. We
take an arbitrary cylinder Q(p0, rQ) = \\x - Xq\ < p0; t0 < t < tQ + r0i € Qj and
in (2.12) we put 7) (at, t) = £Z(x, t)«^(*, *) = £2(*, e) max iu^(ae, t) - k; Oi, where
£(x, t) is a nonnegative, continuous, piecewise-smooth function that does not
exceed 1 and is equal to zero on the lateral surface 5,^ of the cylinder
Q(Pq, tq) and outside it. After substituting this function into (2.12) and passing
to the limit as h 0 we obtain an equality similar to (7.6). From it, since the
upper limit of integration with respect to t is arbitrary, we will have
§8. LOCAL ESTIMATES OF max|u|
Let us now obtain local estimates for arbitrary generalized solutions
u(x, t) from Vl'°(QT) of equation (1.1) without an assumption concerning their
* iu'v, t)ux, <ar
‘i Ae(f>
+(®i“x + aa — /) (u — fe) S2 — (a — *)2K,} dxdt = 0,
to ^2 <C ”1" T0'
This equality implies tbe inequalities
(8.1)
a
££+*> jl)
££+*> jl)
U. , . .i
-fft2 ff I VdxYdx
U I
r
(8.2)
l)For q 3 oo (which occurs only lor nal), in place of the term on the last line we will
have where for all t for which mcs/4^(i) « 0 it is
necessary to assume that vrai^ *s et5ua^ EO 2ero-
192
HI. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
with the same values of the parameters f,k as in (7.14). The symbol A^t),
as above, denotes the set of points * of 0 at which u(x, l) > h. The constant
yj depends on known quantities in roughly the same way as y in (7.14). The
deduction of (8.2) from (8.1) is the same as the deduction of (7.14) from (7.6).
Only this time we can satisfy condition (7.13), more precisely, the condition
t<*>
'o U* <0 I
V" dt
nk
at the expense, not necessarily of smallness in fg, but, more generally, of small¬
ness in the measure of the set where £(%, t) is different from zero. As was in¬
dicated in Remark 6.4 of Chapter II, from (8.2) we get inequalities (6.9) of Chap¬
ter II, and consequently u(x, t) 6 ^((?(pg, ?q), y, r, k, k) and p0 and satis¬
fying the condition
P li IU r, Q(„, T„ V (*,Po> » < min {i; . (8.3)
This is also true for the function -uix, t).
Suppose u(x, t) does not exceed some number k on F' C Fy. We take an
arbitrary cylinder Qip0, Tq), the intersection of which with Tj. is not empty and
belongs to F’, and a function J)ix, t) that is equal to C^ix. t)u^\x, t) in
QipQ, ?q) H Qj and to zero in the remaining part of Qip§, rg). Such a function is
admissible for (2.12) when k > k, since it is an element of W}ji'liQ{p0, r^) fi Qj.).
Substituting it into (2.12) and carrying out the same estimates as for interior
cylinders, we see that u(x, t) satisfies inequalities (6.10) of Chapter II and is
by the same token an element of & iQip0, ig) H Qj<, y, f, k, K).
The membership of u{x, t) and — u(x, t) in classes 8iQip0, r0h •••),
ro^ Qt' ‘ ’ ‘ ^ ‘“plies, on the basis of Theorem 6.2 of Chapter II, the
following theorem.
Theorem 8.1. Suppose conditions (1,2), (7.1) and (7.2) are fulfilled for equa¬
tion (1.1). Then any generalized solution u(x, t) from V^iQ-p) of it has a
finite vraimaxQ>ju| for any domain Q’cQ separated from Fj. by a positive dis¬
tance d. The quantity vraimax^' |u| is estimated from above by a constant de¬
pending only on n, ||tt|2 Qj,, the constants v, ft, jtj, r and q in conditions (1.2),
(7.1), (7.2) and the distance d. If furthermore vrai maXp, u - k < <*> for some piece
r of the surface FJ, then vraimax^u is finite for any subdomain QJ C Q f
§8. LOCAL ESTIMATES OF ma* |u|
193
separated by a positive distance d from I -p\P' and is estimated from above by a
constant depending only on flulj k, v, p., fij, r, q and d. An analogous fact
holds for the function e).
Tbe character of the dependence of an estimate for u (*, t) from above and
from below on the quantities enumerated in the theorem can be seen from Theorem
6.2 of Chapter II.
Remark 8.1. The assertions of Theorem 8.1 on the finiteness of vraimax^<|u|
and vraimax^tdtii) remain valid if in regard to the coefficients one assumes
only (7.3), (7.4). This is proved in the same way as Remark 7.3 at the end of §7.
Requirement (7.17) Imore precisely, the requirement
where Q(k, pQ, r^) is the set of points (x, t) € QlPg, fg) at which u(x, t) > A]
can be satisfied this time for r j = ~ not at the expense of choosing a high level
k but at the expense of sufficient smallness in tbe radius Pq (for r ^ < oo (8.4)
will hold if mesQ(pty r0) is taken sufficiently small). The influence of tbe coef-
through restrictions (8.4) on the cylinders Q(pQ, rQ) to which Theorem 6.2 is applied. The
other constants are not affected by the characteristics of the 6t-.
Remark 8.2. In conditions (7.1), (7.2) and (7.3), (7.4) of Theorem 8.1 the numbers q
and r can be assumed to be different for different coefficients and free terms as long
as they satisfy requirements (7.2), (7.4).
A different proof of Theorem 8.1 can be obtained by using the idea of Moser
stated at the end of the preceding section. For this purpose it is necessary to
take a chain of coaxial cylinders
k = 1, 2, • • ■, where *0 is the center of the balls K ^ and £{£, if) is some
smooth nonnegative function, not exceeding unity, that is equal to unity for £ < 'A,
rj<Vt and to zero for £ > 1, r? > 1.
a
(8.4)
1-1 ?„ Qi*- P.. t»)
ficients bi on an estimate of vraimaxg, |u| and vraimax^+w) manifests itself only
belonging to Qj- and a corresponding sequence of cutting functions
194
III. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
In identity (7.24) it is necessary to assume, as above, that v « |u|s, s =
(1 +«)*, and to take £(*, t) equal to 2,^_ j(x, t) k= I, 2, •••. Then estimating
all of the terms in much the same way as above in §§7 and 8, we arrive at a system
of recursion inequalities for
* 111 1 %, t, 11 (S+K) ?, (1+x) r, Qp^
namely,
< ck + 0. * = 1 • 2 (8.5)
with a known constant c > 1 determined, as is also k, by the parameters from
conditions (7.1) and (7.2). From it follows, as we saw at the end of §7, the de¬
sired estimate
] .j-x
vraimax|«|= lim ©£l+*r*<e~(ll“ II? ? 0 +1} (8.6)
Qn fi-*co ' *’ ' Vs0 >
§9. ESTIMATES OF SOME NORMS OF ORLICZ
FOR GENERALIZED SOLUTIONS
In the two preceding sections estimates were obtained for the ooaximums of the moduli
of the generalized solutions u(x, t) from of equation (1.1) under condi¬
tions (7.1), (7.2) on the free terms f.. But if these conditions are weakened, then
the solutions still remain better than what is required for the energy inequality
(2.2). Mote precisely, the following theorem holds.
Theorem 9-1. Suppose conditions (1.2)—(1.4) are fulfilled for the coefficients
of equation (1.1), while the free terms satisfy the conditions
ii/iUv
in which
§9. ORLICZ NORMS
195
fcg[-
2n
4 + «{n-2)
r,€[».*)
fc€[i. f]. '•s6[i. -j^e]
. 4]. /■*€[*. -J] for n>z-
for it = 2,
/or » = 1,
JLj_ n
77 + W
with
L 2 + B (n — 2)
«46(».t]. r4e[«.j)
■g"]’ r^[i. TT0-]
■ •§•]’ •§■] for a**s'3’
for n — 2,
for n— 1,
(9-2)
and 0 being some number from the interval (0, l).1^ If M|s j. is bounded and
[«(*, 0)|*/®€ L2(ti), «Aen
i/®
|«(*. 010 € V* (Qr)-
Bui if nothing is known concerning then the membership of |u
P2«?r> holds for any strictly interior subdomain Q'C Qf.
Corollary. From Theorem 9.1 and inequality (3.4) of Chapter II it follows that
under conditions (9-1) and (9.2) any generalized solution of equation (1.1) belongs
to L rlQf) or Lq r(Q') such that
1 n _n ft
7+ 2?~ 4 0"
We will assume that u is equal to zero on T j. (The proof is not essentially
changed if this requirement is replaced by the boundedness of u on Ty.)
Fot the proof we make use of identity (2.12), established for u and any func¬
tion )j from ^2’°(Qt).
Let us take in it
1-2
*](*, t) = uh(x. Of+iSft*. 0]® •
1) As is easily seen, for 1 relations (9.2) go over into relations (1.6).
196
HI. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
where
M for uh^M,
+£’(*. ,)= **(•*• 0 for o(9.3)
0 lor «,< 0.
All of the summands of (2.12), except the first, have a limit for h —*0. But it is
immediately obvious that the first summand can not have a limit, since i»g, in gen¬
eral, does not exist. We therefore transform the first summand into the more suit-
able form
1 f I mnf'2**dt
o a „ 0
j I
0 Q
(9-4>
Q ItmO
In the first summand of (9.4) the integration with respect to t can be carried out
in the following manner:
« 2 1 j|
I J J /
0 Q 0 Q
= (1 — 6) J ^ dx
a l/-o
Therefore uht^^x' fidxdt is equal to
i I “*WS)r’***| ’ —id-8) J l^fdxf \ (9.5)
a '<-« a l/-o
In these integrals it is now possible to pass to the limit as h —» 0. Substituting
them into (2.12) and passing to the limit in (2.12) as h —> 0, we obtain the fol¬
lowing equality for u:
§9. ORLICZ NORMS
197
+ f J f[(«,“ + /»)Ul %
o a o a 2
+(V'J -t-au+f) J dje (9.6)
In it i/fyix, t) is equal to II for u(x, t)>iI, to uix, t) for 0 < uix, t) < M, and
to 0 for uix, t) < 0. It is clear that 0 < t/tyix, t) < |it(x, ()| everywhere, and that
ijiyx = 0 at those points where <lryix, t) / uix, t). In view of this sum of the first
two terms of the left side of (9.6) is not less than (0/2) Jq B~2dx I*1 * -
0)p/®(£*. The third term is estimated from below, using the fact that at
the points where i/iyix, t) •> uix, t)
aiju*t Ur %=(I - 0 ao% u°~ \
2 r/ I \
> (4 - l) 2 = (2 — 0) 0v [Ur1) J ■
and where 'I’yix, t) 4 u ix, t)
ai Jaxj Ur Ix,=auw^T'>v [Ur 0 J •
so that everywhere
ana*j Ur % >v9 [Ur I] •
Thus the left side of (9.6) is not less than
0min|v; |}|jUr’) dx\ *’+ j[US"'),] dxdt |
-4 0)|»*e. (9.7)
a
All of the terms in the right side of (9.6) are estimated from above by considering
all of the expressions, in a manner analogous to the arguments just carried out,
at the points where - u, and where i/f y = const (i.e., where £ u), and
then choosing a major ant, common for both categories of points (*, /). This leads,
as is easily calculated, to the inequalities
\afl
Ur 2L, I < <2 - e> I ■a> Ur') w % I.
198
HI. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
16i v*r * | < ® k I*#* *) («*s‘') *t I •
M^r).
w'te"! I
*1 I
<2i/iil*»r'l lUr'L|.
'..r
Each of their right sides contains tbe function which we will denote by v.
Substituting the inequalities obtained into (9.6) and estimating the integrals in the
right side by Cauchy’s inequalities with arbitrary e > 0, we get
0min|~; v}(||o(x. *,) j* fi + |K||2QJ
< J K2!"/! +l*i|)jWr,|
%
n | 2
+ + \f\\vt*Ux dt + 1J |«(*. 0)|« dx, (9.8)
/-i J u
Let us denote 9min IH; vl by 2vj and take e = ul/2. Then, after reducing sim¬
ilar terms and maximizing over t, we deduce from (9.8) the inequality
ViMq, <2 j(^2+~i]/il*|2(,'9'+l/lb|8‘9V:Crf<
1 «l,' (-1 J
2
■+ J\u(x, 0) I*9 dJf. (9.9)
§9. ORLICZ NORMS 199
whete 3) = (1/2»'])5y cj(2 |o£| + |A,|)^ + M- The terms on the right side are esti-
maced from above by Holder’s inequality, the embedding theorem (3-4) of Chapter
II and the assumptions on the coefficients and free terms of equation (1.1) in the
following manner:
J SDvHx dt <131|?, r> % II • If, j' % < P2 II a ||f, f> ^ \vfQh,
Qi, 11 *'
JS/>
Qt, ■—1
.2 <»-0)
dx dt
,2(1-9)
Hwlii4(i~e). ^(i-O). q(i
<P2"-e,p/2J m^-b\
I*'-1 V «(,
J I /11 c|2-6 dx dt < II / n,3, ,,, 0(i || V lli-p-5 -3 g-e, Qh
<r-e|l/lk.,3.«,,I^V
Here, as elsewhere, the parameters with a bar and without a bar are connected by
the equality q = 2q/(q - 1). For these estimates we made use of the fact that by
virtue of (9.2) the space is embedded in the spaces
L«. r(9t)' ^3,0-0), ;,(i-8) W,)and 5LJL f iziCQt.h
y* 2 * 2
and that for them inequality (3*4) of Chapter II is valid. Because of these inequal¬
ities and Young’s inequality'it follows from (9.9) that
i
in i.2
jbl«„ <P*I®I!«. + I-Ill“(*. 0)1° III
V Q,
< [pJ II 9 II,, r, ,fj + (1 - 9) e^e + (1 - |) «*- •] | v\|(i-HT (f,). (9.10)
where
200
III. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
+ y || |«<*. 0) \i ||, „ + e" V(tr" II / II®, ,r
and ( isao arbitrary positive number. Let us take an f such that
1 2
(i_e)ei-»4-(i -f)e2-9 =
V|
8 ’
and choose /j so small that
PZII ® Ik. M?(, *f"' (9.11)
This is possible since r < oo. For such (j and e inequality (9.10) gives
v»
i •(( i uuv
lQ(l
Letting U go to infinity, we obtain from this the desired estimate for -
maxiu(x, t); 0).
l<«m>*|,li<(^-Jr(*i>)T<c. (9-12)
Proceeding by small increments in t, we obtain an estimate of for
all of Qj in much the same way as was done above in §4. Let us explain.
Inequality (9.12), more precisely, tbe inequality
| («W»)* L <{~F(h-v tkjf (9.12’)
is proved by us for essentially any cylinder Qt^_ j = O x (*£_ j, <j), 0 <
tk- 1 as ^on8 as condition (9.11) is fulfilled for Qi.e., as long
as
ik**-!. , Q <%■ (9.H')
* *>k-i-'k °
The quantity F («^_ i> **) in (9-121) is calculated for Qt^_ j in the same
way as F(t^) was calculated above for j 3 (?o jj- partition [0, 71 into a
finite number of closed intervals [«q = 0, <j], [t j, t^J, • • •, [<s_ j, ** 71 in such
a way that (9-11’) is fulfilled for each of them, with the equality sign in (9.1l')
§9. ORLICZ NORMS
201
holding for ail except possibly the last. In view of this we have
s
(5 -1) (£)' < 2 n' (**_,. t„)=|| as ii; , Qt,
k-l
i.e., the number of divisions s in such a partitioning will not exceed the known
quantity 1 + ((802/i-'1)||5) ||? r Because of this we obtain from inequalities
(9.12*) the desired estimate of for all of Qj> if we note that the quantities
in them are estimated in terms of
and the known norms of the free terms of equation
(1.1).
Thus the membership of t)]*^ in *s established. Analo¬
gously, by considering the function (max {-<*(*, t); Ol)1^9, we prove its member¬
ship in ^(*?y0 and obtain an estimate of its norm in ®y 'be same to¬
ken the assertion of Theorem 9.1 concerning a global estimate is proved. If in
regard to the solution it it is known that |u| | r^ is bounded, then, taking rj(x, t)
in (2.12) equal to uk(ifi^^)2^e~ 2£2 with a smooth £(*, t) equal to zero on Fj.,
and carrying out the analogous estimates, we obtain the validity of the second
assertion of the theorem.
Thus Theorem 9.1 is proved.
If 6 = 0 in conditions (9.1), then Theorem 9.1 implies the summability of the
solution u with any positive power. But if the coefficients of equation (1.1) are
subject to requirements (7.1), then it is possible to say more concerning the solu¬
tion: it is exponentially summable. Namely, we have
Theorem 9.2. Suppose the coefficients of equation (1.1) satisfy conditions
(1.2), (7.1) and (7.2), while the free terms fi and f satisfy conditions (9.1), (9.2)
with 6 = 0. Then for any generalized solution from VI?0(Qj.) of equation (1.1)
with vrai max j-^.|u| s k < <*> there exist constants c > 0 and c^ such that
J exp \c\u(x, t) | ] dx dt < ct.
Qr
with c and Cj being determined by the known parameters in the enumerated con¬
ditions.
The proof of this assertion is carried out in almost the same way as the proof
of Theorem 7.1: we prove a certain integral inequality (of the same type as ine~
quality (5-32) of Chapter II), from which by means of Lemma 5.10 of Chapter II
we reach the desired conclusion.
202
IIL LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
The inequality, as in §7, is deduced from equality (7.6) with the use of
estimates of the various integrals by Holder’s inequality. All of the summands
except those containing the factors f. and / are estimated in exactly the same
way as in §7. And the summands with f ■ and / are estimated in the following
manner:
JL
J \f\(u—k)dxdt^ njs (k) |j/||?J. rv Q, || at*) (|^, jf, Qt .
<?(, (*>
J (k) 5J/y|
0/| (*)»-• i-1 IB,, Qtl
where
2r,
■ 1 ’
-, i = 3. 4.
|i, (ft) = J mes "i A„ (t) dt. Ak (t) = fjcgQ: «(*,/)>*| D.
0
A
As a result we obtain the following estimate for k > max{k; 1}:
J (« — kfdx f ||«,||’ q<
<*)
LM'i)
; f I fv i] S 2 I(u“A)2+k2]
u Af'V ' /-1 i—i *
+ III III,, 0,t ■+ -k vf* (*) II f <V <?,,
dxdt
Ilf-1
k,, tv 0,
(9.13)
(9.14)
The first term in the right side is estimated in the same way as the right side of
(7.8) was estimated in §7, while the integral is estimated in terms
1) For «oo one must assume »l*'m*x(X(<{,me* while for = oo
one must assume that is equal to the measure*of the points of the interval [0, !]] for
which mes/4fc(<)> 0. The case K • oo occurs for r,' “ 1, n > 1, while the case q- ~ oe is en¬
countered only forn • 1, when q{ " L
§9. ORLICZ NORMS 203
- I*(%, , using inequality (3-4) of Chapter II (it is applicable since qj and
Fj satisfy conditions (3-3) of Chapter D). After this we carry out a maximization
over t, taking into account that all of our inequalities are valid for any from
[0, X]. This leads to the inequality
min U; 1} | p < p’ || Sx |1,, r> Q nT (ft) | |
M *1 *1
2Q*x)
+ *2 ll^i I7 (A) + eb2i«<*>|^
1 I i-i
n/4 w.
"<■ V
where
while *?, k and **(&) are die same as in inequality (7.9)- We put < » (l/02)mm|l/8; v/Z\
and find from the condition
P2H^iil».r. <?,_ <*) V mesQ? jj.
A
As a result, for k > max \k; li we obtain
r g(i+x> _2_ I
min {7: 7 } IB<*’ Iq, ^ c I*2** 7 (*> +11/3 (*) + f*/4 (fe)J •
(9.15)
with
c = max 111 S>i SI,,'3. V
a ||
£/?
The conclusionof the theorem now follows from Remade 5.4 to Lemma 5-10 of
Chapter U.
One can also obtain local estimates of fg, exp[c \u{x, t)\\dxdt for solutions
u{x, t) from y\*\Qj) concerning which die boundedness of |i*| on Fy is
known. The necessary modifications for this are analogous to those carried out
in §8 in estimating max^, |u( for any u from Local estimates of
WgrQ' /<>' exp[c|uU, t)\]dxdt are given in [5**].
204
in. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
§10. AN ESTIMATE OF HOLDER’S CONSTANT.
HARNACK’S INEQUALITY
In §§7 and 8 we proved that under conditions (1.2), (7.1) and (7.2) each of
the generalized solutions u(x, t) of equation (1.1) from the space V ^(Qf) is
bounded. It turns out that under these same conditions u belongs to Ha' a/2(Qp)
with some a > 0 and the norms |u|^? of it can be estimated in terms of IMI 2,Qt
and the known parameters in conditions (1.2), (7.1), (7.2). Namely, the following
assertion is valid.
Theorem 10.1. Suppose u(x, t) is a generalized solution from Vof
equation (1.1), the coefficients and free terms of which satisfy conditions (1.2)
and(7.1), and suppose vraimaJtQr|it| = U. Then u€ Ha,a^2CQjr), and the norm
W$, for any Q’C Qj. separated from Tj by a positive distance dt is estimated
from above by a constant depending only on n, M, the parameters v, p., iXy q and
r from conditions (7.1), (7.2), and the distance d. The exponent a > 0 is determ¬
ined only by the numbers n, vs p., q and r.
If in addition it is known that on some piece F * C
satisfying condition (A), ^ then u(x, i) € a^^iQ-p UF#) and the norm |u
for any Q' belonging to Q f and separated from F j\V' by a positive distance d,
is estimated from above by a constant depending only on n, U, v, p., fl j, q, d, /3,
and the constants afl and of condition (A). The exponent a belongs to
the half-open interval (0, /8] and is determined by n, v, fi, r and q.
Thus, suppose u is a generalized solution from Vj,0{Qt) of equation (1.1)
with vraimaXQj,|tt| » M. Equality (8.1) is valid for it. We will assume that £{x, t)
in (8.1) is different from zero only for x £ Kp. From (8.1), using inequalities (1.1)
and (1.2) ot Chapter II, we obtain for k £ [-M, Ml
i
i|! «<*>(*, t)i(x, <J(o+v J J (uMfVdxdt
, *•Kp
< J* J [e,n4^2 + t*(“ — + ei“2^
A*.p(0
+5r £ (fl2/M2+/D & + “ *)! %+2 2) (a*M}+
1 I t
Condition (A) is defined on p. 9 in § 1 of Chapter L
§10. AN ESTIMATE OF HOLDER’S CONSTANT
205
-j-e,«y? +-4^26<(a_ft)2£2 + (,fl|jM -H/D2^2
1 t
-f(« —AjaCKljrfjcrf*. (10.1)
Hence, choosing fj = W2ifi + 2), reducing similar terms and maximizing over f,
we will have
I max || «<*>(*. 0«*. Oil*,* +* f {tfpfVdxdt
P Q(P, t)
<^n «*»>(*. *)C(*. *D)l&/fp+cjV*>)a(£+ S| £,!)<**<«
Q(P. V)
+ (M*+ 1) J 3>x(x, t)dxdt. (10.2)
Q {*, Pi T)
where
c= 2[l + 2|,<,*v-^-)].
a,(X. 0=2(2+t±i)%a* + b] +//)^+
/-1 /-1
+ 4(|«|+|/|)£J.
Q(p. T) = /fpX(/0. «0+T);
Q(k, p, t)={(jr, 06<3(P- t): tt(x, ()>k).
The last integral in (10.2) is estimated, using inequalities (1.6) and (3.5) of
Chapter II, in the same way as the integral J®(%, t)dxdt from (7.7) is estimated
in (7.8) and (7.11). Namely,
. P. t)
2(1 + 10
<^n 7 (k.j,p,xy
(10.3)
\ H /
where
M-t T
Cl “ I! II,. r. Qr■ •*(*■ f* P’ T)= J meS ’ P <*> dt 1>>
*) For 9-9 soo one must understand by fi(k, 0, p, r) the measure of the sec of points
t EroinflQ, «o+H for which mesA^ J,t)>Q.
206 HI. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
while the numbers g, 7, q and f' are the same as in (7.9)—(7.12).
Substituting this estimate into (10.2), we arrive at the inequalities
max |J<jc, /)S(*. 4-v|||al*’|i||^ 9(p, x,
t9+ X P
P QiP. t)
2{1 + X>
-fC,(M2+I)ti ' (ft. jr. p. tj. (10.4)
The same inequalities are also valid for the function -u(x, t). It follows from
them that the solution u(x, t) is an element of class M, y, 9, 8, k) with
8 = °a and y determined by v, c and c ^ from (10.4). By Theorem 7.1 of Chapter
II a will be an element of Ha’ a^(Qp) with some a > 0. But the a guaranteed
by Theorem 7.1 depends on y, while y in our case depends on Jtf and Hy In
order to achieve independence for the HSlder exponent of u from M and n j we
consider (10.4) only for r < p2 < p^, where pQ is determined by the equality
x nx
£-(vW*+Ok/p*,2 =1. (10.5)
For such p and r inequalities (10.4) imply tbe inequalities
J|«<V. 0C(*. 0llUp+v|||fl-*>|?| t)
Q(P. i)
4 , p. t||. (10.6)
By virtue of Theorem 7.1 of Chapter II an<j the remark to it, from these ine¬
qualities it follows that u(x, 0 belongs to with some positive
(in general, less than a) already not depending on M and But tbe Holder
constant <»> ^q! will depend on M and fly Thus the first part of Theorem 10.1
is proved. The two other assertions of the theorem are consequences of Theorem
8.1 of Chapter II and the fact that inequalities (10.4) and (10.6) are valid for
v(x, t) = ±u(x, i) and for the cylinders Q(p, r) intersecting Tp as long as the
levels k in them are subject to the condition k > and all of
the domains of integration are replaced by their intersections with Qj>.
§10. AN ESTIMATE OF HdLDER'S CONSTANT
207
In §10 of Chapter V we give another proof of the Holder continuity of gen¬
eralized solutions, and we give it directly for quasi-linear equations, of which
equations (1.1) are a special case. We will illustrate it here by the example of the
simplest representative of equations (1.1), namely, the equation
assuming only that condition (1.2) is fulfilled.
As is known (see Lemma 5.8 of Chapter II), for a proof of the Holder contin¬
uity of u(x, t) and to-obtain an estimate of the Holder constant it is
sufficient to show that tbe oscillation of u in an arbitrary cylinder of standard
form is strictly larger than its oscillation in a smaller coaxial cylinder of the
same form with the same vertex.
Cylinders of standard form are cylinders i|x - x°\ < p, 0 < I - tg < p2\
with some 0 j > 0, which in the argument is fixed, and arbitrary x°, £g and p.
Their vertices are considered to be the points (*°, t0 + 6 jp2). Since equation
(10.7) and condition (1. 2) are invariant with cespect to the transformation of co¬
ordinates x = x - x®/p, t = t - tg/p2 and the transformation of functions
u(x, t) = t) - c), instead of studying u with arbitrary vraimax^^,|»| in
arbitrary cylinders of standard type it is sufficient to prove that for any solution
u(x, t) of equation (10.7) defined in the cylinder *?2 “ ^2 x & “ H*l 2;
0 < l < 61 and varying in the range [0, 1], with die cylinder Q^ x {%$, 0) -
||*| <1, %$<t<61, one has th,e inequality
OSC {a. Qil <(1 —6j,)osc (a, Q5) =s 1 — 60, (10.8)
where 6 and 8Q are some positive constants.
Consider u(x, 0) in Kj. Let
mes|jcg/fi: u(x. 0)< ^ > ymes/fi; (10.9)
then all subsequent arguments will be carried out with the function u(x, t).
Otherwise it is necessary to take the function 1 — u(x, t). Since u{x, t) is a
solution of equation (10.7), inequality (10.4) with £ = £(x) has for it the form
max || «'*'. OSWlIj/f, + v III “it11 Sllj, q(2, «)
o<<«9
0)tWli^t+4n J dxdt,
208 ni. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
from this and from (10.9), by virtue of Remark 7.2 of Chapter II and Lemma 7.1 of
Chapter II, it follows that
mes jjcgKi: «(*, f) < 1} >6mes/C„ <€|0, 6]. (10.10)
with some positive 6.
Let us consider in Q2 the function <f>(u) = -ln[8(l - u + *)], where e is an
arbitrary positive number, which we will later let tend to zero. The function
is bounded from below and is negative for u < 7/8. If we prove that v{x, t) =
<f)(u(x, t)) is bounded in Qj for all f > 0 by some constant JUj, then in Pj we
will have
8(1 - u) > e~Ml
and by the same token (10.8) with Sq = e ^V8-^
For the sake of simplicity we will assume that it(x, t) is a classical solution
(in Chapter V a direct examination will be^conducted for bad solutions). We mul¬
tiply (10.7) by 0 '(b) £(x, t), where £ € W ^K2), and, integrating the result over
K2, we obtain, after an integration by parts in die second term, the equation
1 {v&+a<,v*/s f a‘iv*S)dx ^ °- (1<U1)
From (10.11) we easily obtain for v the inequality
o max J) t Ifa. 4- v il I o**11 £ I!,, Q (2, e,
<max(l.-3i) j («“»)*( | ?, P-t t\i,\)dxdt.
Q (2. «1
which implies, by Theorem 6.2 of Chapter II, the boundedness of v from above in
<?(1, 0/4) - X j x (30/4, 0), as long as the norm |(u|| 2,Qd/2, 30/4) is fi"ite- In
order to obtain an estimate from above for ||t>|| 2 Qd/2, J0/4) we Put ^ ~
5R2(*) y2(t) in (10.11) where the function y (() is equal to zero for t < 0, to 41/6
lot 0 < t < 0/4, and to 1 for 0/4 < l < 0, while 3U*) = ^((*1) is a monotonically
increasing function of |x| for |*| € [0, 2], that is equal to 1 for |*| < 3/2 and to
zero for (zj = 2. Let us integrate both sides of (10.11) with respect to t and in
!•) The idea of introducing similar functions <£(«*) for proving inequalities (10.8) be¬
longs to J. Moser < see (87 a], §5 of Chapter IX in **], and also the end of §8 and the end
of §10 of the present chapter).
§10. AN ESTIMATE OF HOLDER'S CONSTANT
209
the first summand carry out an integration by parts:
r r
f vWi2 dx I + J J aii!ixvx Wtfdxdt
0K'«
— J J [*9l22xx' — ativx 25191 Xs] dxdt.
OK, '
Hence, using inequality (1.2) of Chapter II, we obtain
aifv ,tr Wtfdxdt
IJ Xj
e
<E> J J ctjvxtvx/lV dxdt+ir f / a, ft, W,?dxdt
0 Kt 0 K,
9
+ j f (w>m\2 -f SKY*) dx dt. (10.12)
0 AT, 3
For subsequent estimates of the terms of (10.12) we use condition (10.10) and
inequality (5.4) of Chapter 11. The numbers fj and (2 chosen as follows:
£ j = «= vb2K2/A^2> where /32 is a constant from inequality (5.4) of Chapter
II. As a result of this and die reduction of similar terms in (10.12) we deduce the
desired estimate:
#
J J (V + v'ty dx dt < c.
£ K,i,
*
This, as mentioned above, completes the proof of inequality (10.8).
We have stated here two methods for obtaining estimates of |u|^. Another
proof of this estimate for equation (10.7) follows from the Hamack’s inequality
established by Moser [87b]. This inequality asserts that for nonnegative solutions
in Qp of equation (10.7) we have
vtai max u (jc. t)^c vrai min u {x, t), (10.13)
Q7 ©+
(> vp
where the constant c depends only on v and fi from condition (1.2),
<Zp-Kj,X (<o + Ip2. *o + P*).
Q» — x(?. f P2' \ j >
J «SPx* dx j + J* J
210
III. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
*o < 4o < ‘o + p2,
A M A
and Qp - Kpx(t0 - p , t q + p )C Qf. From (10.13) we easily obtain an inequal¬
ity of the form (10.8) for arbitrary solutions of (10.7), and therefore also an esti¬
mate of the Holder norm |u|^? for any domain Q’ C Qj.
The basic step in Moser’s proof consists in the following. First, by the
method described briefly at the end of §8, one deduces the estimate
vrai maxaMMU <?- < c«aHe. £ > 0>
Qp
Qp (^o* V (*<j) X ~~ P2* tQ>.
Then in an analogous fashion one obtains the estimate
«“lU«p(^',^)<cvral mln“-
%
Finally, for certain e > 0 one establishes the inequality
11 “ !*«■ % (V 'o) C 11 “ I* «p (-V W
Its derivation constitutes die most complicated part of the entire proof.
Moser’s idea has been developed in the papers of S. N. Kru£kov [S2b— d]^
A. V. Ivanov Aronson [132], Kurihara [133"], and Trudinger [136] along lines
that apply to linear equations of general form (1.1), and also to certain classes of
parabolic quasi-linear equations. In [51e], for nonnegative generalized solutions
of a certain class of parabolic quasi-linear equations and, in particular, linear
equations of general form, the following generalization of inequality (10.13) is
established:
vrai max a(x, t) C\ /vrai min a (x, t) -f c2p,'\ ,
I QP /
where d £ (0, l] and where Cj and C2 are determined only by known quantities,
with Cj = 0 in the case of a homogeneous equation.
§11. AN ESTIMATE OF max^loj AND
We will estimate max^,|ux| and for the solutions u of equations
(1.1), restricting ourselves on die whole to interior estimates and assuming for
the sake of simplicity that u belongs to C2’^(Qj-), Let us prove the following
theorem.
§11. AN ESTIMATE OF raax^, |B;t| AND 211
Uf II. . _<»*>• (H-l)
Theorem 11.1. Suppose the coefficients and free terms of equation (1.1) sat¬
isfy conditions (1.2) and
|| da/i 8ai dfi
1 dxk ' 0i' "3*7 ’ *’ a’ dx„ ' J“ ’ llj,, j,, qt
where q and r are arbitrary numbers satisfying relations (7.2) for n>2 and the
relation
0<5tf<i. for n--j. (11.2)
Then for any solution u of equation (1.1) belonging to C2,l(Qj.) the quaitity
is estimated from above by a constant depending only on max^ j.|u|, the
constants v, fi, (ip ? an^ r °f conditions (1.2), (11.1) and the distance from Q ‘
to Fy- The exponent a is determined by n, v, ft, q and r.
Remark 11.1. In conditions (11.1) q and r can be different for different
functions. But merely for the sake of simplicity in notation we will assume that
(hey are the same.
Proof. We differentiate equation (1.1) with respect to each of the and
write the result in the form
“*< + 1 = i [6?/+-£~]> <»•»
where
for m = 1,• • •, re and /4*Q = da./dx^ — 5*a, while S* is the Kronecker delta
symbol. Relations (11.3) and(l.l) are regarded as a system of equations for the
it^, 4 = 0, I,■ • •, n. Its distinctive feature is the fact that the principal part
consisting of terms of the form d/th — dia^.d/dxp/dx- is diagonal and is the same
in all equations of the system. Such systems have been successfully investigated
in as much detail as a single equation of second order; §§1—4 of Chapter VII are
devoted to them. There we obtain, in particular, estimates of max^, |u^| and
{uk)$ in terms of X|.q ||u^]| 2 Qy> the distance from Q’ to rj and known para¬
meters (Theorems 2.1 and 3-1), which when applied to our case gives an estimate
of \uj$ in terms of ||«J2tQr
212 HL LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
Let us give another derivation ot the estimate ol without using
the results of Chapter VII on systems. We first estimate the norms ||ux||s qi for
any s not exceeding some number s^. For this purpose we consider the equality
~ j 2 <“•«>
Q, *-l
where £(x, t) is a smooth nonnegative function, not exceeding 1 and equal to
zero in the vicinity of Sp, and we transform it by means of a double integration
by parts to the form
| |ax(x. <)P+2£2(*. f)dx+ J
-+ (v»,+a‘u+f‘)t (I l2sU*F)X
' / Xk Xt
- {blUxi + aaJrf)-^{ | «x P* dx dt = 0. (Il-K
Relation (11.5) bolds for u from although in deriving it (in tbe inte¬
gration by parts), we made use of the derivatives uxt and D^u. Indeed, let us
take the sequence of infinitely differentiable functions um(x, t) converging to it
in the norm of where Q' is the subdomain of Qf on which £ is dif¬
ferent from 0. For them the left side of (11.4) is-equal to the left side of (11.5),
and as m tends to °° they converge to tbe same expressions for u. Consequently,
if the left side of (11.4) is equal to zero for u, then the left side of (11.5) is
also equal to zero for u.
We leave the trivally nonnegative terms in the left side of (11.5), while the
rest are transferred to tbe right and estimated from above in the following manner:
STS J I «x(*. 0|2s+sS2(*. I)dx+ J(VV
a Q,
4 2tau | ux J21-2 usuvu <uXlXp) dx dt
= ^f2^ux(x. 0)f+2?(x,0)dx- j |_-J_|aj2j+2K,
+[(tS*-,+*a.+S*+S)
- 6* (bjaXj + au + /)] (| ax t uxfi3)^ | dx dt
J l*x(*- 0V^+\Hx, 0)dx+ J 4
a Q,
§11. AN ESTIMATE OF micq, |uj AND (o^
213
+ |l«*l+KI KI+ l^lM + |-J§ I
+ |6,||HxH-|«|M+|/|j|(|«jt|ls «^2)Xj | ] dx dt. (11.6)
We choose f = vl2 and, using the condition of ellipticity (1.2), we pass fiom
(11.6) to die inequality
* j iux(x, t)f+2v(X. odx+j
n Q, <-
+ 2«V | ttx f_2 J] Qjj £2j dx dt
<srb J °)l2*+2^*- ®dx
•+ J [|“,I2s+2(t^tI^ 4
L
where
H-a»(|«,|+ i)S|(i«, I5' %S2), I
i, *1 '1
/.*Ly
-+-|iSI+ IM+M^ + I/IJ.
We estimate due last term in the right side of (11.7) by means of Cauchy’s in*
equality (1.2) of Chapter II, relinquishing a small e to terms with second deriva¬
tives u Namely,
3> (I *> I + I) 21 (I“* I2* Uxp)Xi|
«i»(|«jr| + I)2||«,I**«V|P
+ 2s | a, |2l“2 2 H'pxrf.p -t 21 a,!2*
dxdt, (11.7)
I
214 III. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
■+ Ji«,|2s'2£2+S!j(| «,!+ if ^ | U, fsl?
+1ux|2J+2tl + n^(\ux\+if\ uxJ**£2.
We substitute this estimate into (11.7) andreduce similar terms, assuming € = u/4,
while Cj = v/n. This gives
j(t, 27TT2 J I “Ax. f)f+2t?(x, t)dx
fl
+J [t “L i«,f «i “.r-2 Vvi)*p]dxdt
< 2^P2 J i a Ax. 0)f+2£2(x. 0)rf*
Si
+ J[i«,r2(74ri^i?+(i+^)^)
+ „2(l±i + i)3(»(|«x|+ l)2|Ujcf e2]<ix^
'^■15s'X2 J I *•»(■** 0)|2i+2£2(Jf> 0)dx
a
+cJiiaxr+2(ie,is+^)+(s+i)^2(i^r+2£j+£2)]^*. (us)
where
c = max|l 4«2(l +-^)}-
The term containing 2)^ is estimated by Holder’s inequality:
j j^2(K l*+3P+*i2)rfx<«
<2t
< II Ilf. '• 9, III *x + £2 ll_? r_ „
0-1 ' r-I ' v<
<c1(|||«J:r+,5|far zr n + mes'at'~A (11.9)
\ frr • 7^T • 0/ J
where Cj depends only on fij from condition (11.1), n and ,1/. The first term of
§u. AN ESTIMATE OF ma^,|u*| AND215
the right side is estimated in the same way as the expression
||(«-A)2|| ? ,
■jTT- 7=T,Q<(*>
in (7.9) was estimated in §7. Namely,
IlKr’SlI!*. * q
ff-i * 7^7 *
<lll«Ji+1dl|,7,<? I Jmes? Qdt
2_ 2^
r r
0
2 2_ X- 2
<P*!KrU mes« ? fu.10)
Here the numbers q, 7, q and r1 are constructed from q and r in the way de¬
scribed in §7, and we were justified in using the embedding theorem (3-4) of
Chapter II.
The expression /(£, s) is estimated from below, using the relation
K i«*r‘ £),P=i [(»+ ui*, r •n^r-^+J
<2(s+1 )>,r2i(J; vv,) e+»i«.r’c? ^
in the following manner:
+ §• f«L|^r^rfJcrff-TFT 1 I ^f+^xdxdt- (11.12)
«( 0/
From inequality (11.8) by virtue of inequalities (11.2), (11.9) and (11.10) we
obtain
i) We note that by virtue of die assumptions of die theorem for (1.1) the numbers T,
q, 9 and q are finite for all n.
216
HI. UNEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
ill G+i!(i«,r'oji2%j
+ f J ulAuJ’Vdxdt
< J I M*. 0>l*+2S2(*. 0)d*
Q
+[c+itt] J i“,r+2(K.i5+^>^^
Q,
_2__ 2 J !_
+ e(* + 1)c,p»|Cj" mes7 TQt ~ ~ ^
+ c(s+l)c,mes « Of r. (11.13)
For positive satisfying the condition
2 _2 _2
t(5-|-l)g|P,me8<r~» < -”l^ -)) . (11-14)
(11.13) implies the inequality
iw ,+,Mp + J
u %
< ink'll I «■<<*• 0)i"‘^- °>B.U
1 + J|«Jtr+2(|£,|£+-Si)rfA:^
+ (s+ !)2c.
(11.15)
with the constant C2 depending only on mesQ, and the numerical parameters
in conditions (1.2), (11.1). The number s can assume any of the values s -
0, 1, 2,--.
On the other hand, for an arbitrary smooth function we have the inequality
jK|'2i+^d*<
$2
< c3(s) max Iuf J |**P + |«x|!l+,(^ ■+ P)] dx, (II. 16)
a a
following from inequality (5-8) of Chapter II, with the constant c j(s) depending
§11. AN ESTIMATE FOR roax^, |iij AND 217
only on -s and n. Therefore, assuming maxq^. |u| = M to be known, from (11.15)
and (11.16) we obtain
Jl
ttJ**<gdxdt <c„(s)
c4
i2i + 2
i ua*. o) r+ls(x. o)i
■+ J Krw+e+p)*'**1
Q,
(11.17)
with a certain known constant depending on s.
We fix Sq > 0 (its value will be chosen below as a function only of the para¬
meters q and r of (11.1)) and in accordance with it we partition the closed inter¬
val [0, T] into a finite number of closed intervals [«° = 0, t1], [i1, t2],.. •
• • ■, [t"1-1, tm = T) of length f j/2 (the last interval can have a smaller length),
where tj satisfies condition (11.14) with s - sQ. An inequality of type (11.17)
is valid for each cylinder Q** =* Q x [t* *]. Choosing £{x, t) = £(x)y(t),
in this cylinder, where for k > 2 y(t) is equal to zero for t = ^ and to unity
for t C [tk, t*+ *], but for the time being is arbitrary on the first interval, and
summing the inequalities (11.17) over all (overlapping) cylinders Q*, 4 = 1, 2, • • •,
we obtain
J K (So)f|||«x(x. 0) r‘S(*. 0)||2%
Qt I
(i£,[c+s+s8)^^ + r
Qj
i 2s+2
+ J l“*
xi
<ir
(11.18)
for s =0, 1,• * sq and any smooth nonnegative function £ not exceeding 1 and
equal to zero in the vicinity of s J. These inequalities permit us to estimate an
integral of \ux\2s* * in terms of an integral of |ux)24+2 over a somewhat more
extensive domain. As shown at the end of §2, the integral 2dxdt,
(3(0) C Qp, is estimated from above by a constant depending only on M (even
just on ||ttl2 Qj,), the constants in conditions (1.2)—(1.6), and the distance from
Q(0) to r p. From (11.18) for s~0 it follows that the integral fQ^^y\ux\^dxdt,
9(1) C Q(0), is estimated in terms of fQ^\ux\2dxdt and the distance from Q( 1)
to rj-(0) (Fj-(O) is the lower base and lateral surface of Q(0)). Contracting the
domain step by step, we estimate the integrals
J |ax|Jl"f',rfx:rf<<e6(So). * = 0. 1 sa, (11-19)
<?{*+!>
218 in. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
where Cg(s0) depends only on s„, M, the constants of conditions (1.2) and (11.1)
and the distance from Q(s + 1) to F(which, as always, is assumed to be posi¬
tive). if DM*, o)|so+1<ru 0)1) 2 Q < then one can take the cylinders Q (0) D
9(1) 3 • • • 3 (?(s0 + 1) with the same lower base as that of Qj-, i.e. Q(s) -
fKs) x [0, 71, (1(0) Dfl(l)3..., and diereby obtain estimates (11.19)also for
such cylinders.
Let us return to the system (11.3). Each equation of it can be regarded as an
equation relative to uj. = ux^ of the form
d dF*, (x, t)
U»—Tx7(fl,A^)= dx, ■ <n-»)
while Ufa is like a generalized solution of it from V(it is even better:
Ufc is a continuous function in Q* with continuous derivatives ukx in <?'). It
follows from assumptions (11.1) and inequalities (11.19) that
II ^ 1I** ** «• < ■**• <1Il-21>
where Sq can be taken so large that the numbers 3X1 ^ r2 sat*sfy db* conditions
lf?2 + ^
Va€(-5-• ?)• '"26(1. r) for ft>2,
?26(1.?). /"2 € (1 • r) f°r «=1.
Let us take such an Sq (it depends, obviously, only oo q and r). Then Theorems
8.1 and 10.1 guarantee estimates of maxg,, \u.x\ and ^ux)|)2 for Q"C Q' in terms
°f fi “jcII2 Q’’ *cnown constants, and the distance from Q" to tbe lower base and
lateral surface of QThese same theorems and inequalities (11.19) for the
cylinders 0(s)x(0, T) give estimates of max^» |u^ | and for Q" -Q" x
(0, T) in terms of j qi , Q' = Q ’ x (0, 7"), inaxjj, |u(*, 0)| and 0))q?
respectively and the distance from Q" to the boundary of O'. Theorem 11.1 is
proved.
It is more difficult to obtain estimates of max^^lu^l and for all of
Q-p (i.e., in the vicinity of Sj.). We will do this in §§4 and 5 of Chapter V directly
for quasi-linear equations.
Remade 11.2. The estimate of (and analogously of U^l^.) in terms
of known quantities can also be deduced differently: from Theorem 9.1 of Chap¬
ter IV on an estimate of |b(, uxxtq Qr 10 tenns II d/(-/<?*,• - /q^. and from a
§12. SMOOTHNESS OF GENERALIZED SOLUTIONS 219
theorem of Chapter II for embedding the space into (n+2)/2> 1 (b+2)/29(^_)
for q > n + 2 (see Lemma 3*3 of Chapter II). However, in this connection one
must assume in contrast to Theorem 11.1 that the coefficients a.Ix, t) are con-
v
tinuous in (xf 0-
§12. ON THE DEPENDENCE OF THE SMOOTHNESS OF
GENERALIZED SOLUTIONS ON THE SMOOTHNESS
OF THE DATA OF THE PROBLEM
We will show that the smoothness of generalized solutions from of
equations (1.1) in an open domain QT is determined only by the smoothness of
the coefficients and free terms of these equations; it is increased with an in¬
crease in the latter. Let us suppose that u(x, l) is an arbitrary generalized solu¬
tion ftom P*’°(<?r) of equation (1.1), the coefficients and free terms of which
satisfy conditions (1.2)—(1.6), while (5'is an arbitrary subdomain of sepa¬
rated from Fj. by a positive distance.
It was proved in the preceding sections that (i) if in place of (1.3)—(1.6) the
stronger conditions (1.3), (1-4) and(9.1), (9.2) are fulfilled, then the norms
with 6 from (0, 1) are finite; (ii) if conditions (7.1), (7.2) are ful¬
filled for the coefficients, while conditions (9.1), (9.2) with 0 = 0 are fulfilled
for the {■ and /, then tbe integrals fg, exp \c\u(x, t)\\dxdt with some c >0 are
finite; and (iii) if conditions (7.1), (7.2) are valid, then the norms |u|^> with some
a > 0 are finite, with all of these norms being effectively estimated in terms of
Ml2 Qj,, the distance from @ 1 to Fj- and the known parameters in the corre¬
sponding conditions on the given functions (i.e. the coefficients and free terms of
(1-1)).
Suppose that the given functions possess better properties, for example, that
they satisfy conditions (11.1), (7.2), (11.2). Let us show that the norms
with some a > 0 are then finite. For this puipose we take advantage of the a
priori estimate of this norm established in §11. It is obtained by us, however,
under the assumption that u € C2,1(Qj.) and therefore can not be applied directly
to the generalized solution in question, concerning which it is known for the pres¬
ent only that it belongs to Ha'a^2(Q-j-) and has n% from Lj(Qj).
Let us consider in some cylinder Q't = Q 'x Ug, tj), where SI' is an in¬
terior subdomain of fl with very smooth boundary S', while 0 < tQ < ij < T, the
auxiliary problem
220
III. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
m (auev.r, + V + Ap)
+
®lr; ,="pir
+ Vx, + V + 4
*C*
(12.1)
where F," ,j is the lateral surface and lower base of the cylinder t^, while
o.• ,• •. , u are averagings of a. , • ■ ■, u over x and t with an infinitely differ-
J?
entiable kernel of radius p (see §4 of Chapter II). Problem (12.1), as will be
proved in Theorem 5-2 of Chapter IV, has a unique classical (and even infinitely
differentiable in Qt ( ) solution vp. It is known that the up converge to a in
the norm of '°(Q't0 tj) *ot P will show that the vp also converge to
u(x, t) in the norm of ,() (and by the same token that the \vp\ni ®e
0,1 vn>,*l
uniformly bounded). In addition, from the estimates given in the preceding sec¬
tions and the uniform (in p) boundedness of Ione has the uniform bound¬
ness of the norms and !'\Q" with some /3 > 0, in which Q" t
is an arbitrarily chosen cylinder belonging to ^ and separated from
by a positive distance. By virtue of this, for lim^ gf'7, i.e. lot u, the norms
l“vlo" will be bounded and the estimates established for them in §11 are
tQ’tl x
valid. Analogously, by using the a priori estimates in §10 of Chapter IV, we
establish a subsequent improvement in the differential properties of solutions u
within Qj undet an increase in the smoothness of the coefficients and free terms
of equation (1.1).
Thus it remains for us to prove that |vp - uJ<t» —> 0 for p ~* 0.
Let us consider the integral identity (1.15), which is satisfied by a general¬
ized solution it, and put where £(*, t) is a function from ff'|’1(97’)
that is equal to zero in the domain Qy\Qtg t, , and its p-neighborhood.
Transferring in the integrals the operation of averaging from £ to its coef¬
ficients and carrying out an integration by parts in the term /- up£tdxdt, we
arrive at the equality
“j- {bfttxt -\~o.u ~f~ fjp £J dx dt = 0.
§12. SMOOTHNESS OF GENERALIZED SOLUTIONS 221
Taking equation (12.1) into account, we can write the following relation for the
difference w * m
o
J ( wfi+[{auu^\ ~ au&, + (fl/u)p - VI ^
Qi» >
+ [ (biax)p — V'S, + (a“)p — “p®”] C) rfjf = o. /„ </</,.
It is not difficult to see that this equality is valid for any function £U, t) from
'hat is equal to zero on St' (see in this connection the derivation
of (2.12)). We put w and under the integral sign we add and subtract the terms
aijpupxjwxi’ aipUpWXi’ bif>1Pxiw’ apupw and wtite the result in the fo™
~ J w2(x, t)dx -f J a,jpWx.Wj,.dxdt=—jl—j2, (12.2)
/
/, = J (ail/wwJlj -J- boxful + Op®2) dx dt,
Q‘» t
J\= J {[(fl/y«^)p —ai/pawJWxj4 [(«.«)„ —O/pBpjWjr,.
<>
+ [(*/“*,)p — *fp“p^]« +■ Ka“)p — flp“pl dx dt-
The integral admits the estimate
dx dt
< . I-(■„;_+ i I®II..,ll-l|
( 2 2 2 2
, mesQ'« ?(/—*„) r r.
v4. ‘
where f € (0, 1), S * 2"_. j + 2 |o^|, while the numbers qt Tr 9
and P have the same values as in estimate (7.9).
For /2, by making use of the property of averagings, conditions (7.1), (7.2)
and the boundedness of it is not difficult to deduce that
IA!<6(P)M0; (<TJ62(P) + el®|2<?; >
222
III. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
where 8 (p) —» 0 for p —* 0.
In equality (12.2) we estimate the second term from below with the use of
condition (1.2), and the integrals /j and jj from above, as indicated above, and
then we maximize the obtained inequality over t 6 1*0' *1^ a result we get
J 2_ 2_ _2_1
X(mesQ') ? ~ ? (/, -#0) 7 ‘ ? J+i6*(p), v„=min. v[. (12.3)
Let us take e » i/q/8, in (12.3), and fix so small that
2 2 j_
1. %(meS Q/)7 " ? «2-4)
Then (12.3) will yield the estimate
I®Ip; ( <16v-*fi»(p).
In this way we have proved that = \u — v^ss, 0 for p —► 0 as
v«<Mt P
long as — £q is subject to restriction (12.4).
As indicated above, this fact and the estimates established in §11 are suf¬
ficient, as long as conditions (11.1), (11.2) are fulfilled, for one to determine
whether or not the generalized solutions of class e<Iuat*on (1-1)
possess Holder* continuous derivatives with respect to x.
If the coefficients and free terms of equation (1.1) satisfy, in addition to
conditions (1.2)—(1.6), the conditions of Theorem 9-1 of Chapter IV guaranteeing
the estimate (10.12) of Chapter IV for ||ut» q, in terms of |/- df^dx^ q
then, upon applying this estimate to the vp with ^ C ^ and observing
that the constant in it does not depend on p, we establish that for the limit func¬
tion u there also exist derivatives il% and u%% belonging to ^q(Qt0 jj)- If *b*
functions a..} da.^/dx., a., da./dx., b., a, f and df-/dx. are elements of
ff2m +a, m+a/2(£^ m > 0, then any generalized solution of equation (i.l) is an
element of //2«+2 +a, m +1 +a/2(Qy,)m Xhis follows from the fact that by virtue of
estimate (10.5) of Chapter IV the norms in the space #2ro +2+a, m+1+a/2^),
are uniformly bounded for the v^t and consequently are also finite for u.
In this way we prove
§12. SMOOTHNESS OF GENERALIZED SOLUTIONS
223
Theorem 12-1- Suppose u is a generalized solution from V\'a(Qf) of equa-
tion (1.1), the coefficients and free terms of which satisfy conditions (1.2)—(1.6).
If furthermore conditions (11.1), (7.2) oiuf (11.2) are fulfilled\ then uxbelongs to
Ha> with some a > 0 determined by n, v, fi and the numerical parameters
q, r from conditions (11.1), (7.2) and( 11.2). If the coefficients and free terms
satisfy the conditions of Theorem 9.1 of Chapter IV, then u has derivatives ut
and uxx that are qth-power summable on any domain Q' separated from by
a positive distance. //, finally, all of the coefficients and free terms, and also
their derivativesy in equation {X.X) are elements of m*a/2(Q m >0,
then u will belong to H2m*2+a’m*l+a/2(QT).
Hie a priori estimates of the present chapter and Chapter IV obtained for the
whole domain Qj permit us to establish the membership of generalized solutions
from Vl2'°(QT) in the spaces H2m*2*a'mtl^/2(QT), //2m+2+a*m+1+a/2((?rUr'),
m m>l, etal. if, in addition to the smoothness of the functions in
equation (1.1), a corresponding smoothness is possessed by the boundary values
and the surface Sj> (or part of it), and if also the compatibility conditions on
F* fl Sq are fulfilled up to the necessary order.
We will not cite here the exact formulations of all of these propositions since
it is not difficult to do so independently and the number of different possibilities
is rather large.
We stress only the fact that the smoothness properties of generalized solutions
from of equation (1.1) have a local character: the smoothness of u in
some interior subdomain Q C Qj is determined only by the smoothness of the
functions in Q ‘ making up equation (1.1), while the smoothness of them in domains
Qf adjacent to F#C Fy., is determined, in addition, by the smoothness of F*, the
smoothness of the values of u on F' and the compatibility conditions on F' HSq.
Let us consider the analogous questions for the equations
& /,==«, -a.j (x, t)a 4-b,( x, t)ux.
*ix} 1
a(x, t)u=~F{xt t). (12.5)
For these equations it is proved in Chapter IV (see Theorem 9.1) that if their
coefficients a., are continuous, 6■ € Lr(Qf)f a € LS{Q^) with certain r and
s > 1, while F £ L^($7*), > I, then the basic initial-boundary-value problems
are uniquely solvable in (naturally, for a specific surface S and
boundary function). The differential properties of such solutions are improved as
224
III. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
the coefficients and free term of (12.5) are improved, and this improvement has a
local character. Thus, for example, on has
Theorem 12.2. Suppose u(x, t) is a generalized solution from *2q'hQT),
q > 1, of equation (12.5). If the coefficients and free term of (12.5) satisfy the
conditions of Theorem 9.1 of Chapter IV with index q^ > q, then uix, t) will be
an element of in any subdomain Q' of the domain Qj separated from
Fy by a positive distance. But if the coefficients and free term of (12.5) are
elements of then uix, t) will belong to H *a/^iQj,).
This theorem is proved in the same way as Theorem 12.1, it being only
necessary to use, instead of the energy inequality, inequality (9.3) of Qiapter IV.
§13. ON DIFFRACTION PROBLEMS
By proper diffraction problems we mean boundary value problems in domains
consisting of two or mote heterogeneous media. On an interface of these media
certain compatibility conditions must be fulfilled. There are two such conditions
for equations of second order. These are usually the continuity of the desired
solution and the continuity of its derivative along the conormal to die interface
(in the corresponding physical problems the first condition reflects a natural re¬
quirement, namely the absence of discontinuities in the medium, while the second
reflects the requirement that the forces acting on the interface be in equilibrium).
It was shown in the note that problems of this type for equations of differ
ent types can be reduced by means of a simple technique to problems on the de¬
termination of generalized solutions (with finite energy integral) of ordinary
boundary value problems for one equation. The solvability of these latter prob¬
lems is proved in various ways, in the course of which various approximate
methods of solution are justified (the method of finite differences, Galerkin’s
method, et al.). The unique solvability of diffraction problems was thereby proved
for equations of basic types within a generalized statement of the problem involv¬
ing very weak assumptions on all of the data.
Moreover, in the papers [65', f; 124f] techniques are described for investiga¬
ting the improvement in the differential properties of these generalized solutions
as the differential properties of the coefficients and free terms of the equations,
the interfaces et al. are improved (this is stated in more detail in [9l3,b))- Later
these techniques were substantially augmented by new methods of investigation
that permitted one to prove the classicalness of generalized solutions under
§13. ON DIFFRACTION PROBLEMS
225
nearly necessary conditions for them [65r].
We will dwell here only on how the existence of generalized solutions of
diffraction problems for parabolic equations is deduced from the theorems proved
above on the solvability of ordinary boundary value problems. But we will not
carry oat an investigation of their smoothness, referring instead to [4 ''] since in
essence it is similar to what was stated in §§6—12, while being quite cumbersome
technically.
We begin with the following case. Suppose the domain Qp = 0 x (0, T) is
partitioned into several domains k = 1, 2, • • •, N (so that Qf * U • • *
... U<**>), in each of which there is given an equation of the form
J&M0-fb*(x, *)«,f
+ a*(x. *)« =/*(■*. t) (13.1)
with smooth coefficients and free terms. We wish to find in Qp a function «(*, t)
satisfying: (i) in A = 1, •••,#, the corresponding equation (13-1); (ii) on
the lateral surface Sj. of the domain Qp one of the basic boundary conditions,
for example,
«jSr=0, (13-2)
(iii) on the lower base of Qp the initial condition
and (iv) on the interfaces of the domains Q^k^ and k, / = 1, • • -, /V,
the two compatibility conditions
and
-0. (13-5)
Here dufd-sV = a.,u_ cos (n, *-), n is the normal to directed into Q(l) I
v */ * f
while die symbol [t>] denotes the jump in die function v as it passes through
&k’l) from &k) into <?«>, and
226
III. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
The diffraction problems usually considered are a special case of this problem.
Let us show that this problem is itself a special case of an ordinary boundary
value problem for equations (1.1) with arbitrary discontinuous coefficients, that
is studied in the present chapter. Indeed, we consider equation (13-1) as one
equation
J?u=ss.ut — ~(ail(x, t) at[
-j-a(x, t)u~f(x, t) (13.6)
with coefficients and free term equal in each of the to the corresponding
coefficients and free term of the equation iS^u- = (their definition can be
arbitrarily completed on S^'l\ for example, as half-sums of the limiting values
on if they exist, of the corresponding functions), and for this equation we
pose in Qf the first boundary value problem (13-6), (13.2), (13.3)- As proved in
the preceding sections, this problem has a unique generalized solution u in the
class H Ha,a/2(@y) for any 0()('r) £ LjiQ), and in each of the domains
its smoothness is completely determined by the smoothness of (he coefficients
and free term of the corresponding equation (13-1) in All of the requirements
formulated above for a diffraction problem, except (13.5), are satisfied in an ob¬
vious way as long as the coefficients and free terms of (13.1) in $k\ k = 1, • • •
• • •, N, and also the a\jxi are> ^or cxampie> elements of #,°’a/^((?^). Let us
show that condition (13.5) is a consequence of the integral identity.
J (— “It + ai/^yTU, -I* biUx.r\ +■ aur\ — fr\) dx tit
Qr
J 0)dx = 0, (13.7)
which is satisfied by a generalized solution u of problem (13-6), (13.2), (13-3) as
long as in each of the this solution belongs, for example, to the space
■ j'1^**), and the surfaces 5**'^ are piecewise smooth without tangent planes
parallel to die plane t»0. For this purpose we transform (13.7) into the form
2 J[&]n*«o. (i3.
A-1 Q<») *, l-l S(M)
which is possible by virtue of the assumptions made on a and die 5^*’^. But
from it, since the choice of i){x, t) is sufficiently arbitrary, we obtain equation
8)
§13. ON DIFFRACTION PROBLEMS
227
(13-1) and equality (13.5). Thus, in fact, a generalized solution from j)
of problem (13.6), (13.2), (13-3) is a solution of the problem (13.1)-(13.5) form¬
ulated above as long as it possesses the above-indicated smoothness and the
S<^> do not have horizontal tangent planes. As is easily seen, we also have the
converse assertion: a classical solution of problem (13.1)—(13.5) (if it exists) is
a generalized solution from of problem (13.6), (13.2), (13-3). And
since the latter is unique, we can replace the original (classical) statement of a
diffraction problem by a generalized one: the determination of solutions from
V\of problem (13.6), (13.2), (13.3). By the same token the unique solv¬
ability of problem (13.1)—(13-5) in this generalized statement follows from
Theorem 4.1, die assumptions on all of tbe data of the problem being reducible to
the conditions of Theorem 4.1. The theorems of §§6—12 give sufficient condi¬
tions for when this solution belongs to the classes HCL,a/2(QT), Ha,a/2(QT), and
alsoto Hl'l/2(#k)\ l> 1, and to IT*- l/2«?kr), I > 2, k = 1,. •., N, where
is an arbitrary interior subdomain of QBut an investigation of the be¬
havior of the b near the surfaces requires special considerations, which
are developed in [65r]. Since all of these investigations bear a local character,
die assumption that the boundaries have die form x (0, T), where
the S<M> are smooth surfaces in the x space that divide 0 into the domains
QW, k = 1, • • •, /V, does not constitute a loss in generality: in the small any
smooth piece of not having normals parallel to the t axis can be con¬
verted by means of a change in the variables (x, t) of the form yt = t - t£t(*),
e = 1, •••,«, r - t, into a piece of a surface (and even of a plane) with elements
parallel to die t axis, that does not violate the character of the problem. We cite
here one of the basic results on the solvability of problem (13-1)—(13-5)-
Theorem 13-1- Problem (13.1)—(13-5) has a unique generalized solution
#(*> t) from if if>0 6 an^ ^e coefficients and free terms of
equations (13.1) satisfy the conditions of Theorem 4.1. This solution u is the
same as the generalized solution from of problem (13.6), (13.2), (13-3).
For it one has the assertions of Theorems 7.1 ojtd 9.1 on when it is summable on
Qf with larger summability powers or when it is Hdlder continuous in Qf or Qj
et al. Moreover, outside of the interfaces its smoothness is guaranteed by
the theorems of §§10-12;. in particular, it belongs to the classes #2+a>1+a/2((j(^)
in each of the domains as long as the coefficients and free terms of equa¬
tions (13.1) and also the °^jx, are elements of Ha’And near any twice
228 ni. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
continuously differentiable piece of the surface not having normals
parallel to the t axis^ the derivatives of the solution u are (Hdlder) continuous
on both sides of 5^*’^ right up to the surface itself as long as in some
neighborhood DjX [t j, t^ of it one has
II daij day db, da df ||
II IF' ’Sxi ’ ‘‘ HT' “■ IF' f' ~SF L D< consl
for q> n when t € [tj, (j] and if, as always, the condition of ellipticity (1.2) is ful¬
filled. Thus in this case both, of the requirements (13.4), (13.5) are fulfilled in the
classical sense.
In view of the stability of tbe boundary value problems with respect to vari-
tions in the coefficients and free terms of the equations, the solutions of diffrac¬
tion problems can be obtained as limits of good solutions of equations with
smoothed coefficients. Thus tbe solution u of problem (13.1)—(13-5) or, equiva¬
lently, problem (13.6), (13.2), (13-3) (under the conditions of Theorem 4.1) is the
limit in the norm of of the solutions um of the boundary value problems
(aljmuz^ ■+• bimuXi + amu —
“I Sr = °* H.0 = 1’0m(*).
in which the aijm> b-m, am, fm, 'l’qm are smooth functions approximating tbe oy,
if, o, /, and <Cq respectively in the norms of those spaces to which these (latter)
functions belong (an exact formulation is given in Theorem 4.5). If furthermore the
conditions of Theorem 9.1 are fulfilled, then the um will converge to u uniformly.
We note that the case of nonhomogeneous conditions on Sy and the ^
is reduced to the homogeneous case by introducing in place of u a new unknown
function v(x, t) = u(x, e) - t), where <£(*, t) is some smooth function sat¬
isfying on Sj. and the the same nonhomogeneous conditions as
u. The other boundary conditions on Sy and the conditions
[■w-HU„-° (,3'9)
in place of (13- 5) are considered analogously. An important fact concerning the
present method is that condition (13.5) on a discontinuity has a special form
connected with tbe principal terms of the equation; namely, it involves tbe deri¬
vatives du/dJi along a conormal but not along arbitrary directions I in the x
space. By virtue of this fact the terms on an interface that are singled out in
1) These conditions are weakened somewhat
§13. ON DIFFRACTION PROBLEMS
229
passing from tbe equation to tbe integral identity either completely vanish (see
(13.8) and (13.7)) or the principal terms among them vanish.
Analogous considerations permit tbe following generalizations of problem
(13.1)—(13-5)- Suppose instead of equations (13-1) we have the equations
in conditions (13.4) and (13-11), the jumps are determined in the same way as
above, while in the second problem they are determined differently, namely,
Let us show how it is possible to transform these problems into problems on the
determination of generalized solutions of a boundary value problem for one equa¬
tion with discontinuous coefficients and to thereby replace the classical state¬
ments of these problems with generalized ones.
For problem (13-10), (13.2)—(13-4), (13-11) we have the determination of a
generalized solution u from of the first boundary value problem (13-2),
(13-3) for the equation
t)ut — ~(aktj(x, t)ux^a«.(x, t)u)
-f- 6* (at, t)ux-\- a* (x, t)u = /k(x, t), (13-10)
the coefficients and free terms in which are good functions in each of the
having discontinuities of the first kind in passing through the with the
a*(at, t) being strictly positive, and instead of conditions (13-5) we have the con¬
ditions
or the conditions
[bau\ cos(n, 0— p«(/a^-t-fo,a]oos(n, •*/) |s(t, 0 — 0, (13.12)
in which b (x, t) is a smooth positive function in each of the having dis¬
continuities of the first kind in passing through the S^’^. In the first problem,
in (13.4), and also for [ba-ux. + ba^l^ ^ in (13.12), while
230
III. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
j&xui^baut — -j£-t(baljuXj-\-baiu'} + Bjatj+Au==bf, (13 13)
in which B■ = a.,-4„. + 4,4, A = a-b„. + ab, while the functions a(x, t), a.Ax, t)
1 (IX k s4k\ lf
et al. coincide with a' '{x, t), a?Gc, t) et al. in the domains </■ ’■ Such a solu¬
tion, in accordance with our conception of the definition of generalized solutions
of boundary value problems, is a function from satisfying condition
(13-3) and the integral identity
J [6au,T) -)- (batJar 4 bap} % -j- B.ux i) -+- dx dt
I ‘ )
= J bfr\ dx dt (13.14)
O
for any Tjix, t) from ,0(<3t")- It is not difficult to see that there are two pieces
of information stored in identity (13.14): in the the function u satisfies
equation (13.13) or, equivalently, equation (13-10), while on the 5^*’^ it satisfies
conditions (l3.11). The remaining requirements of problem (13.2)-(13.4) are con¬
tained in tbe fact that u must be an element of satisfying condition
(13.2).
Problem (13.10), (13.2)—(13-4), (13-12) is reformulated as a problem on de¬
termining a generalized solution u from of the first boundary value
problem (13-2), (13.3) for die equation
-2V- -357{bailu*i + K“) + BjaJ'/+Au~bf (13.15)
with the same B . as above and with S = ~(ba), + a,b,. + ab. This means that u
/ 0 tlx,
must be an element of V^'u(Qy) satisfying the identity
J [ -6o«n, 4 (baijuXj4- bap}^4 Bf r\ + Aax\}dxdt
Qr
— f 6«|>0 t\dx I = f bfi\ dx dt (13-16)
i lf-° 4
for any r/{x, t) from t^*at *s equal to zero for t » T. In the present case
identity (13.16) contains equation (13-15) or, equivalently, (13.10), and conditions
(13.3) and (13-12).
§13. ON DIFFRACTION PROBLEMS
231
The existence of generalized solutions of both of the problems just formula¬
ted can be proved by different methods, the first being based on inequality (6.6)
and the second on an energy-type inequality. We will show how this is done, us¬
ing as an example the ability to solve tbe first boundary value problem for equa¬
tions with smooth coefficients (the corresponding theorems are proved in §5 of
Chapter IV). We assume for the sake of nonessential simplifications that all of
the known functions in identities (13.14) and (13-16), except f and are
bounded in modulus by some number U. We approximate the coefficients ba,
batj, bat, B., A and A by smooth functions (ba)m, lba.j)m and so on in Qj- in
such a way that the conditions of Theorem 5.2 in Chapter IV are fulfilled. More¬
over, this is done in such way that we have the inequalities
V!l2<(&a/;)m^y<Mii*. v,>u.
r
r i d j
max —Ft—\dt < (13.17)
* * 01 1
with one and the same positive constants and fi^y and that the approximations
for all of the other coefficients are uniformly bounded, while the (bf)m and (^o)m
converge to bf and b^Q in £2(Qy) and L2(Q) respectively. Then for the whole
family of solutions um of the auxiliary problems we will have io the first case
inequality (6.6), which gives the uniform boundedness of the norms
11 (1316)
and in the second case inequality (6.3) guaranteeing die uniform boundedness of
the
As a result we can pass from identities (13.14) and (13.16) for the um to the
same identities for the functions u that are the limit functions for iu”| in the
o o
weak topology of W2’ (QjO and W2’°(Qj.) respectively. These limit functions it
will also be the desired generalized solutions.
The uniqueness of the generalized solutions of these two problems is
232
HI. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
established on the basis of the fact that these problems are the adjoints of each
other. Namely, suppose u is a solution of the first boundary value problem with
/ and <A0 equal to zero, so that identity (13-14) with /= 0 is valid for u. As rj
in it we take a generalized solution y) of the problem
4 ba.n 4 <4n= P(x. t), (13-20)
<
Tlhr=°- tll,-r = °-
where F(x, t) is an arbitrary smooth function in Qy. Tbe existence of at least
one such solution is proved above. From a comparison of the integral identity
(13-14) for u with f & 0 and r) equal to a solution of problem (13-20), and the
identity (13-16) for rj, as solutions of problem (13-20), in which the solution u
is taken as an arbitrary function, it is seen that
J uF dx dt = 0,
Qr
and thus, since F is sufficiently arbitrary, we have u & 0. The uniqueness of a
generalized solution of die second problem is proved analogously. For the sake
of simplicity in the calculations we assume in this connection that in each of the
domains of all the coefficients of and the function b(x, t) together
with the derivatives of it encountered in our arguments are bounded.
The classicalness of the obtained generalized solutions is established in
principle in the same way as the classicalness of the generalized solutions of
problem (13-1)—(13-5).
Let us sum up what has been proved in the form of a theorem.
Theorem 13-2- Problem, (13.10), (13.2)-(13.4), (13.11) has a unique solution
from while problem (13.10), (13.2)—(13-4), (13-12) has a unique solu¬
tion from Wj,0(Qy) for <»>y / from L2(Qy), with <A0 from L2(tl) in the first
problem and <J/q from IPjWi) ‘n the second. Concerning the functions a-., a^t, a(,
aie b., b-e a, a, a(, b, bx, b( and b%( it is assumed in this connection that they
are bounded by an absolute quantity in each of the domains (f-*\ that a and b
are strictly positive, and that the a;. are subject to inequalities (1.2).
§14. FUNCTIONAL METHODS FOR BOUNDARY PROBLEMS
233
§14. FUNCTIONAL METHODS FOR THE
SOLUTION OF BOUNDARY VALUE PROBLEMS
In this and the following sections of Chapter III we will set forth different
methods for proving the solvability of boundary value problems for linear equa¬
tions. All of them are applicable to the Cauchy problem. The majority of them
can be used for the actual determination of solutions (more precisely, for the
determination of the approximations um that converge to the desired solution for
m —. oo). In §4 of the present chapter Galerkin’s method was investigated. In
Chapter IV two other Methods will be given: the classical method of the theory
of potentials and one of the latest methods, which is based on exact a priori
estimates and the construction of the regularizor, i.e. the operator inverting the
linear principal part of an equation with frozen coefficients. In the present sec¬
tion we describe three methods, which are called functional in view of the fact
that they make use of only the most general characteristics of the operators
corresponding to die problem, and not their particular functional structure. These
methods were used to prove the unique solvability of the abstract Cauchy problem
+ = —»o, (14.1)
in which »q is a given element of a Hilbert space K, «(/) and fit) are unknown
and known functions of t with values in H, while S(t) is a family of semi"
bounded linear operators on H depending on the parameter t. An effort was
made to formulate the conditions which were imposed in this connection on S(£),
fit) and Uqt in such a way that they would be fulfilled in the basic boundary
value problems for linear parabolic equations. Thus problem (1.9) can be inter¬
preted, for example, as problem (14.1) with H *» /^(O) and with S(t) defined by
the elliptic operator
t)uXj + at(x. f)K)
+ t)uXi+ a(x. t)u
O
on the set 3)(S(«)) = W^(Q)m The condition ajsj, = 0 is "absorbed” into the
designation of the domain of definition of S( i). The solution u(x, t) and the free
term F(x, 2) = df.(x, l)/dx^ - f(x, t) are regarded as functions of t with values in
£2(fl), while x) is regarded as a known element of L2(Cl) defining die initial
value of u(x, t).
234
III. LINEAR EQUATIONS WITH.DISCONTINUOUS COEFFICIENTS
The first two functional methods for (1.9) and for (14.1) were given by M. I.
Visit [l24b] and O. A. Ladyzenskaja (for detailed publications and subse¬
quent generalizations of them see [65e,f; 124c] an(j t],e snrVey [124f]).
After that several other methods were proposed ([S5f: 73; 124f] et al.; the
book [7 5] is mainly devoted to abstract methods for the solution of the Cauchy
problem for equations of form (14.1)). The results obtained with these methods for
parabolic equations differ from each other both in the assumptions on tbe known
functions and in those properties of the solutions that have been established with
these methods. Some of them give more exact information on the solution, while
the others are somewhat simpler, with the alternatives involving less (or more)
assumptions on the data of the problem. We will not present and exactly formulate
everything that has been obtained with these methods in connection with the prob¬
lems of interest to us in their general form. Instead we will set forth the basic
ideas of these methods (in the present and following sections) with the example
of an elementary problem
^0u~at — Att = /(jc, t), «|rr = 0, (14.2)
or, in chose methods in which problem (14.2) is considered already solved and is
used as a starting point, with die example of a somewhat more general problem
«==«,— -£—(<!;,•(*, t)uXj)-\-Xu — f (X, t), “Iiy — 0. (14.3)
We note only the following: we chose Galerkin's method as the basic method in
the present chapter because it does not require that the space in which the solu¬
tion is found (the space **e either a Hilbert or a reflexive space. And
the choice of the nonreflexive Banach space ^^(Qy) was determined by the
desire to maximally extend the class of equations (i.e. to weaken die assumptions
on their coefficients), but in such a way that the existence and uniqueness
theorems are still retained for it. The result established by us in §4 does not
follow'directly from the other methods.
The method proposed by O. A. Ladyzenskaja {6 5d-f] in connection with prob¬
lem (14.2) goes as follows. In the Hilbert space H = L-^Q^) one considers the
operator A associating with each function u(x, t) from an<* e<3ual to
zero on Pj* the function Au t). We denote its domain of definition by
35(/t), and its range by 31 (vt).
From the relation
§14. FUNCTIONAL METHODS FOR BOUNDARY PROBLEMS
235
J (Auf dx dt — J [a? + (Aa)sj dxdt +■ J u\ (x, T) dx
Qr *2r s
>c(lt“ IPqP2, C — const > 0, (14.41
it is seen that the closing of the operator A (that it is closable is easily verified
directly or is established as a consequence of the fact that the domain of defini¬
tion of its adjoint is dense in L^Qy.)) reduces to the closing of its range, i.e.
Let us show that 'fU/T) coincides with all of L2(Qy). Indeed, there would
otherwise exist an element v(x, t) £ L2(Qj0 orthogonal to all of ${/)), i.e.,
such that
j vJFfjU dxdt = 0 (14.5)
Qr
for all it € 3)(A).
Let us construct with respect to v(x, t) a function w(xy t) as a solution of
tbe problem
— &W(X, t)~ J v(x, t)rfT, «)|Sr = 0.
(14.6)
It has been proved ([(S5b]; see also [c5q], §9 ot Chapter III) that this problem is
uniquely solvable in for any right side from L2(0). In view of this w -
(—A) lfg v(x. t) dr belongs to ®M), and hence in (14.5) one can substitute u>
lot u. We write the result of this substitution in the form
J v (—A)-,i>-f- J vdx
Or
dx dt = 0,
from which
(—A) 2v II2iQ7.+y
J
dx
0 »2, Q
and consequently v ~ 0. Thus it is proved that
M(A) = Ll(QT).
But this also means that &e equation
236
in. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
Au = f (14.7)
has a solution u from $(/4) for any f from
From relation (14.4) it is seen that the elements u(x, t) from 3)(A) have
generalized derivatives u(, ux and ux from L2(QT). It is assumed here that
the boundary S is a surface of class 0^.
The operator A acting on them is calculated as an ordinary differential opera¬
tor £qU, in which all of the derivatives are regarded as generalized, while the
equation is satisfied almost everywhere in Q-p. And the boundary and
initial conditions on V^ are fulfilled "in the mean*’.
Thus it is proved that for any f(x9 1} from problem (14.2) has a solu¬
tion u belonging to ’*(@7*) Uf $•€ 0^). It is clear that this result is exact
and that the solution is unique in this class of functions. Moreover, we have
actually proved a significantly stronger uniqueness theorem: uniqueness for gen¬
eralized solutions of problem (14.2) (more precisely, its adjoint
— v, — Ao = t|>, v |Sr = 0, v (x, T) = 0,
which goes over into problem (14.2) if t is replaced by r = T — t) from class
L2(Qt). Such solutions are determined in the following manner:
By a generalized solution of problem (14.2) from the class L2(Qf) we mean
a function u(x, t) from L2(Qf) satisfying the identity
f «(— Htydxdt — J f i) dx dt (14.8)
Qr Qt
for any 7/ from W2’^(Qj.) that is equal to zero both on Sj. and for i = T.
The method proposed by M. I. ViSifc [I24b,f] js based on the following easily
proved proposition of functional analysis.
Lemma 14.1. Suppose A is an operator in a Hilbert space H with a dense
domain of definition $(/!) and with a bounded inverse on its range 51C4). Then
the range fR(/4*) of its adjoint coincides with all of H.
Indeed, let us take an arbitrary element / from H. The equation A*x = f, by
virtue of the definition of the adjoint operator A', is equivalent to the identity
(x, Au) - (f, u) for any u from 53(A) or, what is the same thing, to the identity
(x, v) = (/, A 1 u) for any v from 3? (j4). But die latter is solvable for any / from
H, since (f, A *v) is a linear functional over the v’s (its linearity is obvious,
while its boundedness follows from tbe boundedness of the operator A 1 on 5U/4)),
§14. FUNCTIONAL METHODS FOR BOUNDARY PROBLEMS
237
and hence by virtue of the theorem of Riesz on the form of a linear functional tbere
exists at least one element x from 91 (j4) for which (/, A 1ti) = (x, v) for all v
from 5f (4). This element also gives one of the solutions of the equation A*x = /.
The lemma is proved.
It can be applied in a different way to prove the solvability of problem (14.2).
Visik takes as tbe basic space H the function space sca^ar prod-
uct (u, v)^ = Jqj,(uv + vxvx)dxdt. Equation (14.2) and the initial condition are
replaced by the integral identity
/0(h.»))s= f (— m\ -f Hri|r) dx dt = f fi\dxdt, (14.9)
Qt Qt
where r) is an arbitrary function from the set S) consisting of all elements of
ff,2,1(9j-) that are equal to zero both on Sj- and for I = T. The function u from
V\'°{Qt) satisfying (14.9) for all such ij’s is called in accordance with §3 a
generalized solution of problem (14.2) from the space
The integral lg(a, rj) for any fixed rj from $ defines, as is easily seen, a
linear functional over the u’s in H, which by virtue of the theorem of Riesz can
be written in the form of a scalar product of u by some element At; from H, i.e.
to (“• 1) = («. -411V. a £ m. (14.10)
The operator A defined by this identity has as its domain of definition !D(<4)
the set 3). It satisfies the conditions of Lemma 14.1 since, substituting rj for
u in (14.10), we obtain
Ch. — n)= J (— nil, + *£) dx dt
Qt
= j II n (•*. 0) II* a + II n, II* Qt > e II n ll^f. c = const > 0.
from which it follows that
The right side of identity (14.9) can also be represented in the form of a scalar
product in K:
| ff\dxdt =r (F, n)*..
qt
238
IH. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
where F is uniquely determined by f, since for any / from L2(Qf) the integral
fQj.fr/dxdt defines a linear functional over the rf s in H. Hence (14.9) takes the
form
(a, Avf)^ = (F, il)^. (14.11)
If we now apply Lemma 14.1 to the operator A, we will find that the equation
A*u = F or, what is the same thing, identity (14.11), has a solution u from H
for any F from H. By the same token we have proved the existence of a general¬
ized solution from W2’°(Qj.) of problem (14.2) for any / from Qj)-
As H one could take the subspace of die Hilbert space consisting
of functions that are equal to zero on Fj. In this connection it is necessary to
replace equation (14.2) by the identity
J («,£, + UjAxi) dxdt = J fa dx dt- (l4-12>
Qt Qt
in which £{x9 t) runs through the whole set j) of elem«its of M having the deri¬
vatives £ t € L2(QT)f and then to transform (14.12) into an equality for scalar
products in K:
(«. Ojff- (14*13)
The solvability of (14.13) is deduced on the basis of Lemma 14.1. This reasoning
yields the solvability of problem (14.2) in the space IPj for any f from
L2{Q7).
Lions establishes the solvability of boundary value problems on the basis of
a theorem proved by him on abstract bilinear forms (Theorem 1.1 of Chapter III in
his book [73]). In connection with problem (1.9) his approach essentially coin¬
cides with tbe method just mentioned for proving solvability in the space
on ***e basis of Lemma 14.1.
We note once more that all of the functional methods presented here apply to
problem (1.9) (and to several other boundary value problems) in its general form,
with the assumptions on the boundedness of the coefficients of minor terms (made
in the references given above) being replaceable by their membership in certain
Lg oJ^Qf) (tbe tfs are different for different coefficients and in different methods).
In between the first of the stated methods and the method of continuity in a
parameter presented in the next section, there is the following method, which is
based on the same a priori estimates as the first two [i44a]. Suppose S € 02 and
§15- THE METHOD OF CONTINUITY IN A PARAMETER 239
o
suppose it is known that problem (14.2) is uniquely solvable in for any
/ from L2(Qf) (we recall that is the subspace of the space W2’ HQj-)
consisting of die elements of that vanish on Sy). Let us show that the
same is then true for problem (14.3) as well. For this purpose we make use of a
generalization of (14.4), consisting in the fact that for any function v from
J ^S’lVjS’oV dx dt > C! (IIV III Qry, ct = const > 0, (14.14)
Qt
as long as the k in (14.3) is a sufficiently large positive number (cf. §6). It can
be assumed without any loss in generality that the A in £j is such that (14.14)
is also valid.
In addition, £j satisfies a relation of type (14.4), namely,
j (S'iv)*dx<lt'^-c20lvl\g)QTy, c2 = const > 0 (14.15)
Qt
O
(see (6.10)) for any v from From (his it follows that the operator
is closed on W2'*{Qt), has a bounded inverse on its range 91 (£j), and 35(£j)
is a closed subspace of L2(Qf). Let us show that X(Jlj) coincides with all of
L2(Qt). Suppose v(x, t) belongs to L2(Qy) and is orthogonal to fR(£j), i.e.,
J S’xuv dxdt = 0 (14.16)
Qt
for all u 6 r^HQj.).
We construct with respect to it the element w => £q1v £ and put
u » w in (14.16). This gives
J J&xwJ&’o'W dx dt = 0.
Qt
But by virtue of (14.14) it follows that w = 0, and hence v = £gw ~ 0. Thus
S(£j) = LjiQf), i.e. problem (14.3) is uniquely solvable in II2’*(Qf) f°r any f
from
§15. THE METHOD OF CONTINUITY IN A PARAMETER
Let us prove the solvability of problem (14.3), assuming that problem (14.2)
has already been investigated. For this purpose we consider the family of prob¬
lems
240
III. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
~[xaii (*• 0 “Xj + (I — t)«,.] + l.iu = /.
«lrr = °. 0<t<1. (15.1)
Suppose S € 02, f € L2(Qf), and the a-(x, t) satisfy, besides the condition of
ellipticity
viJ< 0,71,1/Kill2, (15.2)
the conditions
Then the problems (15.1) are uniquely solvable in the space
The proof (it is taken from [6 5']) is based on the exact estimate (6.10) of the
present chapter, which is uniform for the whole family of operators and which
for the solutions ur of problems (15.1) has the form
lrt’or<‘l!/lb,Qr. (15.4)
and on die fact that for the operator = d/dt — A the solvability of problem
(15.1) in the class is considered established for at least a dense set
of functions / in L2(Qj>), The latter follows, for example, from the results in §6
of the present chapter, the results of Chapter IV, from die first functional method
presented in §14, and from Former’s method, which will be described in §17. But
if problem (15.1) for r = 0 is solvable in j*) for a dense set of /*s from
then by virtue of (15.4) for r - 0 it is uniquely solvable in fot
any / from LjiQf) [because the convergence of fm to / in L2(Qj<) implies
according to (15.4) the convergence of the corresponding solutions um in
8^2’*(@7') to some function u from ^2'HQf)* which will obviously be a solution
of the problem corresponding to f]. Thus the operator establishes a one-to-
one correspondence between the subspace W of the space consisting
of the elements of W2**(Qy) that are equal to zero on Fand all of L2(Qj<),
Let us show that this is true of all the operators For this purpose we write
problem (15.1) in the form of an operator equation
ft(^, /6t2(Qr), u£W, (15-5)
which, by virtue of the just-proved property of £Q, is equivalent to the equation
« + t.SV ' [JT, - -?„) u = ' / (15.5')
§16. ROTHE’S METHOD; FINITE DIFFERENCES
241
in the space W. The operator — £q) is bounded in V by virtue of assump¬
tions (15.2), (15-3) and inequality (15-4) for t = 0. Therefore equation (15-5) is
uniquely solvable (at least by the method of successive approximations) for r <
and by the same token the operators for r > 0
establish a one-to-one correspondence between IP and In view of this,
equation (15-5) is equivalent to the equation
« + (x — X,) JSTj (JPi — ^) u = _2\'7 (15.6)
in W for any fj < a. The norms of the operators *(£j — £q) in W for all r
from [0, 1] for which exists do not exceed a certain number Oj that is inde¬
pendent of r (which follows from (15.2)-(15-4)), and hence equations (15.6) are
solvable for all r from [rj, fj + l/Oj). By repeating this argument, we come after
a finite number of steps to the operator and we establish the unique solvabil¬
ity in W of problem (15-1) for r = 1 (or, what is the same thing, problem (14-3))
for any / from i2(9r^
Continuity in a parameter can be realized in another way, not with respect to
the operators £ but with respect to bilinear forms corresponding to them (see
§16. ROTHE’S METHOD AND THE METHOD OF FINITE DIFFERENCES
Rothe’s method and tbe method of finite differences may be used for solving
the Cauchy problem and the initial- boundary-value problems for equations of gen¬
eral form (1.1), where the domain Q of variation of (x, t) can be noncylindrical.
The first of these methods was proposed by the German mathematician Rothe in
the year 1930 [I01] and was demonstrated by him in connection with a one-dimen¬
sional parabolic equation of second order. This demonstration was given by him,
naturally, within the limits of classical solvability under the condition of suffi¬
cient (but not overly excessive) smoothness of the coefficients. Subsequent re¬
sults on the solvability of boundary value problems for many-dimensional elliptic
equations permitted one to carry over without difficulty the results of Rothe to the
case of parabolic equations (1.1) with any n > 1 (see [*5h], jn which this is done
tor a more complicated object, namely a quasi-linear equation with noniinearity
in a, and also [36:!] and [] 2 3 a]). At the basis of the proof of classical solva¬
bility by Rothe’s method there lies (as in many other methods) the maximum prin¬
ciple, which is involved in the approximation of parabolic equations proposed by
242
III. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
Rothe. We will describe Rothe’s method by using as an example the problem
a, —A « = /(*,*). «|sr==0. a |, _0 = %(.*) (16.1)
in the cylindrical domain Qr = £lx [0, T\. In this section we will denote differentiation
with respect to x. and I by d^xt and dBt, and use the symbols (■)*. and (-)( for denoting
divided differences. We divide the (*, t) space by die planes t**kh,k=*0, ±1, • • •, into
layers and assume for the sake of brevity in notation that T/h = m is an integer.
Suppose is the intersection of the plane t = kh with Qj, while Sk is its
boundary. On all of the Sl^, k m X, 2, • • •, m, we consider the problems
ur(x, kh) — Au(x. kh) — f(x, k), 0^ = 0, (16.2)
assuming
u(x. 0) =$„(.£). (16.3)
Here
uT (x. kh) = -i[u (x, kh)—a (x, kli — ft)],
while
f(x, k) — f(x, kh).
Problems (16.2) are solved consecutively from the layer to the layer Qfc+j, begin¬
ning with ixl, where each time (even in the case of the general equation (1.1)) there
exists a unique solution of the corresponding boundary value problem for an ellip¬
tic equation owing to the presence in the equation of the term (l/A)it(*, kh) with a
large positive coefficient A~*. It is proved that in the limit as A —» 0 the func¬
tions u(x, kh) [or, for example, their linear interpolations u\x, t), (x, I) 6 Q f,
coinciding with u(x, kh) for ( ■> kh and linearly depending on t within the layers
kh <t <(k + l)A] give a solution u(x, t) of problem (16.1). For this purpose
estimates not depending on h are established for various norms of u( z, kh).
Some of them are based on the maximum principle inherent in tbe difference-differ¬
ential equation (16.2). In its "pure” form this principle holds, as always, for the
homogeneous equation when / = 0. In this case the maximal and minimal values
of the functions a(x, kh), k = 0, 1, • • •, m, occur either on the lower base fig or
on the Sj. Indeed, if a greatest value were to occur at some interior point Xq €
Qj for k>l, we would have at this point X*o> kh) m 0, -Au(xq, kh) > 0 and
it-(*0, kh) > 0, and hence by virtue of equatioa (16.2) kh) *= 0, i.e.
u{x0, kh) = «(*0, kh - A). (Thus a strict maximum at (xQ, kh) is impossible.)
§16. ROTHE’S METHOD; FINITE DIFFERENCES
243
Now taking the point (*0, kh - h) and carrying out the same arguments as at the
point (xg, tg), we get u(xQ, kh — h) - u(xq, kh — 2h) and so on up to a(*0, 0).
It is thereby proved that on the S^, k = 0, 1, • • •, m, or on Qq there exists a
point at which the functions u(x, kh), k = 0, 1, • • •, m, take their greatest value.
An analogous assertion also holds for their least value. By complicating the rea¬
soning somewhat it is not difficult to show that if / *= 0, then the greatest and
least values can not be taken at interior points it u d const [this is done in (he
same way as for the usual differential equation of the form
du , d!u . du „
it - a>^x- +a< sr, = °>
see in this connection [90a]].
The maximum principle gives an estimate of
max \u(x, kh)I
jttfl
kmQ, 1, . . m
in terms of the known quantities in problem (16.1). The case f&0 is considered in
almost die same way, with a preliminary replacement of the function a(x, kh) by
v(x, kh) e^k>>, X > 0. An estimate of max |<Jix(a:, lch)/dx\, is more complicated, but it
too is based on the maximum principle. For equations with coefficients not de¬
pending on * this is comparatively easy to perceive, since the derivatives
du/dXj satisfy the same system of equations as a, namely,
/ du (x, kh) \ A I du (x, kh) \ df (x, kh)
\ Jj- \ dx)—) SI—'
ft = 1, 2 m.
In view of this the quantities
are estimated in terms of
max
x( S3
*-o, 1
max
du (x, Hti)
dx/
df (x, kh)
and upper bounds on the Sk, 4=0, 1, • • •, m, and 0Q. The upper bounds on fl0
are known, while those on the Sj are estimated in the same way as for a differ¬
ential equation, with the use of barriers (see the proof of Lemma 3-1 in Chapter
IV; it simplifies in an obvious manner for the linear case). In the case of equations
244
III. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
with coefficients depending on x one cannot compose equations from which it
would be possible to estimate the magnitude of each of the derivatives du/dx.
separately. But one can do this directly for all of the derivatives du./dx., / = j,..
• • >, a, taking the functions
v(x, kh)
k — 0, 1 m,
with sufficiently large numbers ft and \ and composing equations of type (16.2)
for them. This technique, proposed by S- N. BemStein (see [•*]) for estimating
max |uj for the solutions u of linear parabolic equations of second order, is also
suitable for the difference-differential equations (16.2) (see §17 of Chapter IV).
Higher-ordered derivatives of u(x, kh) with respect to x are estimated analogously.
Thus, for derivatives of second order one must consider the functions
*(,, **)-[,» +
with sufficiently large numbers n and A. On the other hand, tbe smoothness with
respect to t is established directly from equation (16.2). We will not explicitly
carry out all of these estimates and the passage to the limit h —♦ 0; instead we
refer the reader to the paper [63h] see also [6 3c]). We note only that since in this
paper an equation depends on u nonlinearly, in order to estimate max \du/dx\ one
must introduce a more complicated function v than the one mentioned above. But
an estimate of max \d2u/dx*\ is carried out with BernStein’s technique in the way
just described. Here the difference of equation (16.2) from a differential equation
appears to be somewhat greater than in the above estimate of max |u(%, kh)\.
The final results on the classical solvability of boundary value problems in
an investigation by this method turn out to be somewhat cruder than those already
established and presented in Chapter IV.
Rothe’s method makes it possible to very simply, and under comparatively
broad assumptions on the data in boundary value problems, prove the existence of
their generalized solutions (this is done for quasi-linear equations in t65hD- How¬
ever, the process of making these assumptions as broad as possible (as done in
§4 of the present chapter) requires additional arguments. We will show bow to
prove the presence of generalized solutions with Rothe’s method by using as an
example the problem
§16. ROTHE’S METHOD; FINITE DIFFERENCES
245
V = /(*. t). (16.5)
«lSr = 0. a I<_o “ (■*) •
foe i/'g 6 L2(Q), f 6 L^iQf) and the «^-(*> *) satisfying the condition
v|s < a.v (*, t) 1,1; < (i|2. v = const > 0. (16.6)
We replace equation (16.5) by the difference-differential equation
% (k) — {kh) a (ft} ™ (x, M)
<i6-7>
in which
kh
f(x, *>==4' I ft*- T)rfT-
Mi-h
and
kh
ah[}(x,k) =z~ J atj(x, t)dx.
kh~h
To it we add the boundary and initial conditions
u(xt 0) = %(*) and u(xt kh)\sr*s=Q- (16.8)
Since f\x, k) 6 LjiO,), on each layer problem (16.7), (16.8) is uniquely
o j
solvable in W^ (&)• Let us obtain for its solutions u(x, kh) an energy-type esti¬
mate not depending on h. For this purpose we multiply (16.7) by 2hu(xf kh), sum
the obtained equality over k from I to some k^ < m and integrate the resulting
equality over Q. After an integration by parts in the second term of the left side,
this brings us to the relation
2ft ^ J uj(x, kh}u(x, kh)dx
*-i Q
a*
+ S J au^x' ^Xt kh)ax (xt kh)dx
ft-i u 1 f
“2A S J I* ®u(Xt kh)dx' (16.9)
ft-1 £
246 III. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
For a transformation of the first term we use an elementary but highly useful iden¬
tity, indicated in [65*• ] for investigating Rothe’s method and an implicit difference
scheme for equations of parabolic type,
2* 2 (A)«(A) = a2 (A0) — (0) 4- A * 2 («r (ft))*, (16.10)
which is easily verified directly. In addition, we estimate the second term from
below with the use of (16.6). This brings us to the inequality
Ih (k0) == J «j(jc, k0/i)dx + 2vA i J {i£Ydx
a »-1 ot
« i* »
-f-As S J (“r)Srfjc< J “’<*• 0)dJC + 2ft J fudx
A-1 fi ft —1 Qj
. ^0 0 a
< J «*(x. OJAe+AjJ J </*>*** +*2 J “3rfjc> <I6-U)
9 *-i a* *->
from which one readily obtains the desired estimate for ZA(&0):
/*(A0)<t(||'lV,|||,fi-+-||/l!s%M). A0 —0. 1 (16.12)
with the constant c not depending on A. Because of it one can pass to the limit
as A —» 0 and prove that the limit function u(x, t) for {u(x, kh), 4 = 0, !,•••, mi
is a generalized solution of problem (16.5) from Fo* this purpose we
denote by u*(*> I) the function that is equal on each layer x 6 Q, l€
[kh - h, kh), k = 1, 2, ■ • -, to the function u(*, AA - A). It is easy to see that the
functions uHx, t) € are equal to zero on S j, and by virtue of (16.12)
have uniformly bounded norms in W^^iQ^)'-
Because of (16.13) we can choose a sequence Aj, / « 1, 2, • • •, tending to zero,
for which the functions iu**!i and their derivatives ic'Rt>l/dxjc\ converge weakly
in L2(QT) to some function a(x, t) from **1*t *s equal to zero on Sf,
and its derivatives du/dx^ respectively. Let us show that «(*, () satisfies tbe
integral identity
§16. ROTHE’S METHOD; FINITE DIFFERENCES
247
qt
— f (x, 0) dx — I jx\ dx dt (16.14)
Q qt
fot any r/ix, t) from that is equal to zero both on Sj and for t = T,
and is thereby a generalized solution of problem (16.5) from *s suf"
ficient to establish relation (16.14) only for continuously differentiable tj(x, t) 6
rJ’kCy) that are equal to zero in the vicinity of the layer t = T. For this pur¬
pose we multiply equation (16.7) by krjix, kh), sum the resulting equality over k
from I to is and integrate over Q. We transform the first term of the resulting
relation by means of the formula for "summation by parts”
m m-I
h 2 «T(*)»J(*)=—A 2 a(k)r\ (k)+a(m)n(m)—«(0)T)(0), (16-15)
ft— 1 1 k~Q *
which is easily verified directly. In it is used the notation uix, t) -
(l/k\u(x, t + h) - itix, t)]. And in the second term we transfer the derivative
d/dx. to T). In this way we obtain from (16.7) the identity
— h 2 J" «n« rfJc — j* (x, 0) dx
*-0S2k Q
+ h fi I fi J fhr'dx-
1 k~lQk
in which we take 77 (x, t) to be equal to zero fot t> mh h. Let us rewrite it in
the form
j
I
J fix. t)i\(x. t)dx dt, (16.16)
h Q
where the line over some of die functions means that at the point (x, t) of each
layer x € Si, t € [kh — h, kk] the function in question is equal to its value at the
248 III. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
point (*, kh — h). For a continuously differentiable function r/(x, t) the piece-
wise-continuous functions rj(*, t) converge uniformly to r)(x, t), while the func¬
tions (t),(x, t)) uniformly approximate its derivative du/dt. And for the functions
(A (x, t) and fHx, I) one can choose from (Aji a subsequence 1 Aj I, p = 1, 2, • • •,
with respect to which these functions converge to a^(x) and /(x, t) almost every¬
where and strongly in L2(Qf) respectively,1^ with \Jf-1 < fi. In view of this and
the weak convergence of the ul{x, t) and the (<9itA(*, t)/dx^) to u and du/dx^ in
(16.16), one can pass to the limit with respect to the subsequence [hi i and there-
A »
by show that the limit function u(x, t) for die it (x, t) actually satisfies identity
(16.14). Thus the presence of generalized solutions from f'J'^^j.) is proved.
Because of the uniqueness of a generalized solution it from ^|’®(@r) of problem
(16.5) the entire aggregate of the it* converges to u. Using (16.13), one can also
deduce a stronger assertion, on the strong convergence of the (X, l) to u in
L2(Qt) (see in this connection the paper [65m]), and thereby prove that the solu¬
tion u obtained by us actually belongs to V^Qj.).
Let us now proceed to the method of finite differences. For equations of
parabolic type there exist explicit and implicit difference schemes (see [15 ■ 30;
5 2; 6 5e, f, i; 97; 99; 104a. b; 105; 12 1] et al.) leading to the solutions of bound¬
ary value problems under specific relations between the lattice spacings A* and
At. We will consider here only one of the implicit schemes, which is analogous
in a certain respect to Rothe’s method and converges under any relation for tbe
spacings A* and A/. The proof of its convergence is very similar to the proof
just given for the convergence of the approximations u* to a generalized solution
of problem (16.5). Let us show this by using as an example problem (16.5), as¬
suming that condition (16.6) is fulfilled and £ L2(0), fix, t) € L2(Qj.). We
divide the whole (*, t) space by the planes x. = A.A*, £ = 1, • • •, a, t = kQ&l,
ka = 0, ±1, ±2, • • •, into elementary cells, namely parallelepipeds with sides of
length A* in the direction of die xk axes and of length At in the direction of
the l axis, where for the sake of nonessential simplicity in notation we will as¬
sume that m - T/At is an integer. We denote by the union of all elementary
The last integral on (he left side and the integral on the right side can be written
as integrals over all of Qf, potting V (x* t) “ 0 in them for t € [0, A). This extension of
the definition of r), as is easily seen, "disappears” in the limit, not having manifested
itself on the other terms of die identity.
§16. ROTHE’S METHOD; FINITE DIFFERENCES 249
cells belonging to Qj, by the intersection of Qj with the plane £ = kAt, and
by the boundary of £1^. The same notation Q^, Jlj and is retained for
aggregates of lattice points (i.e., the points with coordinates (k^Ax, , kJSx, kgAt)
belonging to the sets Qf, and respectively). We replace equation (16.5}
at all interior lattice points of Qj by the difference equation
u,(x, t) — {af}(x, t)ux(x, /))” —/A(x, tj, (16.17)
in which the following notation is used: u~-(x, t), as above, is equal to
(l/&()[« (*, t) - a (x, t - At)],
uti(x, = x,+Ax. .... xn, t)
— u(x Xi xj) ],
{{x, t) — -^j{a(x k„, t)
— u<xx x, — Ax x„, 01.
X, *n ‘
J ••• J J /a L. *)d£dt,
Xj-Ajr Xn-&X t~&t
and the a^j {x, t) are defined in the same way with respect to the a^x, t). If
a.Ax, t) or f(x, t) are continuous functions of x, t, then in place of a£•(*, t) or
1
/“(*, t) one can take their value at the lattice point (x, t). To relations (16.17)
we add the equalities
«ls&=0, ft=l and u(x, 0) = <|;£ (x), (16.18)
where tfrix) = (A*)"" f^-Ax •" Ax *0^1.• • •. <^i • • • at the in-
terior points of and -0 at the points of Sq.
Relations (16.17) and (16.18) permit us to calculate an approximate solution
u^ at all lattice points of successively layer by layer: t m kAt, Ul, 2,*"
* - *, m. These values of are determined uniquely (in the case of a general
equation they will be unique for all At not exceeding some number in our
particular example they will be unique for all A;).
Indeed, for determining uA on the &th layer (on O^) we have a system of
linear algebraic equations containing as many equations as unknowns (their num¬
ber is equal to the number of lattice points of 0^ *^>e corresponding homo¬
geneous system has the form
250 HI. UNEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
■~^(x, /)—(«* (x. 0)- =0, (jc, t)£Q%—S£. (16.19)
We multiply this equality by (Ax)nu\x, t) and sum over all points of 0^, taking
here and below to be equal to zero on Sj^ and outside of Oj^. The second
term is transformed by the formula for "summation by parts”:
(A*)" 2 “ = — (Ax)'12 vux,' (16.20)
which is valid for any functions u and v, one of which is equal to zero on Sj
and outside of The derivation of formula (16.20) is given in [65c]; it is
easily done on one’s own, however, using the formula for calculating the divided
difference of a product:
U (JC) V (X) ) — 8 (x) VXl (jc) 4- ttX[ (JC) V(XU ...
.... jcH-Ajc, jc„). (16.21)
Thus, owing to (16.20) we have from (16.19)
and consequently /sO on , i.e. the homogeneous systems corresponding to
tbe systems (16.17) have only trivial solutions on each layer Q^, and hence the
complete nonhomogeneous systems are uniquely solvable for any right sides. In
order to obtain a uniform estimate for all u for various Ax and At we multiply
(16.17) by 2(Ax)nAcu^(x, t), sum over all interior points of and transform
the first term by formula (16.10), while the second is transformed by formula
(16.20). This gives
(Ax)“2(«A)’-(A*)»E^+(A')s(M“ 2 2(“f)s
a* Q„a *-•
*.
+ 2 At (Ax)" 2 «?,«? = 2 At (Ax)" 2 2 (16-22)
V lJ Xi X1 b a. I A
0.A ; * u£
From here it is not difficult to obtain die basic estimate
max (Ajc)“ 2 («a)2 + (Aflf* 2 2 («?)*
o* 1-1 Q*
’ (II
4-A^(Ax)" 2 2(“*)2<c (16.23)
§16. ROTHE’S METHOD; FINITE DIFFERENCES
251
with die constant c depending only on l/ll 2 Qj" 1 ^oH 2 0’ l^e numbers v and n and tbe
height T, and thereby a general estimate for the whole family of solutions lu^l
for various Ax and At. Estimate (16.23) makes it possible to pass to the limit
as Ax —>0, A{ —»0 and prove that the limit of is a generalized solution of
the problem. A detailed description of this with the necessary auxiliary proposi¬
tions concerning the interpolation of functions on nets and embedding theorems for
net functions is given in §6 of Chapter I, §2 of Chapter III and §2 of Chapter IV
in the book [s 5c]. A rough picture of it is as follows: with respect to each of the
iA a continuous function (x, t) is constructed in Qj that coincides with
at the vertices of the net, is linear in each argument when the others are fixed,
and is equal to zero for * € 0^, t 5 [0, m A {]. The functions (x, l) are ele¬
ments of v*rtue of (16.23) they have uniformly bounded norms
(1624)
Because of this one can choose from iu^ I a subsequence with A^x
and Aj< —> 0, which converges weakly in ^’^(^r) ®° so,n* function u(x, t)
O | H
from This function u is a generalized solution of problem (16.5). In
order to prove this we introduce a different interpolation of the functions
giving step functions u (x, i), that are equal on each elementary cell of the
net to theit values at one of the vertices of the cell (suppose for definiteness at
tbe vertex having the smallest coordinates). It has been proved th»t the u Hx, t)
converge weakly in L-JiQf) to t*ie salne function uix, t) as the u^(x, f), while the
(u^k), j_ n, converge weakly in L2(Qf) to its derivatives du/dx^ We
take an arbitrary smooth function ijix, t) that is equal to zero both near Sj. and
for t > T - f, < > 0, and multiply both sides of equations (16.17) by At(Ax)nv(x, t).
Then we sum the obtained equalities over all interior lattice points (x, l) of Qj
and in the first and second terms we carry out a summation by parts in accordanc e
with formulas (16.15) and (16.20). This gives us the identity
—A*(Ax)”2 2 «An,—(Ax)n 2 %n (*. o>
*“° a* e*
+ A/ (Ax)" 2 2 =m*xr 2 2 /** «6-»>
Q& l ft-1
252
in. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
which, upon using the above-mentioned step interpolation, can be written in the
form
J —uA (tj,) dx dt— J lfoirjtx, 0) dx
Qt
t T
+ J J “ J jf^dxdt. (16.26)
4(0 Al Q
In this identity we can pass to the limit with respect to the above-chosen sub¬
sequence &k, since the u^k and (u^f) converge weakly in L2iQj) to u and
du/dxt respectively, j;, (*lx) and (jj() converge uniformly in Qj- to r/, drj/dx.
and drj/dt, the oS” (more precisely, some subsequence of them), remaining uni¬
formly bounded, converge almost everywhere to a-, the i/t^ converge in L jXl) to
while the converge in L ^Qf) to f. In the limit we arrive at identity (16.14)
for u and thereby see that u is a generalized solution of problem (16-5). By die uniqueness
theorem for generalized solutions of problem (16.5) in the class W we can assert
that tbe whole set of approximate solutions lu^l has as its limit for Ax —>0 and
At —> 0 the function u.
The difference scheme considered by us, as well as the method ot proof of its
convergence, can be applied to general parabolic equations. It is well known that the method
of finite differences permits one to observe an improvement in the differential properties of
solutions in relation to an improvement in the differential properties of die known functions
defining die problem. Tbe dependences established in this way are not exact but are not far
from it. As to how it is done, the reader can become acquainted with it in the book [65cj.
§17. ON FOURIER’S METHOD
Let us consider the problem
= — cMu — Q, b|s< = 0. «|,_0 = <|>o(*). (17.1)
where 3Hu = (a.lx) u ) - a(x) u.
If Xj xt
As is well known, its solution can be formally represented in the form of a
Fourier series
CO
«(*,()=S¥"Vt»W. (17.2)
*-1
§17. FOURIER’S METHOD
253
in which the a^ = {i/iq, s Jjjtl'Q'l'kdx’ w^ile the and the —Xk ate die
orchonormal functions and their corresponding eigenvalues for the elliptic problem
<*»»=■-*»|,-=0. (17.3)
Each term of the series (17.2) is a solution of equation (17.1) satisfying the
boundary condition, while the whole series is for t = 0 an expansion of the func¬
tion ifrffx) in the functions i!>k(x). For a justification of Fourier’s method it is
necessary to investigate the convergence of the series (17.2).
Depending on the character of the convergence of the series (17.2) its sum
u(x, t) will be a generalized solution from this or that function space. Thus, for
example, if the series (17.2) converges in L^Qj), then u(x, t) will be a general¬
ized solution of problem (17.1) from L^Qy); if it converges in W then u(x, 4) will be
a generalized solution from K'j,ll(l?j'), and so on. An investigation of the conver¬
gence of the series (17.2) reduces to a study of the expansions
OO
<PM— 2a»<M*)- a> — (<P- >M. (17.4)
»~!
of arbitrary functions 4> W *n ^lc eigenfunctions ifr^.
The convergence of (17.4) in L2((l) for an arbitrary function <f>(x) from L2(Q)
is a consequence of general facts from the theory of selfadjoint operators and a
theorem proved by K. Friedrichs in the middle thirties on the possibility of ex¬
tending an operator.^, prescribed on a set of smooth functions equal to zero on
5, to a selfadjoint operator. From these same propositions it is also easily^de¬
duced that the series (17.4) converge in ^(Q) for any function d>ix) from
These propositions ate in turn a development of investigations on the convergence
in L2(Q) of expansions in eigenfunctions of the Laplace operator and the one¬
dimensional differential operator(d/dx)(p(x) d/dx) + q(x) (see [15b]).
The convergence of the series (17.4) in the spaces ^"(0) for any integer m
was investigated in [a5c]. In particular, the following theorem is proved in Chap¬
ter II of that work.
Theorem 17. 1. If </>(x) is an element of BjKQ), m > 0, satisfying the bound¬
ary conditions <£|j = = • • • = =. 0 (for m m 0 all of the conditions
drop out, while for m = 1 the first is retained: $|(- = 0), then the series (17.4)
converges to it in the norm of In this connection the boundary S and the
coefficients a.^(x) and a(x) are assumed to possess a certain smoothness.
It is easy to see that this result is exact and convertible (i.e., from the
254
III. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
convergence of the series (17.4) in the space IP"(Q) it follows that its sum <f>ix)
possesses the properties Indicated in Theorem 17.1) This follows from two conjec
tures proved in[^5c]> j) for ^ sum Gf series one bas the
estimates
S «*+*(■*)
k~\
(m)
^ Cm
2,0
2«?(IMm+>)
ft — 1
(17.5)
and 2) for any function <f>(x) satisfying the conditions of Theorem 17.1 the coef¬
ficients 0^ = (<f>, 0^) are subject to the inequalities
m / oo
|M'Tc y ^oj <B9I8I<<,. (17.6)
The constants c , c^, depend only on m, S and the coefficients of 311.
The numbers as is well known, tend to +<*> for k —* ».
In view of this and the estimates (17.5), (17.6) the series (17.2) will converge
for t > 0 in the norm of any W™(Ji), m > 0 (if only S and the a^, a are suffic¬
iently smooth) as long as 6 £.2(0). And its convergence for t = 0 (and
thereby for t > 0) is determined by the properties of the function <1>q(x) as form¬
ulated in Theorem 17.1.
Remark 17.1. We note that if one confines oneself to the convergence of the
series X“=1 a^i/i^ix) in ^”(0') fot any strictly interior subdomain 0' of the do¬
main (1, then it is not necessary to impose any smoothness conditions on the
boundary S. Indeed, in deriving inequalities (17.6) oaly the formula for integra¬
tion by parts is used, while inequalities (17.5) can be replaced by the inequalities
* r co -ly
<*«(IC,|) 2U!([**r-H) • (17-7)
2,Q L*-1 J
(m)
2 <**♦*(*)£(*)
*-!
where £(x) is an arbitrary sufficiently smooth function that is equal to zero near
all of the boundary S, and these inequalities are derived without making use of
any properties of S at all (in connection with inequalities (17.7) see, for example,
§7 of Chapter III in the book
The paper [4*] is devoted to an investigation of the convergence of the
series 1^-^a^tfr^x) in domains with poorly behaved boundaries for the case
when Si is the Laplace operator. The question of convergence of the series
(17.2) in L2(Sl) for the case Sin - (of jMux,)x. + b^x)ux, + a{x)u is investigated
§18. THE LAPLACE TRANSFORM METHOD
255
in the papers [72]. For an investigation of die convergence of the series (17.2) in
the spaces ^"(Q) it is necessary to take into account the inequalities
P*(-*)BgS>< + M2,Q]. (17.7*)
ll«Wtl^Q+I)<cm[l(o^'”+1«. | + ||a|| 2 Q], (17.8)
proved for nonselfadjoint ® and for various homogeneous boundary conditions
(see Chapter II of [55c]). In connection with the series X“=jOjq!'^(x) they give
an estimate of tenns of H2"- lV&M2,Q
and an estimate of |2J=1oa^4||(|^+1) in terms of |(2“= f + Vt. S“= (a,, A£ty4)|
“d
Fourier’s method also yields a good representation for a solution of die
inhomogeneous equation
tt,—a^a = f(x.t), (17.9)
as long as 3fl, as in equation (17.1), is a formally selfadjoint operator with coef¬
ficients not depending on t. A solution of this equation that, is equal to zero on
Fj- is given by the series
i
<)>* (jc) J e-x^‘-^ft(X)dx. (17.10)
jt-i o
where and Aj have their former meaning, while f^(t) = (f(x, f), ifr^x)), The conver¬
gence of such series in can be investigated by using the propositions formulated
above for the series Analogous investigations are carried out in [6 5 c] for
the boundary conditions du/dN + <ru|j = a..{x)ux.cos{a, x■) + ou|j = 0, in which n is the
normal to 5
§18. ON THE LAPLACE TRANSFORM METHOD
For solving die Cauchy problem and die boundary value problems for the equations
u, — aUus&ttt—(aij(JC)uX) -f «, (jc)u)^ 4 bi (jc) a,(
4-0(jc)# = /(jc, 0 (18.1)
with coefficients not depending on t, one often makes use of the Laplace trans¬
form in t, which reduces them to corresponding problems of elliptic type with a
complex parameter A. Let us describe the basic plan of attack for this by using
as an example the first boundary value problem
“(*•
256 III. LINEAR EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
tt, — o#K = f (x, t). h|^ —0. (18.2)
in the semi-infinite cylinder Q = Q x (0, <x>).
We multiply both sides of the equation by where A « Aj + £A2 is a com-
pier number with A^ > 0, and then integrate over t from 0 to ». Assuming that
all of the integrals converge, we transform the first of them as follows:
00 00 oo
J ute~Mdt = ue~M J ue~udt. (18.J)
« o o
We dejiote the Laplace transforms of u and f by
U(X, l)~ J «(jc, t)e~Udt and/(AT, J.)= J f(x, t)e~
0 0
udt
and take into account that by virtue of the assumed convergence of the integrals
we have ue —» 0 for I —> oo, while by virtue of the initial condition ue —»
i/i0 for t —> 0. In this way we see that S' must satisfy the equation
Xu—oHa = / + <lv (18.4)
In addition, since u(x, t)\$ -0 it must satisfy the boundary condition
tfls-O. (18.5)
It is well known that problem (18.4), (18.5) is uniquely solvable for all values of
A on the complex A plane except the spectral values, which in the case of a fin¬
ite domain Q (and reasonable properties of die coefficients) form a countable set
going off to -oo. In other words, the operacor 3R - A has an inverse in L^Sl) for
all nonspectral A. This inverse operator (5H — A) 1 is called the resolvent of the
operator 3R, and with its use the solution of problem (18.4), (18.5) can be repre¬
sented in the form
u(x. X) = - (a* - X)-1 / - - X)-' Ifo. (18.6)
If tiix, A) decreases sufficiently fast as A2 —> ±°o when Aj > 0 is fixed, then
uix, c) can be recovered from it by means of tbe inverse Laplace transform:
kf+loo
u(x, t) = -^~ J u(x, %)eudh. (18.6')
Xj — ioo
§18. THE LAPLACE TRANSFORM METHOD
257
This formula gives the desired solution of problem (18.2). Such is die formal plan
of attack for the Laplace transform method. In it the functions u (x, l) and also
/(x, t) are actually continued by zero into the domain of negative values of t.
This permits one to take Xj > 0 and achieve convergence of the integral
ae^dt even in those cases when the solution u(x, t) increases as t —*+•».
But such a continuation is not very good when ^0(*) i 0, since it leads to a func¬
tion that is discontinuous at t = 0, while the Laplace transforms of nonsmooth
functions decrease quite slowly as X2 —> ±°° (and conversely, a slow decrease
in S' generates a nonsmoothness in u). In order to avoid "spoiling” problem
(18.2) in this way it is better to first reduce it to one with zero initial condition
(and, if possible, with f and its derivatives with respect to t equal to zero at
t = 0) and then, after continuing u and / by zero into the domain I < 0, apply die
Laptace transform.
A justification of Laplace’s method consists essentially in a study of the
smoothness properties of the solutions "uix, X) of problem (18.4), (18.5) with
respect to x and the manner in which ti(x, X) (as an element of some function
space of functions of x) decreases for X2 —* ±“> and fixed Xj > 0. This was done
for equations of parabolic and hyperbolic types, and also for strongly parabolic
and strongly hyperbolic systems, by one of the present authors (a detailed dis¬
cussion of the hyperbolic case is given in [< 5 c]) and for the parabolic case by
the authors of [fi7]. This method is also analyzed in many papers on the abstract
equations (14.1) with operators S not depending on £ (see [43‘ 73> 70; USa.b
127 et al.]).
CHAPTER IV
LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
In this chapter we consider the Cauchy problem and the basic boundary value
problems for parabolic equations of second order with, on the whole, smooth coef¬
ficients.
These problems are considered here by means of two distinct methods. In
§§1-9 we present a method dealing with exact estimates of a solution in terms of
the data of the problem and involving the construction of a special operator, which
is an analogue of the regularize! in the theory of elliptic systems [11*; 23] and
permits one to solve the Cauchy problem or a boundary value problem "in the
small” with respect to t. It is then possible, by making repeated use of a theorem
on the solvability of a problem "in the small,” to construct a solution in any
finite interval 0 < t < T without loss in the exactness of the result.
By means of this method we study the Cauchy problem, the first boundary
value problem and the problem with directional derivative in the classes H
and The construction of the regularizor is based on the idea of Korn and
Schauder of "freezing” the coefficients of an operator b, and hence this method
demands a detailed study of the so-called model problems: tbe Cauchy problem
and boundary value problems in a half space for equations with constant coeffi¬
cients without minor terms. Such problems are reduced by a linear transformation
of the coordinates to analogous problems, for the heat equation, which are con¬
sidered in § 1. This section has an auxiliary character. In it are cited the well-
known representations in terms of heat potentials of the solutions of the Cauchy
problem and of the first boundary value problem. In addition, an analogous repre¬
sentation, first mentioned in [9 2], is deduced in § I for the problem with direc¬
tional derivative. In §§2 and 3 exact estimates are proved for all of these poten¬
tials in the norms of H*>^2 an(j jpZm.m Some definitions and auxiliary proposi¬
tions that are necessary for what follows are contained in §4. The basic results
259
260
IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
on the solvability of the Cauchy problem and of the boundary value problems for
parabolic equations of second order in classes H^ 2 are formulated in § 5 and
proved in §§6-8. In §9, using as an example the first boundary value problem, it
is shown how analogous results in the spaces Wqm,m are proved by means of a
general plan set forth in the preceding sections. §10 is devoted to exact estimates
of a solution in subdomains Q' C Q J-, both purely interior and adjacent to the
lower base or lateral surface of the cylinder Qj.
It should be emphasized that the indicated method is used to prove the exis¬
tence of solutions of the Cauchy problem and of the boundary value problems in
classes of functions all of whose derivatives entering into the equation, or even
derivatives of higher order, are Hdlder continuous or summable with power q > 1
in the whole domain Qj .
The other method, which is presented in §§11-17, is the method of potential
theory for equations with variable coefficients. The central feature in it is the
construction of a fundamental solution Z(x, t, r) of a parabolic equation of
form (2.2) of Chapter I. This was first done in the papers of F. Dressel [21], who
constructed a fundamental solution by means of the classical method of E. Levi
[69], The essence of this method consists in the fact that a fundamental solution
is looked for in the form of a sum of two terms: a fundamental solution 20 of the
equation without minor terms and with frozen leading coefficients (it is written
out in explicit form and is the principal term with respect to the order of the singu¬
larity at the point x <* £, t = r) and an additional term in the form of an integral
operator with kernel Zq and a certain density Q' determined from an integral
equation. In the parabolic case this integral equation turns out to be of Volterra
type with respect to the variable t, and consequently is solved by the method of
successive approximations.
The restrictions that must be imposed on the coefficients in order to construct
a fundamental solution depend in much on the choice of the principal term For
example, if one chooses for ZQ a fundamental solution of an equation whose coef¬
ficients are frozen at the point (*, t), then it is necessary to require that the lead¬
ing coefficients a(;- of the operator L have derivatives d2a^ /dx^dx^ and
day /dt satisfying a HSlder condition. Indeed, under these conditions a fundamen¬
tal solution Z is constructed in [21] and shown to have the same singularity at
the point z - f, t « r as the function Zq. If one chooses for Z0 a fundamental
solution of an equation whose coefficients are frozen at the point (£, r), then
§ 1. THE HEAT EQUATION
261
all of the subsequent constructions are possible for HSlder-continuous . This
is done in the papers of W. Pogorzelski [94a,c] (for elliptic equations such a
method was proposed still earlier by I.. Lichtenstein; see in this connection his
paper [71] and also the monograph of Carlo Miranda, Equazioni alle derivate par-
zioli di tipo ellittico). By freezing the coefficients only with respect to x, one
can waive the Hdlder condition with respect to t for the a.. [25b, Part II]. A fur¬
ther significant weakening of the conditions on the a., is achieved in [79],
§ 11 of the present chapter is devoted to the construction of a fundamental
solution. For the sake of simplicity in the presentation we do not reduce the re¬
strictions on the ay to a minimum and we assume that these functions satisfy a
Holder condition with respect to all of their arguments. Under these conditions we
prove exact estimates for the derivatives Dx Z, O^Z, DtZ [25a] and their differ¬
ences with respect to * and t. As pointed out in [79], exact estimates of differ¬
ences werf; first obtained in an unpublished work of S. D. Ivasisen.
In § 14 the function Z is used to investigate the Cauchy problem in the class
of bounded functions and a still different method is employed to prove exact esti¬
mates of its solution in the norms of 2.
In §§15 and 16 a potential theory is presented for the first and second bound¬
ary value problems in domains with bounded boundary S. This theory is a generali¬
zation of the classical theory of heat potentials [SJb; 86; 119b], It permits one to
prove the classical solvability of boundary value problems, assuming only con¬
tinuity of the boundary values. However, in using a double-layer potential for
solving the first boundary value problem one must impose additional conditions on
the leading coefficients of the equation: the differentiability of these coefficients
with respect to the and the Holder-continuity of their derivatives. W. Pogor¬
zelski [94d] got rid of these additional restrictions by constructing a Green’s
function for the first boundary value problem with the use of a single-layer poten¬
tial. We present his method in § 16.
Finally, in §17 we present estimates of S. N. Bernstein for the first deriva¬
tives ux of solutions of the first boundary value problem.
§1. THE HEAT EQUATION AND HEAT POTENTIALS
In this section we examine some model problems for the heat equation
Aa(x. 0“=/(*• 0. (l.l)
Let us begin with an examination of the Cauchy problem
262
IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
Au = / (#> 0, —oo < / — 1 »),
«U = ?(4 U.2)
Under certain natural restrictions on the {unctions f and <t> the solution of
this problem is explicitly written out in terms of potentials, the densities of which
are simply expressed in terms of the functions { and <f>, while the kernel is the
fundamental solution of the heat equation
F (je, t) as=
JL— f « (x* = 2.*') for t > 0.
(«)* (L3)
0 for <<0.
which for the heat equation plays the same important role as the function \%\* n
plays for the Laplace equation. The potentials formed bom the function (1.3) are
called heat potentials. We consider at once those heat potentials that solve the
Cauchy problem (1.2). They are the volume heat potential
t
(r*/) = J jT(x—y, t — T)/(y, t)dy, (1.4)
Bn
playing the role of a newtonian potential for equation (1.1), and the potential
(F * t<|>) — Jr(* — y, t)<f(y)dy. (1.5)
Bn
The integrals (1.4) and (1.5) converge if the functions f and <f> satisfy cer¬
tain conditions at infinity: they do not increase too rapidly fot |*| —» °* (not
faster than ecx2 ) and the function f decreases as a power of I for 111 —» -
Since the results of this and the following two sections will be applied below mainly
to potentials with densities having compact supports!we will not consider increasing
f and tj>.
Hie potential (1-5) is for t > 0 a solution of tbe homogeneous equation
In addition, if the function <j> is continuous at the point x, then
lim(r * jtp) = (p(x). (1.6)
t-+o
In order to prove this, we make use of the following easily verified property
§ 1. THE HEAT EQUATION 263
of the function F: for f > 0
f DrtDsxT(x, t)dx = \ *• lt r~\Sr0' (1‘7)
J 10, if
en
From this property it follows that
lim(r*tq)) = lim f T(je — y, <)lq>(y) — y{x)]dy+y(x)
<->0 <->0 *
en
Jl_ _ 2*
=(4it) 2 lim \e TUx—z V~t)— qj (x)J dz -|-<p (*)—<p (x).
/-+o J
En
Let us now consider the volume potential
g(x, 0 ==(!’»/).
The kernels T{x y, t - r) and dT(x~- y, t - r)/dx. have weak (locally inte-
grable) singularities sit the point y - x, t =* r and therefore
dx, [dx, ')
It is impossible, however, to differentiate the volume potential twice with
respect to x or once with respect to t in this manner. Let us calculate these
derivatives, assuming that fix, t) satisfies a Holder condition in the variables x
and is continuous in t. We consider the sequence of functions
t-h
gh(x,t) = J dx J r(x~i>, t — x)f(y,.x)dy.
-oo Bn
It is evident that lim^^gg^ix, t) = gix, t). Let us prove the existence of a limit
for the sequence dg^/dt. We-have
l-h
dgh
dt ~
J J —'t—iy, x)dy
En
J r(jc — y, h)f(y, t — h)dy
t h ^*
J dx J h.‘~:0 [/(y, t>]dy
-oo £„
+ J r<^ — y, h)/(y, t — h) dy.
264 IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
Taking into consideration (1.6), we conclude that
—l|m i£*.
dt dt
Analogously
j dx J ... T) [/(y, x)ldy -{-/(jc, t).
(1.8)
<?»g . ,|m &gh
dxx dx, ^Tx&T,
" J * J ~Wdx) J) I/O- *> ~ /<*■ ■*» ** ^
“oo £
«
From the last two relations it follows that
(£-a) (r •/)=-/. (i.io)
If the function / satisfies a Holder condition in the variable t, then dg/dt
has the representation
J rfc J *ru~£t-x) t/(y< T)_/(J,_ 0Irfy> (M1)
£n
Let us express the solution u{x, t) of the Cauchy problem in terms of the
potentials (1.4), (1.5). We assume that the function / satisfies a HSlder condition,
while tj> is continuous. We extend f(x, t) in some manner into the domain t < 0.
By virtue of (1.10) the function
v(x, e) «= u(x, t) -g (x, t)
must be a solution of the problem
-£.-a,=o,
H_0=<p (■*)—«■(*• o) **♦(*)•
Consequently,
v(x, <) = (r
and thus
u(x. t)—(r */)■+<r* i4>). —(T */)!,_0. (i.i2)
§L THE HEAT EQUATION
265
We will normally make use of this formula by setting the function f equal to
zero in the domain t < 0. In this case
<
g(x, <)= J dx J Y(x — y, t — x) f (y, x)dy,
0 En
and ♦<*>-*(*>
t
u(x, t)= j dx j r(x — y, t — x)f(y,x)dy-j-(T*l(p). (1.13)
0 En
Besides the Cauchy problem, we examine in this section two boundary value
problems for the heat equation in the domain xn > 0, t > 0 of the space
the first boundary value problem
£_a«=o.
«Uo = 0' “Ir.-o-«*»(*'. *> (1-14)
and the problem with directional derivative
4~- — Av = 0,
cia5>
i-l *
Here x* «= {x., • • •, x .), while the 6. are certain real constants with 6„ ?£ 0. We
l n l i n
will suppose that > 0.
Both problems are formulated for the homogeneous heat equation and homo¬
geneous initial conditions since inhomogeneities in the equation and initial con¬
ditions are removed by means of the potentials (1.4), (1.5).
Let us introduce the notation
OO
(G » 2<t>°)= J dx J 0(x' — y', x„. t - t) 0° (/, x) dy',
Bn-i
© for t > 0.
0 for t < 0.
The solution of problem (1.14) is the double-iayer heat potential
266 IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
ar
t
“=-2(3
t
== —2 J dx J — (x'~y'd’x*n' *(/, x)dy‘. (1.16)
In order to verify this it is sufficient to demonstrate that it satisfies the necessary
boundary conditions for xa - 0. Since
« e.~,
oo x2 ,g
J J I. (1-17)
(4n)T ° t
fc/»-1
it follows that any function 0(1', t) which is continuous at the point (/, 1)
satisfies the relation
-2 lim
*_->o
• • *«-.
x y, /—t)—©(x\ 0) dy'=®(*'. <)• (i-i8)
Consequently the potential (1.16) satisfies the necessary boundary condition
for any continuous
The solution of problem (1.15) is expressed in the form of a potential
V = (G * 2$^)*
In order to calculate the kernel G of this potential we take in (1.15) the Fourier
transform with respect to the variables x and the Laplace transform with respect
to (. The functions transformed in this manner will be denoted by the symbol ~
placed over them:
t!lO
7ft. P) = W f J «''x t/(x'. t)dx' (1-19)
(2n)~ 5 *«-.
(here and below x £= *a£a)-
§ 1. THE HEAT EQUATION
267
Under such a transformation problem (1.15) goes over into the following prob¬
lem for an ordinary differential equation in the variable xn:
£l—(p+.g)v = Q, (1.20)
dx‘
U-21)
a«i
la this problem we are for the moment short one condition for the function tT,
which can be formulated on the basis of the following considerations. The function
v is expressed in terms of D* by means of the inverse integral transform with re¬
spect to (1.19), i.e.,
0 + 100
«(jf. o — J «ijr'C</C J p, xn)epldp, (1.22)
(2n)_r~2ni En-\ a~ic°
if the integral in the right side exists. Equation (1.20) has two linearly indepen¬
dent solutions: e~Xn and eXfl where by yjz is meant the value of the
root corresponding to a variation of arg z within the limits — n < arg z < it ; the
function S' is a linear combination of these two solutions. In order for the inte¬
gral in the right side of (1.22) to converge for any xn > 0 we must require that the
function v be expressed in terms of only the first of these solutions. This can be
achieved by imposing the condition
(1.23)
This is the missing condition for the function v.
Thus we must solve the problem (1.20), (1.21), (1.23). It is evident that
p. xn) = /=•(£. p)e-*«V^:
Calculating the function F from condition (1.21), we obtain
P~X Vp+l*
V&. P. x„)= —— <&i(S, P)
and consequently
268
IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
V(x, t) = J ,/x-t dt J e”'
(2n) s 2niEn-t a~la>
e-xn Vp*t‘ „
X —nTi ——-<$(£.
a-l
By virtue of the well-known formula for the Fourier transform of the convolution of
two functions we conclude that
v{x, 0 = (0*2^. d.24)
where
. '
0(x. fl. -1— f e"'t(S f e"—ilZ±Jl-~dp (1-25)
' ' (2n) 2n< J ^ fl_Jjo5 /n-^VT+F
(we have introduced the notation 6' = (/> j, ■ • •, &n_j))-
We choose the number a in such a way that the path of integration lies in the
domain of analyticity of the integrand. This will be tbe case for a = - + a2,
a> 0.
From the analyticity of the integrand with respect to p in the half plane
Rep > -^2 it is not difficult to see that G(x, l) = 0 for negative values of t.
One can calculate the integral (1.25) for positive t. Let us introduce the new
variable of integration s «= s j + is2 connected with p by the relation
Vp + - s.
Then
J it't — inVp4-V J t»i — bns
where W(a) is the right branch of the hyperbola s*~ $2 = a2. It is evident that
the value of the integral in the right side is not changed if the path H (a) is re¬
placed by the path Res = o. Let us make this substitution and put (3a= ba/t>n,
P = (PV---, £„_]:). We get
§ 1. THE HEAT EQUATION 269
Let us again introduce a new variable of integration v - s - if}( and interchange
the order of integration with respect to £ and v. We will have
0(x. t) = -T f —
X J exp|/*'£— & — (&)**+2Wft —
J-f/oo
J
‘-/I-1
1
.a-io
n-I
a-I a—too
(1.26)
where
7= J exp {- (£2 + <f£)2] '+<(*'+■ — jc„P)4| dl.
We assume that j8 ^ 0 (if /3 * 0, then it is easily seen that / =
(n/t)in~W2e~*'2/4t).
Let B be an orthogonal matrix of order n - 1 with elements /3jy , where
P;., = J!'. (J — 1 B — 1),
and the remaining elements are chosen arbitrarily. Let us introduce in the inte¬
gral / the new variables of integration i). (j = 1, • • •, n - l) connected with the
by the relation £ «= 817. Then
P£ = |f*|%.
t2+(PD2 = *l2 + P2n;.
x% = y'r),
where
n-i
y;=Sp6i^. d-27)
and consequently
270
IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
fl-1 oo
>-2 -oo
oo
X J *-(,+0sK'+,(n+aei0i'-v*i)1,i dI)i
if-I
fn\ 2 1
:(t) /T+P5exp
i V..i (yi + 2rt|PI-jf«mi)»
4t
ff-t
' a VT 1
■w
VT+F
7 eXP
/-'■i
n~ I
4<(l + F>
"W+FT i+P’
..i,m ipi
■•ii*1 r+> ■
and by virtue of (1.26)
G(x. /) =
I
exp
04 <«»
a-1
(4jtf)'T" VT+F
J_ V _ (yi-^iPi)a
' 41 Zuyi
J-i
4/(1+^)
x I
a-loo
n—I
, Vs d „v„ 1 v..« <y,-*«iFi)t
2u 41 2u?l 4/(1 +PJ)
a-i L 3
a+tm
v f t vH vx» gy, IP l\ rfp
X J PIt+F TqqF TTV) v
a—too
The first integral in the right side of (1.29) is easily calculated:
A+JOS
/ *
J M
a~laa
-v\ vx»
I+F T+F TTFr
(1.28)
(1.29)
§ I. THE HEAT EQUATION 271
—O©
r _ / ■-v Ri' (Xfl 1I ^t)*
= iy «(n-ii»i , (1,30)
The second integral, which we denote by K, is most simply calculated by means
of the obvious relation
dK 1
dxn ~ T+F
Since
•fOO
iaH-ax„-ay,\^\\ [ wH \ dw
1 + pa 1 + pjJ ,
it follows that lim^ _ + 00K = 0, and consequently
*=T+F J 'dz~lY~t(\ + P) Jexp[— dx
*n *n
oo
= 2 l\Tx J e~*dz. (1.31)
JK/p+l?)
Substituting (1.30) and (1.31) into (1.29) and making use of tbe fact that by virtue
of(1.27)
—ffr ■ y,‘
and consequently
i-i
(yi-^IPI)8
/-2
ITF
we obtain
272
IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
0(X. ()-
(4ixt) ‘ bn
n-1
x‘
+ -^rSP«^e'4‘™r f
f » TP n-i
2y_(——
(4n<)~ b„
X>
~xr
, 2VT x*h _±.rv*-vsr I
+ iir2i ?^ J * "
bx
Wfm
(1.32)
Here 6 = (6 j,*■ •, bn) and bx - 1,"=1bjxi. Carrying out the differentiation in the
second term, we get
0(x./)*= l—.hy-i
*?r . _ *
— bn (bx)
(4a<)T **
|»IVT
jr> <ȣ}*
^✓e~T? 4/W
“ I
J
ȣ_ j
(1.33)
bx
2VT|»(
This function, being a solution of the homogeneous heat equation, has the property
jfl
(4 *,T#
TT g ff (f, <)
3*; ■
(1.34)
and therefore by virtue of (1.18) it is clear that the function (1.24) is really the
solution of problem (1.15).
A special case of problem (1.15) is the second boundary value problem for
which b j = • • • = bn _j = 0, bn = 1. Its solution is a single-layer heat potential:
I
v(x, f) = —2(r*2<t>§)= -2 J dx J r(*' — /, x„, t — x)%(y'. x)dy'.
§2. ESTIMATES IN HOLDER NORMS 273
§2. ESTIMATES OF THE HEAT POTENTIALS IN HOLDER NORMS
In this and in the following sections exact estimates will be obtained for the
basic potentials (V * f), (T * j 0), (G * 2 40, introduced in § 1, in terms of their
densities.
It is convenient to introduce the standard notation foe the domains and spaces
in which the functions f, <j>, $ and the potentials are defined. We will make use
of the following notation (some of which we have encountered above):
£n+j is the euclidean space of points (atj,— , %n, t),
En is the space of points (* jf - • •, *B),
En is the subspace xn = 0 in £„+1.
En _j is the space of points *'<=(* j, •••,
Dn+ j is the half-space t > 0 in £n+1,
Db + 1 is the half-space *„ > 0 in En + p
D is the half-space x > 0 in
ft r n ft
«v 'V
D is the half-space t > 0 in E.
n r n7
R is the domain > 0, t > 0 in £„+j-
We will denote by {T > 0) the set of points of the domain B that are
situated in the domain I < T, with B standing for any of the domains introduced
above, except En and Dn ■ Thus, for example, = En x (0, T), =
E„ x T), etc.
The following ine<jualities will be proved below:
«r (2.i)
«fl+l En+l
((r*,<p»% (2.2)
vn+l cn
)<'■$ <c3<a>V'>(r, (2-3)
"n+l c«
In ail three inequalities I is an arbitrary nonintegral positive number, T is
any positive number, including + oo (in this case it is natural to take B),
274 IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
while (v) ^ ^ is the norm defined in § 1 of Chapter I. ^ The constants are every¬
where independent of T.
Inequality (2.3) for bn = 1, 6a = 0 (a < n) gives us an estimate of a single¬
layer potential. In addition, by virtue of (1.34)
~2 *!®)(G *2<I))'
and hence (2.3) also implies an estimate of a double-layer potential:
* 2<V)\(n ^ ^ c (2A)
For a proof of inequalities (2.1)—(2.3) we will make use of the following esti¬
mates of the kernels F and G:
|0Kr(*. Ol<fr,,H''r''£exp(-C-y), (2-5)
|o;£>i0(^. (2.6)
Estimate (2.5) is obvious for C <%■ For a proof of estimate (2.6) it is suffi¬
cient to demonstrate that
** , <»*>* ? ,
e «+W j e-*rfz<cexp(—(2.7)
bx
2V*|*t
for any x € Dn. If 6* > 0, then
S3 (6*)2 y j—■ (bx)*
Dx 0
2K<|6|
and
X1 .JtSL T /— -r1 <**>> ,— -r>
e 4, 4ri> J e-fdz^cy je~ v '®r<|/f*‘w,
bx
2Vt\b\
since x2 - (bx)2/b2 > 0. But if bx < 0, then
(hx&
x2 _ L-gL ^ x211 — max cos2 8j.
D Here we suppose PQ a P is + « in (i.ll) and (1.12) of Chapter I.
§2. ESTIMATES IN HOLDER NORMS
275
where $ is the angle between the vectors 6 and * and the maximum is taken with
respect to all x G. Dn such that bx < 0 (i.e., cos f) < 0). The maximum value is
achieved when x = 0 and the vectors x'= (x * ,) and i>‘= (6i> ,)
n L 7 n l I n~l
have opposite directions, with
b'
max cos’d =u- = 1 ■
Therefore if bx < 0, then
(bx?
*3
b1 b1
As a result we obtain inequality (2.7) and consequently (2.6). The constant C'
depends on b2/b2.
Besides estimates (2.5) and (2.6) we will make use of the fact that the poten¬
tial (r * f) satisfies the inhomogeneous heat equation (1.10), while the remaining
potentials satisfy the homogeneous heat equation. Finally, we will make use of
equalities (1.7) and (1.17).
Let us proceed to the proof of inequality (2.1). In accordance with the defini¬
tion of the norm (tr)^ (see (1.10) ff. of Chapter I) it reduces to a proof of two
Ea7
estimates: n+1
(DM (V * /)>^'i;K, < c </>2r,., (2.8)
where 2r + s«2f+ 2, £'»[{]« and
/ l+2-2r-s \
{Dr,Dsx (I’ * /)> tJ) < c {/)% . (2.9)
c«+i
where r and s satisfy the condition 0<l + 2-2r — s <2.
By virtue of equation (1.10) both estimates can be proved by restricting one¬
self to the case r = 0 and estimating only the derivatives (T * /).
Let us prove (2.8) for r = 0 and s = /' + 2. Since
Di(r*/)=Di(r*oi7). (2.io)
it is sufficient to prove estimate (2.8) for r =» 0, V = 0 (i.e., for I = a < i). Sup¬
pose, as in §1, (F * /) a g{x, t). By virtue of formula (1.9) we have
276 IV. LINEAH EQUATIONS WITH SMOOTH COEFFICIENTS
D\g(x. t)-D2x,g(x, t)
t
= j dx J Dir (x—y, t t) \f(y, x)—f(x, t)) dy
*00 j jr-y| <2 {x-jr*
t
-fdx J Dl,T(x' ~y,l-x)
~oo |jr-y|<2U-«'|
XI/ O’. *) — /(*'. x)]dy
I
+ j dx J [£>*r (x — y, t — x)
-00 1J— y|5>V| JT-J-’ |
~£^r(jc'-~ y, ^ — x)][/(y. T) - fix. x)\dy
t
+ | [/ (*'. t) — / (x, t)] a’T X
-OO
X | Dlrix — y, t — t)rfj»= S^i- (2.11)
(,r~y| >2U~jr'| *"*
We estimate the moduli of each of the four terms of the right side with the
use of (2.5)* The first and second terms are estimated in precisely the same way,
for example,
|/il<c J \x — yfdy
I jr-y |<C2| x-x' I
X J flexp (-C dx {f)™E<p t
-CO
1 x—y\ < 2 I x-x’ I B+1
** + l
In the third term, for \x ~ y\ > 2 \x - x*\ we have
|D|r(x — y, t — x) — Dl-r (x‘ — y, i-t)|
< c | x - y | (< - t)'"^ exp (-C
and hence
|/3l<c</)f£<n,U-^l J i x — yfdyx
J*—y |> 2 jx-jr' I
§ 2. ESTIMATES IN HOLDER NORMS 277
X / <#-*)'”exp(-Ci|^Jl)rft
Finally, I^ can be written in the form
*
IA = — J 1/ (*'• T)“/(*• T)] rfT
—OO
W r dr(jr —y, < —T) _
x J wk n>dSv
and since u-n-tix-f1
| ar <jc, t)
I dxk
it follows that
l/J^-^r^/y^n J dSy
n I jr-y 1*2 f jr-jr' l
1 n-t-2 i x-x’
x f (t-%f ^ rft=C|^-x'ia(/y;, (r).
—oo
Consequently,
{ei^r) <c </>£(?) .
C«+l
Applying this estimate to the potential standing in the right side of (2.10) for
0. ~ I ~ , we obtain inequality (2.8).
Let us pass to the proof of inequality (2.9) for r=0, 0</+2-s<2. Fran
the latter inequality it is seen that s > I and consequently £?* (P * /) =
DS%X (r * /)**/), where s j = s - lf satisfies the condition I - V < s I ~ I* +2,
1.e. 5 j can assume two values: s j - 1 and 5 ^ = 2- Thus the proof of inequality
(2.9), in the same way as the proof of inequality (2.8), reduces to the special case
i « a < 1, r=0 and s = 1, 2. Let us prove inequality (2.9) for these values of the
indices.
Using formula (1.9) for the case s = 2, and equality (1.7) for the case s = 1,
we see that for s = 1,2
1
Dsxg(x, t)— J dx J Djr(x — y,t—x)lf(y, x) — /(*. t)ldy.
278 IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
Therefore, assuming for definiteness that t‘ <t <T, we have
t
D‘xg (x, t) — Dxg (x, t')— j dx j Dsxr(x — y, t — x)
r
Xl/(y, r) —/(jr. T)1 dy— j dx j Dx r(x — y, t' — x)
*'-> En
2 t'-t
Xt f(y, x) — f(x, t)l dy + J dx j[Dxr(x — y, t — x)
-oo En
— DXT (x — y, t' — x)] \f (y. x) — f {x. t)J dy.
Estimating the right side of this representation in the same way as the right
side of (2.11), we obtain
\D'xg(x, t)— Dsxg(x, </>*“' ,n .
* £ fl + 1
which proves inequality (2.9) for r = 0, I = a < 1. As a result we obtain inequality
(2.9) in the general case.
Thus inequalities (2.8) and (2.9), and consequently inequality (2-1), are proved.
Let us pass to the proof of inequality (2.2). It also reduces to the estimates
<D^i(r.| ¥)Oh,<*<9>£ (2r+*-0 (2-12)
and
(P'fixfF *, *)>i ‘ <° < *— 2r—5 < 2). (2.13)
Since (F * j <£) satisfies the homogeneous heat equation it is sufficient to
prove both inequalities for r = 0. A fact to be taken advantage of here is
°x(r *is>)=(r «,£>».
This permits us to restrict the proof of inequality (2.12) to the case r = 0, s = 0,
/ = a < 1, and the proof of inequality (2.13) to the case r = 0, s = 0, 0 < / < 2.
Let (r * j ij>) ** h (x, t).
For the indicated values of the parameters inequality (2.12) is proved in the
following way: since T > 0 it follows by virtue of (1.7) that
§ 2. ESTIMATES IN HOLDER NORMS 279
\h(x,t)~ k(x',t)\ = J Tty, 1)\v(x — y)—y(x' — y)]dy
En
< J TCy, f)|<p(x — y) —?(x' — y)\dy
E»
<1 x-x'\a{9)fn Jr(y. 0rfy«l*-*T<*>v
"En
Let us prove (2.13) fot r = 0, s = 0, I = p € (0,2). Since the function I’ is
even with respect to all of the variables x it follows that
h(x, 0 = J r(y. *)9(x— y)tfy = (" T(y, t)<p(x + y)dy
En 4,
J T(y, — y)-f-<p(*-l-y)lrfy
Bn
*= »(•*)+ J J r(y. 01<P (x — y)— 2tp(*)-f tp (* -f- y)]dy
*n
(we have also taken advantage of (1.7)) and
h(x. t)-h(x, 0 = 4 f rr (y. t)-T(y, t’)\
<
X 1<P (X — y) — 2<p (x) 4- f (x+y)] ay
t
*“T /rfT J t<p (*—y)—2,p (-*0+<p o+y)i ■ (2.14)
We now note that
|<p(x—y) —2cp(jc) + <p(jc-4-y)|<c<<p)£’!ji|p. (2.15)
For p < 1 this follows from
<p (*—y)—2<p (*)+<p (•* + y)
= [<p(*- y)-?W1 -H<p(* +y)—q> Ml.
and for 1 < p < 2 it follows from
280 IV. UNEAR EQUATIONS WITH SMOOTH COEFFICIENTS
<p(x — y) — 2<p(x) 4- <p(x -f y)
I
= J ^-[<p(-*-Hy)+<p(-*—*jOi<«
o
* f
= SyJ [vJ'X*+ty)—%(x—ty)]itt.
ui o ' ' J
Estimating the right side of (2.14) by means ot (2.15) and (2.5), we obtain
i
|a<*. t)-h(x, OI<e<«P>|| J ax j iy}»l*LgJl jay
? en
<c{tp)f f x^+^dx<c(t-t')^^)f.
a
Thus we have proved inequalities (2.12), (2.13) and, consequently, (2.2).
Let us pass to the proof of estimate (2.3). We will establish the following
inequalities:
{(Dxa *j<X>»^)B(ni < c (2.16)
(2+L)
<(Dfi *2®))^^ < « Wjjjjip . (2.17)
{(DxO *2®))^, < c « * \ (2-18)
((DtQ *2®)>|^(rj<c (2.19)
*• w fl+ t CB
<(G *2®)), ~(n < e <®>Sr>. (2.20)
in which 0 < a < 1,
In ordet to prove (2.16) we will make use of the representation
00
(JD/j *j®) = J rft J D/Ux’-y'. x„. T) I® (y'. t~x)
# *«-.
CO
— ®(y'. 0M/+ / rft J DxG(x'-y\ x„, *)[®(y', 0
0 E»-.
CO
—®(*',*)]<*/ + ®(*'. OjdT J Dx0(x'.xlt,x)4x'. (2.21)
§2. ESTIMATES IN HOLDER NORMS 281
Let us demonstrate that for all x_ > 0
ft
CO
J* (2-22)
u en-,
It is clear that the left side of (2.22) is equal to zero for i < n. On the other hand
the derivative 6G/dxn is expressed by means of (1-34) in the following manner:
A*»
Therefore
±*L-y*L*°. (2.23)
ia JxZ Za b„ dx, - K
! « J
“ ea-l
—zi*
u £».i
Let *. 2 G D . Let K be the intersection of the ball with center x and radius
*
21* - z| in the space En with the hyperplane (it can be empty).
From (2.21) and (2.22) one easily obtains the equality
I Jr—* I1
= J dT J ^0(*'->^*»-*)i®(/-oi<*/
° B«-i
|jr-*P
— f dx J ^0(z'—y'.zn, x)[<P(/. t—x)—<^(y',l)\dy'
« Bn-1
oo
+ JdT J bSr0(*/~/-*-T)
I*-* I*.
d 0 (*' — /. zn, T)] [<D(y'. <_t)-<D(/. t)]dy'
dz,
fsiJldx-
282 IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
OO
- f !«>(/. 01 dy' f gg..y^ rfT
K 0 *
+ J v 11--''
Z>T)]>«»»(/. 0-«Pu'. Ol*
00
-(-(IDU'. /) 01 J dx J — T) dy'
" V. *
•+■ $gL\®(x'. t)~<S>(z', 0). (2.24)
We will estimate each term of the right side of (2.24) with the use of (2.6). We
note that for i ^ n
0(7 (■<•' — y', x„. t)
dxt
dy'
iX>
j dx j G(x'~ y', t)« rfSy..
where X is the boundary of K, which for nonempty K is a sphere in the space
with center x\ such that if y1 6 £, then Vl** — J |2 + x* = ^1* “ *1* *-et
G* and G “ be the two summands in the right side of (1*33)) whose sum is G. It is
easy to see that
J G'(x'-y'. xn, x)nidSy.= 0
and
so that for i <n
10"(I-
t *
\dx /
da (x' — y', x„, t)
dxt
dy'
< J dx J | G"(x'~y'. x„, t)| dSy <
§2. ESTIMATES IN HOLDER NORMS
<c|x— z |"_I J e
rft
This inequality is also valid for i = n on account of (2.23) and the fact that
fdt J J
0
Taking into consideration this inequality and estimate (2.6), we obtain
0 fa-i
0G{z'— y', za, t)
n 1 00 o
■+ Si-*/-z/i Jdk Jt2 dt
+ l^-55T^I)dy'
/-I
s/ f I <>*0 <»' — y'. “»
* J j du/du/
dy'
u-kx + {l-k) z
+ <*>?>)
oo
Is 1 <?p <*' — T) 1
dxj I
+ J a*/J
dfl (*' — y', t)
d*,*
£n^0|rfT + C|^ _ zT]
i * -«r (rnj =« I x - * r wjjp.
which proves (2.16).
Inequality (2.17) is proved analogously. Since
284
IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
00
(0,0 *2<b) — J dy' J Dfi (x’ — Jl', x„, x) 1® (/, t — t) — ® (/, 01 dx,
K A
c»-l
which is analogous to the representation (2.21), but without the second and third
terms of the right side, it follows that the difference of the potential (Dfi *24*)
at the points x and z satisfies an equality, in the right side of which are found
the first three terms of the right side of (2.24) with kernel dG/dt in place of
dG/dxi, and inequality (2.17) is proved by using exactly the same estimates as
were used in proving inequality (2.16).
Let us pass to the proof of estimates (2.18)—(2.20). We denote tbe kernels
DXG, DtG, G of the potentials being estimated by the common symbol @(x, l).
We have
I
(® * 2®) = J dx J <S(x'— /, x„, t — T)t®(y', !)—®(y', t))d/
-® en~ I
oo
| ®(y'. Orfy' J *„> T^dT
e«-1 0
and, assuming for definiteness that t > we obtain
(@ (x. t) * j®) — (® {x, /') * j®)
I
= J dx f ®(x' — y', xn. /-x) (<£(/. x)~<X>(y'. t)\dy'
— J dx j © (*'-/. <'-t)l®(y', T)-®(y'./')ldy'
en-> ,
2t*-t
+ | dx J !©(*' — y'. xa, t — x)
£«-i
— ® (x' — /, xa. t' — t)l [® (y', T) — ® (y'. Ol dy'
2(1-/')
•+ J 0—®(y'. Olrfy' J ®(*' — y'. xn, x)dx. (2.25)
Ba-1 0
The first three terms of the right side are easily estimated with the use of
inequality (2.6). Only the last term is estimated in a manner that depends on what
is meant by <3{x, t). If ® = G, then
§2. ESTIMATES IN HOLDER NORMS
2 «-n
J (<!>(/. t) — ®(/. t')\dy' J 0(x' — y', xn, x)dx
-1 0
a /a\
<(/—I dT J lO(y\ x„, X)\dy'
’ # A /?
fl + 1 / ff \
2 m)j!ry
2(t-n
For (3 = dG/dt we have
J l®(/. t) — <S>(y\ t')\dy' J dG {X'~£' T) dx
V-. 0
= J O <*' — /. x„. 2 (<—<') )[<»(/. 0-W. OW/
£#-l
<<*-o2 wjjy J i<? (*'-/. 2(/-^))|d/
" £/I-1
a /«+l\
Suppose ® , a < %. Then
2</~n
J l® (/. o - ® (/. Ol dy' J dx
c»-»
= J rft J l®(/. /) —<D(x'. 01
dG (jc' — y', ■»„, t)
k» — » f
J dT J [<P(/. o—Ol <>a (*'~y'{-r"> T> d/
it F
20-0
+ [<x>(*'. o—■®(*'. O) J J
dQ(x' — y'. xm x)
dx,
285
Since
286
IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
j dt J t)dy'
=~ibi" / d% I *- x)d*'-
it follows that
n — I
2 </-/*)
f I® </. Cv'. O) J T) *
8U-0
<2(o^><n j * j i^-/r Tl h'
+ W“’?ft'f (®)pp.
It remains for us to prove inequality (2.18) in the case a > %. We note that
if k 4 ti, then
and by virtue of (2.20)
Further, from (1.34) we obtain
k**l
The terms under the summation sign have just been estimated, while the first term
of the right side is estimated in the following manner:
OP
J dx J T) |d>(x'—y',/—t)—<D(jc'—y.f'—1)| dy'
° S
«!-<•)• wr?,r,j"a, J
u t
§2. ESTIMATES IN HOLDER NORMS
287
and inequality (2.18) is proved for any a < 1, a=^=^ -
Thus inequalities (2.16)—(2.20) are proved.
Let us now use these inequalities to prove estimate (2.3). Like estimates
(2.1) and (2.2) it reduces to a proof of the fact that for 2r + s =* I' + I
(DW (G *2 0)))^’^ < e <0>>|’<n (2.26)
and for 0</+l-2r-«<2
/■ i-U-Zr-j) \
<AK (0 .ff)* < e <®>%. (2.27)
X*"« + I C/»
Since (G *2<1)) satisfies the heat equation, we need only prove inequalities
(2.26), (2.27) for the case when Dsx « Ds, and DJ « » i-e. we must estimate
the derivatives DrtD£r^D^ {€ *2 $) for 4 < 1.
Let us prove inequality (2.26) for such derivatives. We choose numbers r j< r
and s j < s — k so that the sum 2r ^ + s ^ has a maximum value not greater than l'.
Since k < 1 it follows that 2r^ + s j > V ~ 1.
Let r 2~ r ~ r\> s 2 ~ s - k ~ s i* We have
D',DSX~ *D*a (0 *2 <P) = £>?£# Dia (0 *2 Dr,'Dsx'’®). (2.28)
Suppose that r j and s j can be chosen so that 2r j + s j «= /'. Since 2r + s =
I' + 1, we have 2^2 + Sj + k = 1> from which it follows that = 0, s ^ + k — 1-
If on the other hand the numbers r j and s j cannot be chosen so that 2r j +
s j = but can be chosen so that 2r j + s j = I’ 1, then in this case 2r^ + k = 2,
s 2 = 0 (otherwise one could increase s j by unity and achieve the result that
2rj+ = /'), and consequently k = 0, = 1.
Enlisting inequality (2.16) or (2.17), depending on the value of 2rj + i j, for
an estimate of the right side of (2.28), we obtain (2.26).
In (2.27) the sum 2r + s can have two possible values: 2r + s = 1‘ + I and
2r + s = V.
In the case 2r + s = V + 1 estimate (2.27) is established in exactly the same
way as (2.26), with the use of inequalities (2.18) and (2.19).
288 IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
Consider the case 2r + s m I'. If 2r j + s j = I', then r2 = s 2* k = 0, while il
2r j + s j - I' - 1, then r2 * 0, s2 = 0. k = 1, and hence fot the proof of inequality
(2.27) when 2r + 5 = V one should make use of estimates (2.20) or (2.18).
Thus inequalities (2.1)—(2.3) are proved.
§3. ESTIMATES OF THE HEAT POTENTIALS
IN THE NORMS OF W^n'm
Let us prove that the potentials considered in the preceding sections satisfy
the inequalities
<«r •/)»JS£1<«<V»S%+(.
(3.1)
m •« &))?:£, < c
(3.2)
«(<j <<
'** fl+i *' a
(3.3)
(3.4)
where q > 1 and m is a nonnegative integer. The norms in inequalities (3.1)—(3-4)
are defined in § 1 of Chapter I.
It is easy to demonstrate that inequalities (3.1)—(3-4) for any m > 0 readily
reduce to the same inequalities with m = 0. For inequality (3-1) this is implied by
the following property of the volume potential (r * f), which we have already made
use of above:
D'tDj (r * /)=(r * OfOj/).
The other potentials satisfy the heat equation. Thus, for example, instead of
(3.2) it is sufficient to establish the inequality
and instead of (3-3), the inequality
Since
§ 3. ESTIMATES IN NORMS OF V^a'n 289
o;(r,I„)=(r*1o».
°rK (° *2°) - (°
it is evident that inequalities (3.2) and (3-3) with m > 0 follow from the sane in¬
equalities with mm 0. The same applies to (3-4).
We will therefore prove inequalities (3-l)-(3-4) under the assumption that
m = 0. In proving (3.1) we will use a theorem concerning multiplicators in Fourier
integrals [82],
We denote the Fourier transform of a function fix, t) with respect to the vari¬
ables * and t by f i£, £q):
CO
7ft. !o)=—J e~lxidx J e_'<E"/(*. 0*.
(an)"3" Bn -«
£ = fti L.)-
It is not difficult to calculate that
i
A+>
2
(2a) J (i|o + l!)
According to the well-known theorem on the Fourier transform of the convolu¬
tion of two functions,
and consequently
■sr<r-/)-TETF/- —jjj&r/.
The functions AQ(f0, f) = i(Q/{i(0 + ( 2) and Ajki(0, f) = - £k£/i^0 + £ 2)
satisfy the conditions
ft, ff-ar-k* «-5>
... <»S,_ _
for s = 1, • - •, n + 1, ij = 0, •.., n and 4- ij ik /')■ Hence they are
290 IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
mulciplicators of class (£?, Lq) ^ and we have the estimate
(r*/)j| + S ||-3^3I7(r*^|| B
**n+l » i-1 * • W. crt,,
«+l A, ./-I
representing inequality (3.1) with m = 0. Thus inequality (3-1) is proved.
Let us pass to the proof of (3-2).
Let @U, t) be a symbol denoting D(F or D^F. The kernel ®(x, t)
with respect to all of the variables x, and in addition, for any I > 0,
j ® (jc, t) dx — 0.
Therefore
(®»]<p) = — J ®(y, t) [<f(x — y) — 2(p (x) + q> (x + y)) dy
and
where
! (® * i<p) ll9j £fl ^ y Jl®(y- o IN (y) ^
£n
< ~rZT | N(y)dy,
t~*~4
A/(y) = II«p(Je — y) — 2<p(JC-) -+ <p(* + y)|L E .
/*
Applying Hdlder*s inequality, we get
£ \4-/ r _.f xi--;
i>*i<P) l!?, tn < J e~ » N" Cy) dyj J e~il dy j
-Vf f
1+35 J
t
and
is even
1) Although the formulation of the basic theorem [82] being used here contains a
somewhat stronger condition than inequality (3.5), it is completely clear from the proof that
the theorem is also valid for condition (3.5).
§3. ESTIMATES IN NORMS OF W^m 291
1
\7
IK®*!?)!!,,* + <cf
V t, + z£» /
~<(Jl
Q.E.D. The latter in this chain of inequalities is proved for q ^ 2 in exactly the
same way as (2.IS), and for q = 2 is implied by Parseval’s equality. Indeed,
<£(* -y)- l<tt(x) + 4>{x + y)
i— r sin2 — 0(£)rff,
(2n)nn } 2 ^ 5 5
ft
'"W
where 0(f) is the Fourier transform of the function <*>{x) with respect to all of
the variables x. Hence
f —— f\<f>(x - y) - 2<f>{x) + d>(x + y)| 2cfz
J lyl"+2 r
= 16 f|$(f)!2^ f,in4I?-JL..C f|J(OI2lfl2«if
J ^ 2 |y|.+2 J
n n n
For m = 0 estimate <3-3) reduces to the inequalities
|(4r*>)| ~ (3’6)
jj (dI~i,d*i!0 * 2<X))[J <c((<d»('t), a==0, 1. (3.7)
• n+i 1 **
and in the case a = 1 it reduces to just (3.6).
Let us prove (3-6). We note that
292 IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
(§,,«.)= f dx j T* (O (*'-/, t-x)-G>(x'-y’,t)ld/.
» En~!
so chat by virtue of (2.6)
^ a..
<e|e_CTA/,(T)4-,
0 J
where
<*>(■*'. i—c)-Q(x\ oil ...
*’ ‘‘it
Applying Holder’s inequality, we get
2 1
/ 00 \7
u-i-i).
Let us now form the L norm of both sides with respect to the variable xn on
the semiaxis x. > 0. We will have
I(t*<®)| <‘(/«WT5.K‘r’>Y
»•»«+1 \5 t4 + 3 0 XJ J
-(/ W-piJ -
and (3-6) is proved.
In proving (3.7) it should be assumed that » > 1* We use the fact chat
(0’-*DiiiO.j©)= | dx J Dl:kD*G(y\ xn, t)
0 En-1
X(0(V“- y, / — t) — (*', t— x)\dy'.
§ 3. ESTIMATES IN NORMS OF W^n’m 293
Consequently, 2 2
CO ,V
iK^'XG*^)ii,f' <c S-Sr S e'c~7~Nify)a/
u t 2 **~I
f J qggv.f
'■‘-fr-'+'S’ ! U~‘J
*(i —TSzf
1
1 / f tff(y'W V
where
and
iv2(/)=||<i>(JC'-/. 0-«>(*'. *)ILS.
q. an
WiD^D'a.m
'*• UB4.1
/ 00
<c| J Nl(y')dy' J -J-
dXn
i j * - ELXjT I
V*.-. • *.’0',+»9’ ’ V
J jy'j«“2+« \\ "qfX,Ea
£tt-l
Thus we have proved estimate (3»3)*
It remains to prove estimate (3*4), which, like (3*3), reduces to the inequali-
<58>
(S * ’®) ll„ „ *-0. 1. (3.91
Since
ties
294
IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
00
d'T(r.xn, t)
dxndx
I©(*'_/, t-x)
—<S>(x'—y'. ()} dy'.
(3-8) is proved in the same way as (3.6). As to inequality (3-9), it can be reduced
to the already proved estimate (3-3) with m - 0 if one takes advantage of equation
(1.34). We have
Estimating the second derivatives of the potential (C * with the use of
(3-3), we obtain
In this section we gather definitions and auxiliary propositions used only in
the present chapter and partially in §10 of Chapter VII.
We will study below boundary value problems for equations with variable coef¬
ficients in domains of cylindrical form Qj. = fi x (0, D C where T > 0 is a
finite number and 0 is a domain in the space En.
We will formulate at once the restrictions Imposed on the domain.
The domain fi and its boundary S can be both finite and infinite; the boundary
S must be sufficiently smooth. At each point of the boundary there must exist a
tangent plane. Let n(£) be the unit vector of the outer normal to S at the point
f. A Cartesian coordinate system iyl with origin at f, the y axis of which is
directed along n(f), is, as usual, called a local coordinate system. It is assumed
that there exists a number d > 0 such that, in a sphere of radius d with center at
any point f € 5, the surface S is given in a local system at tbe point f by the
equation
n
Estimates (3.1)-(3-4) are proved.
§4. DOMAINS. SOME AUXILIARY PROPOSITIONS
§4 DOMAINS 295
y„ = P(.y') (y' = 0't y»-i)). (4.1)
where F is a single-valued function. We will say that S € Hl (S € Cm) if F{y')d
Hl (K0) (F(y') e Cm(K0)) for any £ £ S, where KQ is the ball |y'| < i/7., and if
the nonas |F(I^IkJ^) are bounded by a common constant. We will consider
that at least 5 € H1+a; 0 < a < 1. Under this assumption, in a neighborhood of £
we have the inequality
|£|<‘i/ia- (4-2)
We will assume that it is possible to construct in the domain tl for any A > 0,
no matter how small, a finite or countable (for unbounded fi) number of subdomains
and Q^) possessing the following properties:
1. (Ju<6> = (jQ(S) = fi.
* tt
2. For any point * € fl there exists an such that x € and the dis¬
tance from x to
^ js not jess than dk.
3- There exists an Nq not depending on A such that the intersection of any
1 distinct (and consequently any N0 + 1 distinct empty.
4. The sets eft) and that are separated from the boundary S by a posi¬
tive distance (we denote the set of their indices k by IS) are ^dimensional cubes
with common center f ^ € fi whose linear dimensions are equal to and 2/SA
(/3 > 0) respectively, while the sets and adjacent to the boundary (the
set of their indices is denoted by ?l) are defined in local coordinates at a point
g{k) £ 5 jjy the inequalities
'|y.l<4 («=i «~1); 0<yn-F(y')<x,
|y0l<x (a= 1 n— 1); 0 < y„-F(/) < 2X.
The set of domains oj^ ' will be denoted by while the subdomains 0^*)
are denoted by
It is possible to show (we will not do this) that in domains with smooth
boundaries satisfying the conditions formulated above one can construct systems
of subdomains and for any A > 0.
The change of variables
(“<«)• 2»==y» — F(y') (4.3)
296 IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
takes a domain * {k £ HI) into the cube
|«„|<A,, 0<*„<2A,,
which is denoted by SJ. Its face |zj < \, zn =° 0 is denoted by a. We also set
sm = a>m n 5. 5w = 0(*,nS ??).
And we introduce the functions £^(*) having the properties
tww= l for *e“w- m
* jo for x£Q\ d'K
By virtue of property 3 of the domains
and hence the functions
H»*> (*) j £-li*L
J
possess the following properties: ^ = 0 in fl\
and moreover,
2 tf*’(*)£**’(*)= i. (4.«
ft
Let us pass to the proof of the necessary auxiliary propositions for the func*
tions from H6lder classes and from classes W^Jtftn (Qj).
We begin by considering the problem of extending functions from these classes.
Suppose y and V ^ are two domains in some euclidean space, with V C V j,
and suppose a function u is given in V and belongs to some function space B(^)
with norm ||u|(y. We say that it is extended with preservation of class into the
domain V ^ if a function u* is constructed in V ^ that coincides with u in V and
belongs to the space B (^ j), with
II “* lly, * II« IIk
where the constant is independent of u.
We first consider the case when the domains V and are symmetric
with respect to some plane. Suppose for definiteness that V is the half space
Dn + i, while V i is the whole space £n+J, and one wishes to extend into £n+j a
function u(xr t) given in #n+|, for t > 0. We extend it into the domain t < 0
§4 DOMAINS
297
by putting for t < 0
(4.5)
where the Af are numbers determined from the linear system of equations
It is easy to verify by means of this system that every sufficiently smooth
derivatives cfcu/dt^ (k = 0, • • •, N - 1) on the plane t 0.
The above construction for extending functions is due to Hestenes and Whitney
and is described, for example, in [28].
It is possible to show by means of quite elementary estimates that the exten¬
sion of functions in the indicated manner is an extension with preservation of any
of the classes used by us. In particular, the inequalities
hold for any Z > 0, both integral and nonintegral, as long as N > [1/2] (/V is the
number of summands in (4.5))*
It is clear that the described method for extending functions preserves class
noc only under an extension from a half plane onto the whole plane but also under
an extension from the domain 0 x (0, T) into Q x (- T, T), as well as under an
extension with respect to the space variables x (below we will extend functions
in this way from the domain into with respect to the variable %n).
This method can also be used to extend functions from domains with smooth
boundaries. One constructs in the given domain Q a sufficiently fine *'par¬
tition of unity* *
(tbe partition of unity (4.4) can be used). Then an extension of a function u re¬
duces to an extension of the function u^ » uijj^ for those k for which the boundary
of the support of the function ^ has a common part with the boundary 5 of
function extended in the indicated manner will have at least /V — 1 continuous
(4.6)
and also
(4.7)
A
298
IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
che domain ft. And this can be done by the above-mentioned method after a
"straightening” of S^.
In this way one can piove the following theorem used below.
Theorem 4.1. Suppose fi C En is a finite or infinite domain satisfying the
conditions formulated at the beginning of the section, and its boundary S belongs
to the class HK Tken every function u(x) € (Q) can be extended with preserva¬
tion of class onto the whole space En.
An analogous theorem is valid for the spaces (it is proved for integral
I in [5*]).
In addition to an extension with preservation of class from a smaller domain
into a larger domain of the same dimension we will make use of an extension of
functions given in the space En into the domain of larger dimension. We
will do this by solving the Cauchy problem (1.2), the solution of which can be con¬
sidered as an extension of the function defining the initial condition into
the domain First we note that the estimates of §2 imply the following re¬
sult.
Theorem 4-2. If f € + j), <j> € ff*+2(£n), then the solution (1.12) of
problem (1.2) satisfies the inequality
{“)V2'<c(V)o ,+ {*%**)
tt *■ i V "a+i a /
with the constant c depending only on n and I.
Proof. We extend the function f from + | into + i by formula (4.5) with
N a U/2); then according to (4.6)
</>SL<c,v>w
By virtue of (2.1) and (2.2) with T = e. the function (1.12) satisfies the estimate
, ++ <^;21
+«r * f)%+n!)) < c{l/)on+l + <<2)).
and the theorem is proved.
(T)
This theorem permits one to construct in the domain Dn+ j a function from
class + j) if the values of it and certain of its derivatives are given for
t - 0.
Theorem 4.3- Let there be given in the space En functions
§4. DOMAINS 299
<frj(x) € (En) (y = 0,• • •, [i/2)), with I being an arbitrary positive nonintegral
number. It is possible to construct a function u(x, t) £ SBe^
-<*,{*). (4.8)
M
fij2" («)
/or arey finite T and
j—Q
m
{*)$<* S(<P/r2/). (4.10)
n*r! y«o a
Proof. Let / > 2. We introduce the functions
i
'M*)=S(0(~1)1AJq)''--'(jc) (^=° [4])' (4-id
5-0
Every function u(x, t) that is given for ( > 0 and satisfies condition (4.8) also
satisfies the condition
(w -*)'«(*. *)|/=0 = <M*) (4-12)
and conversely. Let us verify that (4.12) implies (4.8). Conditions (4.12) can be
written in the form
s-o *■ M”.w J
('-° Ij})- (4-13)
From the equation corresponding to j = 0 we obtain
“l<=o = <*0W-
Using this equality, we get from the equation corresponding to j = 1 the condition
du |
and so on.
We define u (*, t) from the Cauchy problem
0. “i< =o — $0W — (*)•
300 IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
the function u* ^ from the problem
js£L_a««>—*«. »®u-tiw
and so on, while the function js defined from the problem
(4 ~ A) * *“ °- ^ *L=* [4]w>
Since — AVu, the function /) satisfies conditions (4-12) and
consequently (4.8). In addition, by virtue of Theorem 4-3
But (4.11) implies
(4)
2/)
•* f.Q 9 "
so that
[4] [41
and inequality (4.10) is proved.
We have assumed that I > 2. For I < 2 it is possible to take u(x, t) = 4>q(x).
It remains to prove that inequality (4-9) is fulfilled for any finite T. Here we
make use of the identity
+1^i!r I« - l,J *•
This identity implies that for any ot < 1
\g\% <Cx(T)y
rt+1 '**
3'g(-,T)
dr*
(0)
4- Ct (T) (g){* rf!) . (4.14
§4. DOMAINS 301
We take the function DrtD*u with 2r + s < I as g and put
Estimating the second term of the right side of (4.14) with the already proved in¬
equality (4.10), we obtain estimate (4.9).
Thus the theorem is completely proved.
An analogous result for the spaces + j) is formulated in the follow¬
ing manner.
Theorem 4.4. Let there be given functions <f>. € 1(F,n)
(/ = 0,• - ■ , m - I; m > 1). It is possible to construct in the domain func¬
tion u (x, t) such that for any T > 0
and
(*
This result is well known [9; 49c], Theorem 4.3 is also known, but in the
theory of functions it is normally considered as a special case of corresponding
results for spaces H^''2 of S. M. Nikol'skii [89], which we will not use.
Let us now formulate an important definition for what follows.
The set of functions from class Hl’^2(Qj.) satisfying the zero initial condi¬
tions
■»(*-" [I])
dt*
0
is called the space
Analogously, the set of functions from class {Qj.) satisfying the zero
initial conditions
dkti
-= 0 (* = 0 m-I)
ot 1/aO
is called the space (@y)*
302 IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
These spaces possess the following property: the zero extension of functions
from these spaces into the domain I < 0 is an extension with preservation of class.
The following assertion is valid for functions from j.).
Lemma 4.1. Let fi' Cfi and Q'T-Cl' x (0,r). If u £ ^(Q'T), then for all inte¬
gral j < I
(L\
(aVf) <ct~
t.Qx
This lemma is obvious.
Let us take an arbitrary small A and consider the system of subsets 5?^*.
Let Q^ x (0, r). The following norm is obviously finite for every function
u enl'l/2(gr)-.
Lemma 4.2. Suppose
«]$'= sup <«>$*>■ (4.16)
T a vt
A2*, (4.17)
with k < 1, A2k < T. Then in the class IJ^'^2(Qj) the norms °nd ^u^qI
are equivalent:
(4.18)
and the constant c does not depend on A and k.
Proof. The left inequality of (4.18) is completely obvious. It is also obvious
that
It remains to show that (u)^n < c \u\ty . Let tr + s = I'; we consider the
*,Vr“ vr
function
\—r-r\D'tDsxu(x. t)-Dr,D'x u{x\ 0|.
\x—x'
Let Xq, Xq and tQ be points such that
L__ d;d>(*. ol -Dr,D'xit(x.t)|
X~Xo
t-U tmt9
> I sup Ur I D;Oi« <*. 0 - D\D5x u <*'. <) |.
2 X.X-.t I* — * I
§4 DOMAINS 303
If - *q| > d\, then
D\Dsxu (x. t)\ -D',D’xu{x, <)L
1
< s“p i D'0'“ I = S“P S“P I D'‘D*U I-
Now by virtue of Lemma 4.1
sup I £>;d>(x, t)I < cX‘-r->T7~
Qm <?v
t-v
2
<«&V< * > W«t < -^r («)v
But if |*0 - jeJjl < dA, then by virtue of property 2 of the sets 0^ the points xQ
and Xq belong to some one set Q^o^, and therefore
<«)!r%t<2
The lemma is proved.
Lemma 4.3* Suppose a function iff^ix) defined in 0^ possesses the property
|#<M*)l<p- Cs<W41). (4.19)
while the numbers r and A are connected by relation (4.17). Thent for any func-
lion u G Hl-l/2(Q<Tk)),
<«<»>'$*>• (4-20)
in which the constant c does not depend on A and r.
Proof. It is easy to see that
from which, by virtue of (4.19), we get
Ml
<'•’*“>*! 4*)<c 2 {ij (“tl*1 + JP7 <?>!»<*>).
/ HI
< c ( Q<*) + S JFTJ WqI*I
J-9
304 IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
We have made use of the fact that for / > 0
One estimates in an analogous manner. Using Lemma 4.1 for an
*»vr
estimate of (u)^t) and taking into consideration (4.17), we obtain the required
inequality (4.20). The lemma is proved.
In what follows we will frequently have to estimate functions of the form
w(*. 0= 2 «'*'(*. <). (4.21)
b
where = 0 for x
Lemma 4.4.
\<0
Qx < N0 sup W'yQy (4.22)
Proof. Since
- 1 <*<*'>>%> < 2 <*<*'>>>,) < ^0 sup {*>■*■’)%■»
J-I T i-1 vt *' **r
it follows that
w «, = sup o’(*. < Na sup
X * Wt * VT
and (4.22) is proved.
Corollary. Suppose (4.17) holds and
w(x, 0= 2't,<*,(*)«'(*,(*. 0. (4.23)
ft
where while the are functions, each of which is differ¬
ent from zero only in 0*** and satisfies inequality (4.19) Then
{®)«T <csup{w ^'L,
T * WT
with the constant c being independent of A and r.
Let us turn to a consideration of the space (#r). We first note that by
virtue of property 3 of the domains £}*** the norm ||uj|? ^ is equivalent to the
§4. DOMAINS 305
«*>)*•
since
Consequently the norm { \u}) is equivalent to the norm
{{“Yfix < \<mQX <«
In the space W^n,m (Qf) the norms ((u))^2^^ and Ijall*2^ ate equivalent.
This is implied by the following lemma, which is analogous to Lemma 4.1.
Lemma 4.5. If (4.17) holds, then for any u 6 (Q^)
<<“>>*<*> W.24)
and for any u € W^*'m (Qr)
««»*\ < {{“))'?.%' (4.24')
where / < 2m.
Pioof. Let us prove (4.24). If / * 2m' < 2ro the proof is obvious. For j =
2m' +• 1 one must use inequality (3-15) of Chapter II, which in the present case is
valid for 8 = by virtue of (4.17). As a result we obtain
«<^l < + «'* II " H,. Q<?>*
and since for u € f^m’m (<?<*>)
11“ II* || | ^ < «■
ic follows chat (4.24) is proved. Inequality (4.24') is proved in exactly the same
way.
306 IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
Lemma 4.6. If t/r^ix) possesses property (4.19) for s < 2m and (4.17) holds,
then
with the constant c being independent of A and r.
Indeed, with the use of Lemma 4.5 and inequality (4.17) we obtain
2m
Lemma 4.7. If (4.17) holds, then, for a function w(x, t) of (4.21),
i<2mJ ✓.r'v/,/„„(*\»ry-
with the constant c depending only on Nq,
Proof. We have
<[S(i)v^yarfY
<c(N0)
Q.E.D.
Lemmas 4.6 and 4*7 imply
Corollary. //(4.17) holds, then the function (4.23) is subject to the inequality
i*«t <‘[s (4-25>
§4 DOMAINS
307
in which the constant c does not depend on A and r.
Lemma 4.8. Let S € 0^m. Then every function u(x, t) € (Qr) can be
approximated in the norm by infinitely differentiable functions that are
equal to zero for small t.
Proof. We extend the function u(x, t) with preservation of class into the
domain ojf+j, and then into putting uix, t) - 0 for t < 0. Finally, we ex¬
tend it with preservation of class from onto the whole space £n + j- Under
such an extension
The sequence of functions
a„(x, t)s=u(x,
each of which is equal to zero for £ < 1/ra, converges to u (%, () in the norm
ll<£>+1 . Now each function u# can be approximated by averagings of it, viz.,
infinitely differentiable functions that vanish for small t. The assertion of tbe
lemma follows.
If uix, t) £ W^m,m(Qr), then, as is seen from Lemma 3.4 of Chapter II, the
values of u and certain of its derivatives on Sr belong to the classes
r -54 <s■ l/1 '(S.), where s is an integer, and inequality (3-20) of Chapter U
holds. If the function uix, t) is given in the domain Dn+j and u £ ,m(Dn + j),
then, besides inequality (3.20), an analogous inequality is fulfilled between the
principal parts of the norms in (3*20). Namely, we have
Theorem 4-5- For any function u £ +j)
in which 2r + s < 2m and the constant c does not depend on T.
The norms in the spaces H^’^2iST) and Vl^^iST) il a nonintegral number),
as was indicated in §3 of Chapter II, are usually determined by means of a parame-
trization of the surface S., viz., a partitioning of it into sufficiently fine parts
‘ <V
and a mapping of each part into the space £R. In this chapter we will always
make use of a parametrization connected with a partitioning of the surface S into
subsets S(A)=n(A)n5 (we denote the systems of sets by
308 IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
while the sets 5^ are denoted by 6^), and consequently of the surface ST into
subsets S<** = S<*> x (0, r). They are mapped into the space En by the transfor¬
mation of space coordinates (4.3). Under this transformation S&) goes over into
the (n - l)-dimensional cube o
\za\<% (a= 1 z»— 1)
and S<A) goes into oT = <rx (0, r).
The space of functions from Hl,l/2(Sr) satisfying the initial conditions(4.15)
is denoted by In this space we have the equivalent norms
Isup !«<*><*'. o£
and *
{«)» = .up <«»>(*'. /)>*.
in which u^Hz', t) (z' 6 a, k € 30 is the function u{x, l) given on in the
coordinates of (4.3), i.e. in local coordinates at the point Also,
Lemmas 4-3 and 4.4 are valid in the spaces fjl’^*(ST).
Let as now introduce the spaces with nonintegral I > 0. We
base the introduction of these spaces on the following natural principle: if a func¬
tion u € is extended by zero into the domain t < 0, then the resultant
o q n *
function u° must belong to the class Let us determine what condi¬
tions must be satisfied by the function u for this to be true. We first consider the
case when \ = 1/2 < 1. We calculate the norm
/- J W w
"•cn 1 -oo >
WuH/.Q? dy'
dyl'l I |jt'— y' I'-I'l)
T t \7
-4- J dx' J dt J \u*(x', t)~ -„t j
£.-i
This norm is equivalent to the norm For / < 1 this is obvious; for
£ > 1 the second norm contains the terms
§4. DOMAINS
309
J*'
du®{x', i) i9b®(*', «’)
a*,.
a*.
ii+?
i-i
(i= X),
which are absent in the norm [u0]^ . .. They are estimated in terms of this norm
with the use of embedding theorems for the spaces with nonintegral I [9],
We do not present these theorems here; we merely note chat a limiting special
case of them (q —» oo) is given by Lemma 3-1 of Chapter II.
The first summand of the norm is obviously equal to
- •’ .«
SW "'J 1^
(-1 0 £*_] £«_ii 1
dy
while the second is transformed in the following manner:
t T
^11- 1
dylli |je'_y'|»-l+«(«-UI> ’
llowing manner:
J dx' j dt j iu°(x',t)-u>(x\ Ol*
' „ j —CO —OO
t X
- J dx' J dt JI ■“<*'• -«<*'• W ud§w&
*i»- X » 0 11
t 0
+ J **' J!“<*'• or* J wzypg
0 -oo 1 1
T 0
+ J dx'j f T jjpllT
En„ j 0 -oo
T t
/ ‘"7" /ol*
«#»-* 0 0
Bn-1 0
Thus
310
IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
Kcr-Wf)'
j S «■»>
0
Concerning the latter term of the right side of (4.26) one can show the following.
If qX < 1, then
J
£ii-i 0
X
+ "Jr J ‘Ix'j\u(x',t)fdt. (4.27)
If q\ > 1, then this integral is bounded only if u{x, 0) = 0 (such a limiting value
of the function u(x, t) is meaningful for q\> 1), in which case
j
Bn-1 0
«I “ /* /«■»
B„-\ 0 0
But if q\ = 1, then an estimate in the form of (4.27) or (4.28) does not hold (see
in this connection [116b]).
In view of all that has been said it is natural to define the space Wl"^2(D^^l
* o y n
for I < 2 as the set of functions with finite norm
A
For 1/2 > l/q this means that the condition u (x, 0) = 0 is fulfilled.
For I > 2 the space f *’,/2(/><r>) is naturally defined as the set of functions
satisfying the zero initial conditions
d*u
dt*
§4 DOMAINS
i =0 (fe = 0 [41-1}
«*-o v L J J
and, in addition, having the finite norm
K? v^{{uK b»+(/ I
3H
(4.29)
fll
t? a (x', Q
<K'
14]
</je' | .
where A = 1/2 - [1/2] (if A > l/q this norm is finite only for |(_0 = 0).
Theorem 4.6. For any function u 6 W^a,m (R^)
(2m)
(4.30)
(2r + s < 2m).
Indeed, if the function a(*, t) is extended into the domain t < 0 by zero, we
will have, on the one hand,
«u»n«=
«+i
V, *<«
and, on the other hand, by virtue of (4.26)
.,(0
It only remains to apply Theorem 4.5-
In exactly the same way as Lemma 4.7, one proves
Lemma 4-9- Every function u € W^' ^ 2(D^) con be approximated in the norm
g(r) k)" infinitely differentiable functions that are equal to zero for small t.
The space ^ 2 (Sr) for nonintegral I > 0 is now defined as the set of
functions satisfying for I > 2 conditions (4.29) and having the finite norm
where
t)
[41
dz'
312 IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
with {{«)) ^^ being defined by equality (3.17) of Chapter II.
It is not difficult to show on the basis of Theorem 4.6 that the inequality
{O^DsB|(2?_2r's"1/?)<c ««)K2o) (4.31)
t x V»V7
which is analogous to (4.30), is valid for any u 6 For m - r — s/2 -
1/2q - k + l/q, where k is an integer, the norm on the left side is stronger than
the norm \D\ '
It is possible to show that the characteristic given by (4.31) of the properties
of the boundary values of functions from the classes JPjjm’m(Qr) and of their deri¬
vatives on the surface Sr is exact in the following sense. It is possible to find a
function u € whose norm can be estimated in terms of corresponding
norms of a certain number of its normal derivatives on Sf of the order r, 0 < r <
2m - 1, which are given. One such possible extension of a function from 5 into
Qt is obtained by solving a boundary value problem in Qf for a homogeneous
parabolic equation, which will be done below in §9.
Let us at once establish the necessary properties of the surfaces Sr and ro
(we recall that Fq is the set of points * € SI, t *= 0) of a function from W^m,m(Qr’)
that does not satisfy zero initial conditions. We confine ourselves to the case
m = 1. Let SR. = & x (0, r), where SI is a cube in ESI = a x (0, 2A); o C E .
* n n i
is the cube |*a| < A (a = I, • • •, n - l), 0 < xn < 2A.
Theorem 4.7. For any function u £ 1(Rr) for q <3
0, l)-u(x',x, 0)|? (2)
— dxn<c{Mq«■«>
r &/\
!dx'idt j
a 0 0 (* +
furthermore, when q > 2,
2A
jdx’jdt J rf*n<^IWI?,3}f) • (4.33)
rt n ft \* +• *» t*
Proof. We make use of the identity
§4. DOMAINS
u (*', 0, t) - u (*', xn, 0) = u (*', 0, () - 2k|x', , «J + a (*', xn, t)
+ 2 u j*', tj - u j*', -i, oj - [b(*‘, *b, «)- «(*', xn, 0)]
xn
(*', vo>-uj*', -j.ojJ- J -j + >m](7
3X3
I I
2 j“f (*’’ “ /“f *1.’ ~ 2
u(x', xn, 0)
The first three terms of the right side are estimated by means of Holder’s inequal¬
ity in the following manner:
I J J * + *] [j - m]^| < j I *,#&. V
» *- n
*)l dr„
r"dr„
1
xn
c
h?
0 * n
0
xn
314
IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
with the third term being estimated in exactly the same way as the second. Here
and everywhere € is a sufficiently small positive number (e < l/q').
From here with the use of elementary estimates one obtains an inequality that
differs from (4.32) only by the presence in the right side of an additional term
2A
u(x, x , 0)
4*'T’0]
Let us estimate this quantity in terms of the norm {{«(*, 0))) . Let £ (*)
9,5c P
be the ball with center at x = (*', xn) € K and radius p. We have
mea(Kx (x) fl &)
J 0)-u(y‘, yn, 0)]dy
kxjx) nSR
+ J »Cy'. y,. °]
K-wnU
dy
whence
§4 DOMAINS
315
< c
J -jMt-3 J 1“ “<?> °>!9^
kz <*>n5R
■Jpr£rr J (y. <»
» n «*„(*)(!»
dy
< c
jdxj\u(x, 0) - u(y, 0)|9
dy
3? K
l*-rl
n+2« -2
«.
• (4-35)
( 2}
The norm in the tight side is estimated in terms of j|u||^ by means of the
embedding theorem (Lemma 3-4 of Chapter II). Inequality (4.32) is proved.
Estimate (4.33) is proved in an analogous way with the use of the identity
0
For tbe expression in square brackets we have the estimate
T
jd*j \UX U t) - <lx(X, 0)| 9 -Aj. <
SR 0 * 2
"Sr
36)
(4.37)
From it and from (4.36) we easily obtain (4-33)-
Estimate (4-37) is proved in exactly the same way as (4.32), with the use of
the identity
V7
ux{x, t) - uXn(x, 0) = JL J U^x', *„ + £<)- uxy, xn + £ 0)]#
\/< 5
316
IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
1
VT
\fi i
j [“*„(*'> *„ + £>«)- uxJ'X’ xn’ ‘fi = /% (*'• xn+'Jt>
VT
VT
!f J uv(x’’ xn’V)iv-^r ] /“*„*„(*'> *„ + 1.
vr
+ "7= I [“x„(*'’ *n + >?. °> - “*„<*'> V 0 >] drl ■
V* 0
In this identity the arguments of the derivatives of a in the right side can
fall outside the limits of the domain 3?r when (*, t) £ 3Jr, but if I < 4A2, then
they are confined to the domain j?f = S x (0, r), where St = a x (0, 4A).
We extend the function u (x, l) from &r into SRf by the method of Hestenes
and Whitney described at the beginning of the section. The elementary estimates,
which we will not repeat, lead to the inequality
4A^>
jdx J !“*„(*■ *> - “*„(*> o)!9 5
» 0 t
0 , 2
In the case r > 4\^ we also have
<<*»(V
q,r.T
< c
i
jdx j \uxn(x,t)-axn(x,0)\'1 J^<,
K «A2
t 2
KJk+K>o),:,;
r», «<2> i
< c
Estimate (4.37), and with it (4.33), is proved.
Inequalities (4.32) for q ^ 3/2 and (4.33) for q ^ 3 can be deduced from
Lemma 3-4, since under these conditions the integrals in the left sides of (4.32)
and (4.33) are estimated in terms of the sum of the norms ||it(je', 0, l)||^ ^
(or ||uXn(*', 0, rtlj);^’) and ||uU 0)||<^2/?).
We now introduce, for functions given on a surface Pr, namely the sum of the
lower base and lateral surface of a cylindrical domain Qr, the norm
§5. BASIC RESULTS ON SOLVABILITY
317
(4.38)
for l/q < I < 1 + l/q. If u(x, 0) = 0, then it is easily seen that this norm is equiv-
From Theorem 4.7 we obtain the following refinement of inequality (4.30) in the
case m = 1, q = 3/2 and q = 3:
Here the b. are smooth functions given on Ff and having on ST the property
’S.?-1bi(.x,t)ni(x)4 0.
§5- FORMULATION OF BASIC RESULTS ON THE SOLVABILITY
OF THE CAUCHY PROBLEM AND BOUNDARY VALUE PROBLEMS
FOR EQUATIONS WITH VARIABLE COEFFICIENTS
IN HOLDER FUNCTION CLASSES
The basic result of this chapter is a theorem on the unique solvability of the
Cauchy problem and two boundary value problems for a parabolic equation of sec¬
ond order in a cylindrical domain Qj. - 0 x (0, T).
Let us formulate these problems.
We denote by £(*, t, d/dx, d/cit) the linear parabolic differential operator
with real coefficients
alent to the norm {u introduced earlier:
q,Sr
(4.39)
2 • T
2 •
(4.40)
(4.41)
318
IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
ml » & &u V * /v
•&{*' *• dx ’ dt j u~~ dt 2j atj(x’ dxtdx,
i J_l '
i, 1-1
+ 2ai(JCl 0-^- + o(x. i)u. (5.1)
dx,
I •» I
We assume chat this operator is uniformly parabolic, i.e., in the domains where the
above-mentioned problems are solved, inequality (2.5) of Chapter 1 is fulfilled for
any real fj,--., fn-
We assume that the coefficients of the operator of (5.1) arc defined in a
(T)
layer #n + j = En x (0, T). We will consider in this domain the Cauchy problem
•*(*'*' -W' w)u(-x' *)-/<*• o.
(5.2)
Suppose, further, that 0 is a domain satisfying the conditions mentioned in §4.
In the cylindrical domain Qj, « Q x (0, T) with lateral surface Sy. ** 5 x (0, T) we
will consider the first boundary value problem and the problem with directional
derivative:
&{x, t, -6L, -|-)u(jc. *) = /(*. t).
“ Lo=<p to. «Ur=® (•*, /),
«1,-0 ='p to-
*(*.*. .£)«|
(5.3)
l$r
(5.4)
ft
63 2 ~SJT + *(■*. 0 al*_ = ® (■*• 0-
“ ' r
We assume that the functions b-{x, t) satisfy everywhere on Sj. the condition
SM*- 9M*) >*>0- (5.5)
11-1
This condition can be written in the form \b * w| > 5 > 0, where b * (i j, • • •, &B),
and it means that the vector b does not at any point lie in tbe tangent plane to S.
The boundary condition of (5.4) can be written in the form
§5. BASIC RESULTS ON SOLVABILITY 319
IK*. oi-Jr+H.
$T
where 1 = b/|b|.
In problem (5.2), and in the case of unbounded 0 problems (5-3) and (5*4), it
is necessary to restrict the growth of the solution for \x\ —♦ oo. We will consider
these problems only in the function classes H? tjje elements of which are
bounded, and in the classes the elements of which tend to zero in a cer¬
tain sense for \x\ oo.
We will assume that the functions £ 0, $ in (5-3) and (5.4) satisfy the com¬
patibility conditions for x € 5, t « 0. These conditions consist in the fact that
the derivatives cfru/dfi|t-0, which can be determined for t * 0 by means of the
equation and initial condition, must satisfy for x € S the boundary conditions of
(5.3) or (5.4). We introduce the notation
ot lf«0
,*(*./. ^)a= J] alJ(x,t)1J^-~Yial$L-~au. (5.6)
I, j-1 * / — I
It is obvious that the functions «^(#) ik » 0, l) are determined in the following
manner:
alm (jc) = q> (x), ua) (x) = JL [x, 0. -§-}?(*)-+-/(.*, 0), (5.7)
while the remaining functions are found from the recursion relations
£)*e.
=S (*) ■^ (*• ■&) (x)+fW (x)- (5-8>
/-o
where 6^Hx, t, d/dx) is the operator resulting from the operator 3 of (5.6) upon
differentiating the coefficients / times with respect to t.
We will say that the compatibility conditions of order m > 0 are fulfilled for
problem (5.3) or (5.4) if
<*>(,«» = - Jr |f=0 = W - 0 m)
or correspondingly
320
IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
dk
dt*
t -0
x£S
(ft — 0.
/»).
= ©<S) (JC) (ft =
j-0
xZS
0 m).
We will first consider problems (5-2)—(5.4) in the Haider function classes
j]l,l/2_ foralulate directly the basic results on the solvability of these prob¬
lems in the form of theorems which will be proved in the sections immediately fol¬
lowing.
Theorem 5-1- Suppose I > 0 is a nonintegral number and the coefficients of
the operator £ belong to the class H*’l^2(D^+J. Then for any f €■ H*'^2(D*^J,
$ G Hl*2(En) problem (5-2) has a unique solution from the class
jjl+2,i/2+l(p(^)p_ satisfies the inequality
.<(+2) ^ „ h ( !«>._ J_l +
fl \ h+i « /
(5.9)
with the constant not depending on f and <j>.
Theorem 5.2. Suppose I > 0 is a nonintegral number, the coefficients of the
operator £ belong to the class H^^2{Qj,), and the boundary S belongs to the
class H1*2. Then for any /€ HtJ/2(Qr), <t> € Hl+2(0), 4> C Hl*2>l/2n(§T)
satisfying the compatibility conditions of order [1/2] + 1 problem (5-3) has a unique
solution from the class it satisfies the inequality
(5.10)
Theorem 5-3- Suppose I > 0 is a nonintegral number, S € /7^+2, the coeffi¬
cients of the operator £ belong to the class 2(Qj) and, finally, 5(,
b eHl+1’l/2+1/2(ST). Then for any f € Hl-l/HQT), <(>€Hl+2(Q), <t> € Hln’(ln)/2(S7)
satisfying the compatibility conditions of order [(/ + l)/2], problem (5-4) has a
§ 5. BASIC RESULTS ON SOLVABILITY 321
unique solution from the class with
I « Iq+2)<<(I/Iq + |tp|(Q'+2)+|Og+I>) . (5.11)
Since the proofs of these theorems are almost completely identical, we will
confine ourselves to a detailed proof of Theorem 5-3- The calculations in the
proof nf this theorem are somewhat more complicated than those in the proof of
Theorems 5.1 and 5-2.
Theorems 5.1—5.3 are first proved under special assumptions concerning f,
4> and 4, which amount to the following: <f> 0, while f and $ satisfy the zero
initial conditions
%l-° (*-• m
■o.
m =(
and
<?'<P I
L-o’
where i = 0,..., [1/2] + 1 for problem (5.3) and i = 0,.. -, [(Z + l)/2] for problem
(5-4). It is easily verified that under these conditions the functions f and 4>
satisfy the compatibility conditions of the order indicated in Theorems 5.2 and
5-3- In this connection u^H*) = 0 for k = 0,..., [1/2] + 1.
If the functions f, <f>, $ satisfy the indicated conditions, then we will call the
corresponding problems (5.2)-(5-4) problems with zero initial data. By making use
of the notion of the classes introduced in §4, we can formulate these
problems in the following manner.
1. Find a function u € ^ *2^/'2+1 (D^j) such that
4:''4r)a==f- (5.21)
where
2. Find a function u € such that
T
(5-3')
where
322 IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
f£H‘'l*(QT), a>e//+2'*+l(Sr).
3- Find a function u € |/i+2’*/2+1(@y) for which
J>[x, t, -jL. £)««=,/,
/ <m i „ (5.4')
where
. / ... i+1
fzn'^{QT), ®e« ■"*“(«,).
0 o
The chief step in the proof of Theorems 5.1—5.3 is the proof of the following
assertion.
Theorem 5.4. Under the conditions of Theorems 5.1—5-3 problems (5.3*) and
(5.4‘) have unique solutions in the cylinder Qr~ Q x (0,t), while problem (5.2')
has a unique solution in for r < , where Tq is a certain positive number
depending on the coefficients of the operator S., on b^, b (in the case of problem
(5-4*)) and on the various characteristics of the domain 0 and boundary S, but
not on f and 4>. The solutions are subject to the inequalities
l«l£w <«I/I$t> . (5.12)
n+l a+l
1 “ i£2) < * (i / +1 © g;J)). (5. i3>
I« C <c(\f($x + l® If’)- (5.14)
in which the constants remain bounded for r —* 0.
With the use of this theorem one can easily prove Theorems 5.1—5.3 for a
cylindrical domain of arbitrary finite height T.
As will be shown in §10 of Chapter VII, Theorems 5-1—5.3 carry over to para¬
bolic systems (see Theorem 3-1 of Chapter VII). Here we formulate a theorem on
the solvability of the first boundary value problem for a parabolic system of sec¬
ond order, the principal part of which splits into separate parabolic equations:
/-I 1 l-l /-I
m
+ 2a<‘'V. t)u' = f*(x, t).
Its I
§6. MODEL PROBLEMS IN A HALF SPACE
323
«Vo = <P*(*).
uk \$r = <Pft (x, t) (5.15)
(*=1, . . .* /ft).
Theorem 5-5. Suppose S€H^+^ and the coefficients a..t a^* \ a^*r^ be-
long to the class Hl-l/2(QT). Then, for any fk G Hl’l/2{QT), <f>k € Hl*2(tt),
€ H**2' i/2+l(Sf) satisfying the compatibility conditions of order [Z/2] + 1,
problem (5.15) has a unique solution it = (u1,-••, um) with v) € Hl+2,i/2+l(<?y)
and
i<1,*l*+3,+J, |a,‘C) •
The compatibility conditions for problem (5.15) are formulated in exactly the
same way as for problem (5-3), while the proof of Theorem 5-5 coincides verbatim
with the proof of Theorem 5.2. It is only necessary in all of the calculations to
take u as a vector and the a; and a as matrices with elements aj1’ * and
We selected problem (5.15) here because it will be the object of a special
study in Chapter VII.
§6. MODEL PROBLEMS IN A HALF SPACE
We consider the Cauchy problem and boundary value problems with zero initial
data in the domain for equations having constant coefficients and containing
only the leading terms. In the Cauchy problem it is required that one find a func¬
tion w €^i+2,i/2+1(0^j) satisfying in the equation
T- 2 *W iSsr-/. «■»
I, J-i
where / 6 *1) and the a., are constants satisfying condition (2.5) of
O B *1 tj
Chapter I.
In addition, we consider in the domain R^ the boundary value problems
324
IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
tt
do
dt
i. j-i
d*v
V dxi dXf
■/.
«i. t).
da fl*»
~sr~ 2j au
/.
S.
/-i
‘IT,
dxi dxj
= ©a(x'. <)
(6.2)
(6.3)
(the bi are constants with bn £ 0).
It is assumed that f € Hl’l/2(/?^), Oj. € ^+2,i/2+l{pf T)^
^|Je sojutjon cf problems (6.2) and (6.3) must belong to
the class 0»+W/*+l(ji<nj.
Theorem 6.1. Problem (6.1) is uniquely solvable in the class
while problems (6.2), (6.3) are uniquely solvable in the class ^+2>*/2+l(jj(r)) yor ony
T > 0. The solutions of these problems are subject to the inequalities
(6.4)
(6.5)
(6.6)
in which the constants do not depend on T.
Proof. We first prove this theorem for the heat equation (o„ «■ 5^.). As can
be seen from tbe results of §1, in this case the solutions of problems (6.1)—(6.3)
are given by
where
ra— J dx J V(x—y. t—x)f(y. x)dy = (r*/°).
o
f f(x. t) for <>0.
(6.7)
§6. MODEL PROBLEMS IN A HALF SPACE 325
where f* is an extension with respect to the variable *n of the function /(*, f),
given in R^\ into the domain by means of a formula of type (4.5), so that
f*® is the extension of f* into the domain t < 0 determined by formula (6.8), and
Wl = (P. _(r,/•%_„: (6.10)
n
and, finally,
u(.x, /)=(r*/*(,)+(o*2©o). <(U1)
Bj = ®2 - 2 (r *f*)\xn-0-
where
Estimates <6.4)—(6.6) are consequences of estimates (2.1), (2.3)» (2.4) and the
fact that, under the zero extension (6.8) of functions from the classes the
norm is preserved.
For example, inequality (6.6) is proved by means of (2.1) and (2.3) in the
following manner:
< <(r * rYJl+<(0 *2 »5)y'gi < <(r. r0))^
+* < <(r * ?))'$ +c {+<(r. /«)
< c | + (/*9)£<n i | < c | Wjm1 + (/)$n |
The uniqueness of these solutions follows from the results of § 2 of Chapter I
(see Theorems 2.6 and 2.7 of Chapter I). The conditions of Theorem 2.7 are ful¬
filled since one can take as 0 (*) in the half space *n > 0 the function
<P(*«) = -2 + 4mxn+2‘
Thus the theorem is proved for the case when o- * 8-, i.e. when problems
(6.1)—(6.3) are considered for the heat equation. The general case reduces to this
one, since problems (6.1)—(6.3), as is known, can be written as problems for the
beat equation. For this purpose it is necessary to carry out certain linear trans¬
formations of the coordinates. First we make an orthogonal transformation of coor¬
dinates
326 IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
y* — 1(2 fotx/>
reducing the matrix A with elements a-., which wichout loss of generality can be
regarded as symmetric, to one of diagonal form, i.e.
R
i “Al-
It is necessary to take as the (3^ the components of an Zth eigenvector of the
mattix A; the numbers are the eigenvalues of this matrix. Equation (6.1) is
written in the form
dw V */.. «
If one introduces the new variables
„ _ y‘
then this equation goes over into the heat equation
*ZpJl-Aw(z. t) = f (z, t).
In problems (6.2) and (6.3) these transformations rotate the plane » 0 in some
way. Therefore in considering these problems it is convenient to make an addi¬
tional orthogonal transformation of coordinates
![ — 2 Yu**.
which will take this plane into the plane £ = 0. The Laplace operator does not
vary under this transformation. Thus it is possible to transform problems (6.1)—
(6.3) by means of simply a linear transfonnation of coordinates into the same prob¬
lems for the heat equation. In this connection the boundary condition of (6.3) goes
over into
where
§6. MODEL PROBLEMS IN A HALF SPACE
327
6'« y ,
uh **
Let as verify chat the necessary condition
n
*«= £ $Srp*A^0 (6.12)
1,1-1" “*
is fulfilled. The equation of the plane xn = 0 has the following form in the coor¬
dinates ly\ and !z|:
n
2 $kayk “ o,
«
EP*,. —0.
fe — 1
On the other hand it can also be written in the form
Consequently,
from which it follows that
js Ynkzk —
*-1
(6.13)
— 21 a»Pla’
V7<|rf|<vU.
where p and v are the largest and smallest eigenvalues of the matrix A (the con¬
stants in inequality (2.5) of Chapter 1).
From (<S.12) and (6.13) it follows that
bn — -J §kn$kfil 3
1 V • * r is.
d
i.k-l
and
Thus the condition b' £ 0 is proved, and the proof of the theorem is complete.
328
IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
§7. ON THE SOLVABILITY OF PROBLEM (5.4')
In this section Theorem 5-4 will be proved for problem (5.4'). The basic idea
of the proof consists in the following. Let A be the linear operator defined in the
space Hl*2'L/2*l{QT) that associates with each element of this space the pair of
functions (£«, Under the restrictions on the coefficients of the operators
£ and SB imposed in the condition of Theorem 5.4 one has for each u €
ij‘+2’l/2+l(QT) the inequality
I \qx +1 J’4't+1) < « !<^- (7.1)
Let
. / _
§(/’ = H'*(Qt)Xtf+ * 2 (St)
o o
be the Banach space whose elements are the pairs of functions h - {f, $), where
f € $ € £m»<m)/2(Sf), ancj whose norm is defined as the sum of
the norms 1/1^ and ^. Inequality (7.1) expresses the fact that the operator
A is a bounded operator acting from into Problem (5 .4') can
be interpreted as the problem of solving the equation
Au = h, (7.2)
where h £ while Theorem 5.4 can be interpreted as a theorem on the exis¬
tence of a bounded inverse operator .
In order to prove Theorem 5.4 we will construct in this section a bounded
operator R acting from into //^ + 2’^2+1(<3r) and such that for any h and
v€Hl*2’l/2n{Qf)
ARk=*k -(- Th, (7.3)
RAv — v + Wv, (7.4)
where T and W are bounded operators in the spaces and
respectively, the norms of which are small if the height r of the cylinder Qr is
small.
Equalities (7.3) and (7.4) permit one to prove Theorem 5.4 with the use of
very elementary methods of functional analysis.
Let us fix the number r in such a way that
§7. SOLVABILITY OF PROBLEM (5.4')
329
imi < i, m < i.
On the basis of the contraction mapping principle we can now conclude that the
operators I + T and I + W have bounded inverses (/ + T)~^ and (/ + BO-1 defined
in the spaces and gl+2,l/2*l(QT) respectively. By applying the operator
(/ + If')-1 to both sides of equality (7.4) and replacing k by (/ + T) ^h in (7.3),
we show that for any h € and v € H^*2,^2**(Qr)
AR([ + T)-'h = h,
(J -|-Ur)-1 RAv = v.
This means that the operator A has bounded right and left inverse operators equal
to RU + fi-* and (I + W)~*R respectively. These operators, as is known, coin¬
cide:
/?(/ + r)-1 = (/-J-r)"1« = 4~‘. (7.5)
Consequently the operator A establishes a one-to-one correspondence between
the spaces Hl*2'^2+1(QT) and i.e. equation (7.2), or, what is the same
thing, problem (5.4'), is uniquely solvable for any h 6 Estimate (5-14) is
equivalent to the assertion that the operator A~l is bounded. But this follows
from equality (7.5):
IM'MUIk'+rfMIll*!!-
lU-Mklk'+wr'IlM
If one chooses r so small that, for example, J7*8 < Vi, then ||(/ +■ 7V*0 < 2 and
|^*"^|| < 2 \\R§. The norm of the operator R will be estimated below (Theorem 7.1).
Thus everything reduces eo a construction of the operator R> a proof of (7.3),
(7.4) and estimates of the norms |/?||, ||7*j» ||f'||.
Let us proceed to the construction of the operator R. We fix in an arbitrary
manner a small A> 0 and introduce the sets from §4. We introduce
in addition the following notation. We denote by Z% the operator, defined on the
functions u(z) given in the domain §2, that with each such function associates the
same function under transformation from the coordinates jzj, connected with the
local coordinates at the point by formulas (4.3), to the original coordinates
{*}. Further, let
ro { * & 9 \ da \1 / ax
-Erszj'
1,1-1
330 IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
and let be the operator £Q in the local system of coordinates \y\ at the
point , i.e.
_ °‘J
I /—
where
o**> i V
“ dt 2d aiJ dy.dy. '
I, /-> ' '
(A)
with the firi being the elements of an orthogonal matrix connecting the coor¬
dinates \x\ and (y):
in-
/M
It is obvious that the operator XQ also satisfies the condition of parabolicicy
(2.5) of Chapter 1 with the very same jj and v as £q.
Analogously, let
and
®<*> " ' Ij;
where
Let f(kHx, t) (ft € !S) be a function from 2(D^\ j). We denote by the
operator that associates with this function the solution of the Cauchy problem
with zero initial data
-S’o(E‘*). o. ^)t0 = /'*’(*. o.
(7.6)
which, as was shown in §6, exists and is unique. Further, for A; € 91 let
f<kHz, i) € gl-l/2(R&>), t) € and fl(A) denote the
§7. SOLVABILITY OF PROBLEM (5.4') 331
operator that associates with the pair of functions hS^ - (f^ \ $(*)) the solution
of the boundary value problem with zero initial data
-Stf'ftf’. 0, ^-)*<*> (z, 0 <=/'*'(*. 0.
0. = <D,*I(*'. t). (7.7)
The solution of this problem, as was shown in §6, also exists and is unique
in the class
o
Let £^Hx) and rf^Hx) denote the functions introduced in §4. We define an
operator R by the formula
M = 2 rfk) (x) v<*> (x, t), (7.8)
*
where
I /?<*>£<*>/ for k £ as,
0='Utf(6lU4-,e>/. for kex. a9)
By virtue of (6.4) and (6.6)
(*esK). (7.io)
B+I + l
(/?(»>(/<*>. <c((Pk%x] + (Wk'%lX) (7.11)
(*€»)•
with the constants in both inequalities being independent of r and k by virtue of
the uniform parabolicity of the equation and condition C5-5).
We introduce in the space /^+2,2/2+1 ) t[,e nonn of (4.16), and in
the space the norm |]A||^(/) = 1 /1^' + By virtue of Lemmas 4.1 and
4.2 this norm is equivalent to the norm |/|g^ + |#|j + 1*-
Let the numbers r and A be connected with each other by relation (4.17).
Theorem 7.1. The operator R is a bounded operator acting from the space
into the space ff**2’1'' 2+*(Qr), i.e.,
m^Kciihw^, (7.i2)
with the constant c being independent of X and r.
332 IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
Proof. Let us estimate the vector v = Rh by means of the Corollary of
Lemma 4.4. We will have
< c snp <?<*>)<£$.
Using inequalities (7.10) or (7.11) for an estimate of and the obvious
property of boundedness ol the operators Zk and Z£1, we get
(7.13)
for k £ 38 and
J + (7.14)
for k € 31. From here with the use of Lemma 4.3 we deduce
I»)S+*< f (7 <?(*,/><'«) 4- sup ,5<*»cI>>"+,>)
vt \ * »€* ®r I
<c(»fPV>2» + “{
and the theorem is proved.
Theorem 7.2. For any h on</ t> € H^*2’^ 2*^(Qr) relations {7.3) and
(7.4) ore valid, in which T and V are bounded operators in the spaces Srj^ and
^1*1,1/2+1 reSpectively anJ their norms are small for small r.
Proof. Let us introduce fot the minor terms of the operators £ and $ the
natation
It
•S’i~gjA “ = "*■ *)*•
i-i
_#,(.*, t)u = b(x, t)a.
Also, let
^ = (^0®. Is,). 'V = (-?>,
We have
.*v» - s =2c*wv>- ifHjrtn+
§7. SOLVABILITY OF PROBLEM 0.4) , 333
'• -k• °- 4r- £)]«*
If k €®, then
*4™ °-
= -2o(lw. 0.
while if A € 31, then
= ^’o(l<*>* 0. w)ZtRW(-Zi'^
=ZkMk) (|w. o. grad F . |) Zil^)
-^‘>(iw. 0, 4)] «<*>(*-¥*>/. zrW+tf*’/.
since, by the definition of , for z € 51 we have
o. -|, 4)rw(zs-1c(4)/. Z*:
We will always denote the n-dimensional vector with components dF/dz^, •••
•••, 3F/dzn_^, 0 by gradF. Under the change of variables (4.3)
d d dP d , , . d a
■357 = 1^—5^^ -^“157’
i.e.
d * ap &
Sy
Now using the fact that t^£ ^ ^ = 1, and-, for the sake of conciseness,
letting
Zw = R(>,)(z;X<k)f. z;'^®) (ftgw).
we obtal" S’Rh = ft-f-r,A.
334 IV. UNEAR EQUATIONS WITH SMOOTH COEFFICIENTS
where
Tth = 3>xRh + 2 — nw-S>w)
+ Sr»'*)[*?°(*’ t> 4x' W*)--2”o(&<*>- °- -§?' 4")] v(k)
+ S^[^(^0.^-rad Fi-.i)
-^(6W.0. •£.■£■)]*•*>(«. 0- (7.15)
In a completely analogous way we establish that
3SRh\s% — h + T2h.
where
r2h = ®xRh |St + 20sW*V*>- n‘*
+ Sn“’z,[»f>(s»>.o.
0. £)p».
Thus
i4/?A = {S’RH, &Rh |St) = (/. 0>) + (T,A. TjA) = h +■ Th.
In order to estimate the norm of the operator T we express the operators T j and
T2 in terms of the coefficients of the operators £ and S. We have
r,»-(g«,(i1 o-j|-+«(*. <)|m
-£ 2 K<«- ^+^0. o£%)
-£ is «)]■££
ft t, /-I '
*€* La, fi-J p
§7. SOLVABILITY OF PROBLEM (5.4')
m 0 "L if. i!2!L+*<*> _i?L_
00 ' * ' A. A«2 • a«8 i_ A*. Am I
335
r,A =
d*& *4 08 <**« ^ d*n j
o«t a * J
b(x, t)Rh |jt -|- ^ 0 ®<*)L
ktn i-1 T
+si’i"|[*l('.oi^L
*€» /-I T
*€» Q«i B
By taking into consideration the fact that for x € 0^
|«w(*. 0 —«iy(6W. 0)1
<|«y<*. 0~«y(5w. Ol + layfe'*1. <)-«y(lW. 0)|
feX for i>Il<xl-,<!
^teX' for /<lj<C '
and for z‘ € o
cXl for I <
16, to 0-M4w. 0)|<o.
(7.16)
and also making use of Lemmas 4.1 and 4.4, one can estimate the function T
in the following manner:
i(*)
17'ia)qt<c (milC + X S“P (vWfQw +-}J sup
+ K'-» 'tnpWiSp)
< c tT +(.iI+jl+x»-«) sup {v<y$
from which, by virtue of (7.12)—(7.14) and Lemma 4.3, we obtain
336 IV. UNEAR EQUATIONS WITH SMOOTH COEFFICIENTS
{TAqx < e (A+ J + *'-«)|| A (1^
<c(x*
In a completely analogous way one can show that
(r2*}£»<ep+>.)l|Aile(ir
Consequently,
II Th |g() <c{J + X'-m) || h ||^)( (7.17)
from which it is seen that for small x and A tbe norm of the operator T will in¬
deed be small. Since the parameter A can be taken arbitrarily small, it can be
considered that we have obtained the desired estimate for the norm of the operator T.
Let us pass to the proof of (7.4). We have
RAv — RA0v RA,v.
For k € SI
R^&ov = 0v - J?£k)v)
+ Rw(r2’o(x. t.
—^o(|W. 0. -§7, -gj-^F'v + P'v, (7.18)
Since, by the uniqueness of the solution of problem (7.6),
R(k)JZ>„(t« 0.-^.4)twt- = £%.
Further, for k e St
Rik) {Zi 'P^v, |St)
=«<*> [z:' zi' fe(*'S'?-isj
§7. SOLVABILITY OF PROBLEM (5.4') 337
W* °- ^'))‘Z*,£<*)®L 0J + Z*'5W«> (7.19)
since, by the uniqueness of die solution of problem (7.7) (Theorem 6-1),
Ri>)[sf)(lm- 0. -s;>-w)z*y%>
From (7.18), (7.19), (7.8) and (7.9) it follows that equalicy (7.4) holds, and
Wv^RAiV + Y. T{iw&0v — ^oSw«)
+2>w*w[(.2\,(*. *■£.-£)
*£.■£))*%]
+ 2 [-ZiT1 — -2’lmv).
+ £T>(‘)Z^<e)[z*‘* (•*• (*• '• ^-4)
&'(*.(*. *. I?)-jr,(tw. o. -yUw*|sJ
+ 2j [(^f ’ (lW. 0. ^ - grad
Jl
(<’(1-’. o. £-«*'£)
An estimate of the norm (lPt>) ^ 2* is obtained with the use of this representation
in exactly the same way as estimate (7.17). By inequality (7.12) and Lemma 4.1
we have
{RAtV}^ < c || ||#w < c <«* [*)£*.
Making use once more of Lemmas 4.3 and 4.4 and inequalities (7.10) and (7.11),
we obtain the estimate
338 IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
<«(tT {„}«£*> +xsu/<<W+-^s«P(<w
+ X‘-M sup <^>) < C (x'-r + J) M«*»,
which completes the proof of the theorem.
Thus we have constructed the operator R, proved equalities (7.3) and (7.4)
and obtained estimates of the norms ||r||, ||!F'|. As was shown at the begin¬
ning of the section, these results imply the unique solvability of problem (5.4')
and the following estimate for its solution:
Here c is the constant in (7.12). By virtue of Lemmas 4.1 and 4.2 this estimate
is equivalent to (5.14). Theorem 5-4 is proved for problem (5.4’).
As we have already noted, this theorem is proved in a completely analogous
way for problems (5-2') and (5.3'); moreover, the calculations for the proof are
simplified, since there are no boundary conditions in problem (5.2'), while in prob¬
lem (5-3') they have a simpler form.
We again emphasize that in exactly the same way it is possible to consider
problem (5.4') for the very same equation in a cylindrical domain whose lower
base lies in the plane t = < F. The height of the cylinder in which the exis¬
tence of a solution is guaranteed can be taken to be the same for any Iq ; this is
a consequence of the uniform parabolicity of the operator £ and inequality (5.5).
We denote this height by Tq.
§8. ON THE SOLVABILITY OF PROBLEM (5.4)
In this section we deduce Theorem 5.3 from tbe just-proved Theorem 5.4. It is
obvious that this requires reducing problem (5-4) to the problem with zero initial
data (5.4'). For this purpose it is necessary to calculate from the equation and
initial condition the functions u^Hx), k = 0, • • •, 1 + [2/2] (this is done with the
use of (5.7) and (5-8)), and then to construct a function v (x, I) which possesses
the property that
tL=“(,iw i+[{]). (8.D
and to introduce a new unknown function
u'(x, t) = u(x, t) — v(x, t). (8.2)
§8. SOLVABILITY OF PROBLEM (5.4) 339
It must be a solution of the problem with zero initial data
•s’[x- *' 5F- If)
*(*' '• -SrH
where
(8.3)
T
Let us verify that if the conditions of Theorem 5-3 are fulfilled, then a^(») G
fjl*2-24yjj fot k < 1 + 1/2 and
|«W£+*_",<e(|/^ + |»l2+S)) (0 < < < T). (8.4,
This is easily done with the use of (5.7) and (5.8). Since the coefficients of the
operator being derivatives of the coefficients of the operator A, belong to
the class (Qj.), it follows from (5-8) that for k < 1 + 1/2
| < | /*-> |',+2-“> + c S|
<|/l£+*SU(/>ir~2'>- (8-5)
V< Jm.0
In this way an estimate of |«t^| ^ ^ reduces to an estimate of the norm
|«0'>$+2 ^ for / < k. Since = <j> it is obvious that (8.5) implies (8.4).
Thus it is shown that u^Hx) € ffl*2-2k(Q). By means of Theorem 4il we
can now extend the functions u^Hx) onto the whole space En in such a way
that
| Ifrj i(l+2-2*)
| tt '\B < c I« la (8.6)
It
By virtue of Theorem 4.3 it is possible to construct a function v (*, t) €
[{1 + 2,1/2+1 (D^j) satisfying the initial conditions (8.1) and the inequality
1+[41 1+[?1
pn+l n *~0 u
<^(I/Iq+I<pO (8-7)
(we have also made use of (8.4) and (8.6)). Therefore the functions f and 4>' in
(8.3) possess the following differential properties: f' € 4> ' €
yl+Ul+Un^
340
IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
According to Theorem 5.4 there exists in the cylinder a unique solution
of problem (8.3) from the class *2, l/2+l ((^), that satisfies the inequality
i«' <<(ir i(4+1®' i?;”). (8.8)
This means that problem (5.4) also has a unique solution in QT^, with (8.8)
and (8.7) implying that
i«i«' i(«;J)+1 * ig;*<«(i /1? ir2)+1 ® i"*”)-
Let us show that a solution of this problem exists in a cylinder of large
height. For this purpose we fix some point in the interval [0, rQ]> for example
Tq/2, and calculate W*ti/{ft*)|(_r()/,2. Let
i!^j =cp<s>(*).
dtk \,.1l
1 2
Since u €. we have (O). We construct a function
v'(x,t) G where (>' = Q x (r0/2, D, in such a way that for
t < 1 + 1/2
~nrl “**’&>
dt* L Ji
* 2
and
•’(41
\Vf^<c Z (8-9)
A-J
This can be done by exactly the same method as was used above in constructing
the function v.
The function u" m u — v‘ is a solution of the following problem with zero
initial data posed for I = fp/2:
SB*? Is = — •£’*'' Is = ®"-
We have established that a solution of this problem exists for r0/2 < ( < rQ> while
by virtue of Theorem 5-4 it is possible to assert that it exists for Tq/2 <t< ir^/2
and satisfies in the cylinder Q" = fi x (rfl/2, 3*g/2) the inequality
§9. THE FIRST BOUNDARY PROBLEM IN Ivfl(QT)
341
(S,, = 5X(|,^)). (8.10)
Hence, a solution of problem (5.4) exists for 0 < 4 < 3r0 /2; it is easily verified
that (8.9), (8.10) imply
i<£
2 V ~I~ ~T )
By continuing to argue in the same way, we exhaust the whole interval [0, T] and
prove the existence of a solution of problem (5.4) in die cylinder Qj,, and also
estimate (5-11).
The uniqueness of the solution of problem (5.4) or of the equivalent problem
(8.3) <n a cylinder Qj, is an obvious consequence of the uniqueness of the solu¬
tion of problem (8-3) in a cylinder of small height Q established in Theorem 5.4.
In this way Theorem 5-3 is completely proved.
§9. THE FIRST BOUNDARY VALUE PROBLEM IN CLASSES IP2,1((?r>
Problems (5.2)—(5.4) can also be solved in the spaces (q > 1,
m a positive integer). The general plan for investigating problems (5.2)—(5.4) in
these classes does not undergo any changes in comparison with the plan presented
in §§6-8 for application to the spaces We will confine ourselves here
to an illustration of this plan by using as an example the first boundary value
problem (5-3), which we will consider in the classes
As a preliminary, we introduce the norm
ll/C* -WPI/II(9.1)
T VT
where the supremum is taken overjwith respect to all cylindersjq^ -ax (0, T), the
bases |<u of which are the intersection of a domain 0 with some domain of unit
measure,(for example a cube. If the domain £1 is bounded, then the set of functions
with finite norm 1 coincides with Lr(Qj), but for unbounded D it is wider
than Lr(QT).
Theorem 9.1. Let q > 1. Suppose that the coefficients a{. of the operator £
are bounded continuous functions in Qj, while the coefficients a(- and a have
finite norms l°jilan^ ||a|jrespectively, with
342
IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
max (q, n + 2)
for
9*
u-f-2.
n-4-2 -f-e
for
? =
n 4- 2,
/ ^ “f" ^ \
max ^9, J |
for
9 ¥■
n 4* 2
2 *
^+*
for
9 =
«4-2
2
and ( being an arbitrarily small positive number. Suppose, further, that the quan¬
tities ||a-1|and M(1°nC) tend to zero for r —* 0. Let S € O2. Then,
" tvr>Vt,t + T "*.vt,t+r
for my fCLq(QT), <j> 6 W*~2/H0) and <b G r|“1/’'1'1/2«(Sr), with q ± 3/2,
satisfying in the case q > 3/2 the compatibility condition of zero order
*ls = 4>U’ <9’2)
problem (5-3) has a unique solution u 6 V2'* (Qj.). It satisfies the estimate
II “ Co< c ( # •f Kq + II * -* II0 if+ T~l+ ** ) • (9-3)
3
(the last term can be omitted in the case q > -). The constant c = c(T) remains
bounded for finite values of T.
For any f 6 L 3/2^7^ an^ any f ^ ^3/2^®’ ^ ® ^3/2such that the
function
(ip (*) * € fi,
[<&(x, 0 * e S,
given on I'j,, A05 eAe finite norm II’PI!j/jTy °/ (4.38), problem (5.3) Aos o
unique solution u G IT3/2^7^' ^ s<*itsAes *^e inequality
IMP
|.<?r
l/»l0 +}W$ir
i'?r 2’1
(9.3')
From Theorem 9.1 and Lemma 3-3 of Chapter II we have
Corollary. If the conditions of Theorem 9.1 are fulfilled for q > (n + 2)/2,
tften the solution of problem (5.3) satisfies a Holder condition in x and t; when
q > n + 2 the derivatives of the solution of problem (5-4) with respect to the x.
also satisfy a Hdlder condition in x and t.
Indeed, it follows from Lemma 3-3 of Chapter II that for q > (n + 2)/2, q^n + 2,
§9. THE FIRST BOUNDARY PROBLEM IN IV^l(QT) 343
And if q > n + 2, then 2 - (n + 2)/q > 1.
In the proof of Theorem 9-1, just as in the proof of Theorems 5-1—5.3, a basic
sole is played by the investigation of a problem with zero initial data, which is
formulated in the following manner: find a function a £ &<ich that
«L ».4)
..2-' ' '
where
/6W). ®€fr?,‘"w(5x)-
Theorem 9.2. Under the conditions of Theorem 9-1, problem (9.4) has a unique
solution u e jf 2,1(@r) for any f £ L^(QT), <J> 6 I-1/,Z?(Sr) if r < rjj,
where r’0 is some number not depending on f, 4>. In this connection
H“C<?t <‘(tf H„, 0t + mffi)- (9-5)
The index q can have my value greater than 1, in particular, 3/2.
The proof of this theorem requires the consideration of a model problem in
the domain RW):
Tf— £ atJ ~ f (*• 0.
I. /-l
(9-6)
It is assumed that / € LAR^’^), 6 l~l/2q^( T)y tj,e solution must be
q o q n
found in the class W^^iR^^). Let us show chat problem (9-6) has a unique solu¬
tion in this class satisfying the inequality
«W»£V>< e (l! / %, *<n+1® (9.7)
In exactly the same way as in the proof of Theorem 6.1 it is sufficient to confine
oneself to a consideration of the heat equation. It is shown in §6 that for f 6
^M/2yj(D), g^(+2,//2+l(p(7')} y > g) problem (9.6) has a solution in
344 IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
the class +2,//2+l^<7’>j (|lat .g eXpresse(j by formula (6.9). Let us show that
estimate (9.7) is valid for the function (6.9). We extend the function fm0 of (6.9)
from the domain E^\ into the space E ,,, and the function <a? from E^^ into
^ n+1 K n ▼!' 1 n
En, in such a way that
(9.8)
(9.9)
As was explained in §4, such an extension is possible. Using inequalities (9.8)
and (9-9), and also estimates (3-1) and (3.4), we obtain
(6.10) of this function. Taking into consideration (4.26) and an embedding theorem
(Theorem 4.5), and also estimate (9.10), we obtain
Combining estimates (9.10) and (9.11), we obtain inequality (9.7) for the function
(6.9).
Now suppose f € Ly(R^)), 4> £ We approximate
these functions by functions f<»> € Hl'l/2{Ri7">), ¥n) € H1*2'1/2+1(d[T)) U>0).
Let be the solution of the problem with the functions
f^n\ <S>^n ^ in place of f, <P in the right sides of the equation and boundary condi¬
tion. The function is estimated in terms of and
by inequality (9.7), ftom which it follows that the sequence )i/" H converges in
c II / ll», af?)'
(9.10)
In order to estimate the norm of the function <u® we make use of the definition
§9. THE FIRST BOUNDARY PROBLEM IN IV^l(QT) 345
the norm to some limit v € which is the desired solution
of the problem. This solution is subject to inequality (9.7).
The uniqueness of the solution follows from the fact that inequality (9.7)
holds for any solution of problem (9.6) from the class *0?*^), which is easily
demonstrated by approximating this solution with the functions v^n ^ <£
^i+2,J/2+lyj(D^ ajjjj using the fact that the inequality (9.7) is valid for the
functions v^nK
Thus we have proved the unique solvability of problem (9.6) in the classes
|fW”).
The Cauchy problem with zero initial data for the equation
da yn d*w , %
dt V dxi dxj f (9-12)
l, l-l
(T)
with fixed is considered in exactly the same way. For any f € Lq(Dn + j)
this problem has a unique solution w € )> •i*!*
«W>C^:<C|i/,UV (9‘13)
Let us pass to the proof of Theorem 9*2. In the same way as in §7, we write
problem (9-4) in the form
A'u * h9 (9.14)
where A* is the operator associating with each function u € |*he pair of
functions A'u * (£w» u|By virtue of Lemma 3*4 of Chapter II the operator A*
is a bounded operator acting from the space W2* *(Qr) into the space
The solvability of equation (9-14) for any A € , or, what is the same thing,
of problem (9.4), can be proved for small r by constructing andoperator R' that is
completely analogous to the operator R of §7. We define it by formulas (7.8) and
(7.9), but in this section the * will be operators that differ somewhat from the
corresponding operators of §7. Namely, we will now denote by for k C 58
the operator associating with a function € L^(D^j) the solution of the
Cauchy problem with zero initial data
346 IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
■*>(*'"• »• ■£■ 4)'
(9.15)
cr*)6^2.,(£)(t)i)
and for k G -R the operator associating with a pair of functions 4>^),
where G L (R^), <I>^) G 1-1/2? (D^) the solution of the first
* q o q n
boundary value problem with zero initial data
'>=/*’(*• '>.
(9.16)
It is not difficult to verify with the use of estimates (9.7) and (9-13) that if in
(9.15) the function f^Hx, t) vanishes outside of while in (9.16) the func¬
tion /<*>(*, f) vanishes outside of the domain S and vanishes outside of
the domain <7* = o' x (0, r), where a’ Co is the (n - l)-dimensional cube
KKf (a = l, ..., n 1),
then for k G SB
< «!|/W II,. (9.17)
and for k € 31
(9}t = i»X(0. T)). (9.18)
The definition of the domains E, a was given in §4.
We introduce in the space W2,1(0 ) the norm (t>|^2L , and in the space 3
°q XT t
the norm
11*11^1/),.^+
where I /!? ^and are norms defined in §4. Using (9.17) and
(9-18), it is easy to show that for any h G 58
§9. THE FIRST BOUNDARY PROBLEM IN V^CQj.)
(9.19)
547
This inequality is proved in exactly the same way as (7.8), but in place of Lemma
4.4 it is necessary to use Lemma 4.7. It should only be noted that inequality (9-18)
can be applied in the proof of (9.19) under the condition that the functions
£^4? in (7.9) are equal to zero outside of <7' for any t £51. This will be
the case if Zj~* = 0 when z' € for any k G Sft. It is obvious that
this latter condition can be fulfilled without damage to the other properties of the
functions , which are formulated in §4 and made use of throughout.
Let us prove, finally, that for any h € and v € (Qr) onc has tbe
with bounded operators T‘ and W, the norms of which are small for small r. Let
By repeating the arguments of Theorem 7.2 we easily see that relations (9.20)
and (9-21) indeed hold, with
where Tj is the operator defined by equality (7.15), in which R is replaced by
relations
A'R'H = h + T'k,
R'A'v = v + W'v
(9-20)
(9.21)
A'0v = (JS’qV, Atv = (Ji’xV, 0).
T'h — (7>. 0),
R' and
W'v = R'A',v+ J rf5/?'*’(£'*>_TtSo — S’£"}v)
aim
(9.22)
348 IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
tj\
Let us proceed to an estimate of j| 7”A||«n and tlP'u) n ■ We will show that
for any v€W'£’1(Q)) 9 9’ f
(9.23)
where rj(r) —> 0 for r—> 0. We have
l-S’l'l* 0, < S I *■ I? {' ^ + I"!,. Q,
Let us estimate |la; dv/dx^]^ . Applying Hdlder’s inequality, we get
0(*>- (9-24)
r-j ’ T
Here r>q is the index defined in the formulation of Theorem 9.1. If r = q, we as
usual take
IItt-I, = II 4s- II . to* vrai max I 4^-1;
II dx, llJSL £,(*) II dxi llco. Q* _(*) I dxt I
• VT T VT
in this case inequality (9*24) remains valid.
Using Lemma 3*3 of Chapter II and Lemma 4.5, we get
BiS. | <c({v))™ {ku (9.25)
II dx, ||_2_ 0(*i
r-q • wx
and since by condition the functions o; have a finite norm ||o;.||^^ it follows
from (9.24) and (9.25) that
so that
In a completely analogous way
0t <c(|ii«inr0>, +«•»«,)[?
\r-ri
and, consequently, (9*23) holds with
§9. THE FIRST BOUNDARY PROBLEM IN V^CQj-) 349
By virtue of a condition of the theorem this quantity is small fot small r.
Estimate (9.23) together with inequality (9*19) permits us to estimate the first
trams of the right sides of (7.15) and (9-22). The remaining terms are estimated in the
same way as in §7, with the use of Lemmas 4.6 and 4.7, inequalities (9-17)—(9-19),
(7.16), and also the inequalities
|at)(x, <)!<«.
| at) (*, t) - atj (!'*>. 0) | < t,, (X) f(x, t) e O'*1),
where jjj (A) —> 0 together with A. As a result we obtain the estimates
<c (nw + n, <*•>+*■ 4 *T) Mi?<v
From here it is clear that, by choosing sufficiently small numbers A and k, we
can make the norms of the operators T and W' arbitrarily small. This proves
Theorem 9-2 in the same way as in the Holder-continuous case.
Theorem 9-1 is proved from Theorem 9-2 in the same manner as Theorem 5-3
in §8. In order to prove the existence of the solution of problem (5-3) and estimate
(9.3) we reduce problem (5-3) to a problem with zero initial data for a function
ti' ~u - v, where v (x, t) is a function given in Qj. and satisfying the initial con¬
dition
v(x. 0) — <p(x),
with 2
II ® C < c I! <P ll^a ’(9.26)
This function can be constructed by extending <p(s) with preservation of class
onto the whole space En and putting v » (T *j$>). Then estimate (9.26) will be
a consequence of (3.2).
The function u must be a solution of the problem
*{*•
“'1*,. = ®'. (9-27)
“'l(.„ = o.
350
IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
where f' = f- £v and = $ - t»|^ . In this connection, on account of Lemma 3.4
of Chapter II, we have
v\qt < + *»<<?, < m,,gr + o m%£/9\
II V < I11> llJX^ -+ H * (9.28)
+c «■• icw < ii ® Iit7}+c ii»ifc^ •
while by virtue of the compatibility condition (9.2) the function 4>' for q > 3/2
satisfies the zero initial condition $‘(x, 0) = 0.
Suppose to begin with that q £ 3/2; then 1 - 1/2q 4 l/q and as was shown
in §4 (see (4.27), (4.28)),
-H$'L
(9.29)
(the last term can be omitted in the case q > 3/2).
Applying Theorem 9.2 successively for the intervals t € (Arj, (A + l) r^),
k * 0, lA, J, •.. and using (9-29), we prove, in exactly the same way as in §8, the
solvability of problem (9.27) and the estimate
c ll/ll,,Q + II + T~l + (9.30)
This result and estimate (9-28) imply Theorem 9.1 in the iase q 4 3/2.
Now consider the case q • 3/2. Let us estimate the norm
function <J>’ of (9.27). Let
, fo a GO, tm 0
*) (*, t)
Taking into account (4.39), (4.40) and (9.26), we obtain
m
ff) m
W\Yi
f,sr t’1 j
l’r r
iwiiW +11*11 ^
■f ,1 T
|.r r
< c
fWI® + «<„
1 * T i’VT
(9.31)
§ 10. LOCAL ESTIMATES 351
Thus £ jf (Sj.). The solvability of problem (9.27) in the class
2 1
is established by means of Theorem 9.2 in exactly the same way as in
die classes r) with q ^ 3/2.
O 1
Hie solution of this problem is subject to the inequality
"(2> <c
in,
j.Qt 1 -sr
which together with (9-26), (9-28) and (9.31) proves estimate (9-3’).
Thus Theorem 9.1 is completely proved.
It carries over word for word to problem (5.15).
It should be emphasized that the estimates (9-3) and (9-3')> proved for all
indices q, of the solution of problem (5.3) are exact in the sense that their right
sides are estimated in terms of their left sides multiplied by certain constants
which do not depend on the functions f, ifi and $.
An analogous theorem is valid for problem (5.4), but the singular index for it
will be q = 3- In proving this theorem it is necessary to use (4.41) instead of
(4.40).
§10. LOCAL ESTIMATES OF THE SOLUTIONS
OF PROBLEMS (5-3) AND (5.4)
This section is devoted to estimates of the solutions of problems (5.4) and
(5-5) in the norms j u |q? , where Q' = O' x (T v T 2), Q' C (2, 0 < T ^ < T2 < T.
In the right side of these estimates, besides the norms of the known’functions,
will be some weak norm of the solution, for example max|u|, but in a certain
larger domain Q".
We introduce tbe following notation. The intersection S fl Q' is denoted by
S'. Let ODO'D O', with the distance from O' to 0\0“ being positive, and let
S" - 5fl n”. If the set S' is empty, then the domain 0" will be chosen so that
S" will also be empty. Let Q" = 0" x (Tg, T2), with 0 < TQ < T^ if T± > 0, and
Tq * 0 if *= 0. Finally, let S'j, = S' x (T j, T2) and ^1’
Theorem 10.1. Suppose u € Hl*2’l/2*l(Q“) and suppose that in Q"
352 IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
-S’{X' *’ SF’ W')“ = /i
in addition, if Tq » 0, let there be fulfilled the initial condition
«l/_o—<!>(*); (io.2)
and, finally, if S is not empty, let there be fulfilled the boundary condition
“I*: =®c*. o d°-3)
or r*r*
SSu L- =®(x, /). (10.4)
t,t,
Suppose the coefficients of the operator S. belong to the class
and S“ € H**2. If condition (10.3) is fulfilled, then
I “ <*,(|/|$+l* la-+2> +1 ® +c2\u |p,; (10.5)
but if the boundary condition (10.4) is fulfilled and the coefficients b. and b of
the operator 35 belong to the class +D/2(5 ^ ^ ), then
I «l‘£ ’* < c(\ / $ +1 <P &*>+ I® i"Q +c2l«|p.. (10.6)
If S" is empty, then the right sides of (10.5) and (10.6) will not contain the
norms of the function $, and for T0 > 0 they will not contain .
Proof. For the sake of definiteness we will assume that 5“ is not empty,
that the boundary condition has the form (10.3), and that T0 = T j • 0, and we
will prove inequality (10.5). We note that it is sufficient to prove it for domains
n" and O' of canonical form, i.e. for those entering into and
Let the domain 0 * be given in a local coordinate system at some point
( € S by the inequalities
— *i<y0<xi (0=1 a—I), 0<y„—F(y')< 2X„
where Aj is a small positive number. Along with 0“ we consider the domains
qM (0 < A < Aj) defined by the inequalities
- — X) (a = 1 « — 1),
0<y„--F(/)< 2 (?.,->.).
We introduce the notation = O^ x (0, T j), S^ = O ^ fl S, x
(0, T J.
§10. LOCAL ESTIMATES 353
Let (^Hx) be a function having the properties
(1 for JCgQ(M.
?(M= /xn
(o for a:£Q\S'‘2;.
The function v = is defined in Qf^ (it is equal to zero outside of Q") and
is a solution of the first boundary value problem
S’f> = g.
where
g =
On the basis of Theorem 5.2
I * 1^’ < * (i g l&, +1 ft« f3) + l
’+i^rr,
( Q^t </’) O**)
+ 1^1"? I" (10-7)
4?}i
We have
<./-! 7 i.^at J #« 1
and therefore
i^-?*sr.r
o'
fio+2 m+i
ki+2 m+i i
,(r) »-0 q(t) j
In the norms of the functions /we isolate
while the remaining terms are estimated in terms of the norms of the
function u(x, t) by means of relations (10.1)—(10.3)» For example,
m
354
IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
, <2. . \
2 iff 7*1+S jpr J ^ ^(%\
J.j) *-oX Q /
/hi+2 , i*i+2 . \
Treating the remaining terms of the right side of (10.7) in the same way and taking
into consideration (10.8), we get
I a l?J? < I * Q2) < c, { {ff\^ + (tp)*';/’ + {<S>f^
+ c2
[/ +
J®
4?>
\<*j
»w
(10.9)
,w *-•*
The norms of the function b(jc, i) that appear in tbe right side are estimated with
the use of Lemma 3-2 of Chapter II. By virtue of this lemma
0W Q(i) Qw
<“V*l < ft1**-* {«f:i\+*■* mv
,w ,to
(10.10)
These inequalities are valid for S < c Q (A ^ - A/2), where Cq is some constant not
depending on A. One can therefore take 5 = < A with ( small. If we estimate the
right side of (10.9) with the use of (10.10), we will have
I u < e | u |«+*j + c { </)« + «.+J> + <«>>£* j
,(0>
(10.11)
Let
—(/)<?•+(<p>b-+ 2)+■
/V,
= '«C
“i<3*
/(X) = A.'t2Mg+2), /f (A) = rf*X+ Ce%
and
§ 10. LOCAL ESTIMATES
355
The number fi < 1 is fixed so that S = fA < CqCAj - A/2); for this purpose it is
sufficient to take n = min(l, 2l*2c0). Multiplying (10.11) by Ai+2, we get
/w<j/(D+*(*)<4/(!)+*(*.).
Since the function f{A) is bounded for 0 < A < Aj it follows from this in¬
equality that
/-o v '
Tbe function K (A) is monotonicaliy increasing, and hence
oo
-±-K(k) = 2K{l).
j.o
Dividing this inequality by A*+^, we obtain the desired estimate
which is valid for any A < A^. Inequality (10.5) is proved.
We have assumed that Tq « T j = 0. For T1 > Tq > 0 estimate (10.5) is
proved analogously; it is merely necessary to take as the cutting function a func¬
tion depending on x and t and vanishing for i < Tq.
We note, further, that in place of in the right sides of (10.5) and (10.6)
we can put the weaker norm ||tij|r q„ r > 1. We can also obtain estimates analogous to
estimates (10.5), (10.6) in the norms { (u) )^>^- For example, for a function
u € W2,l(Q") satisfying relations < 10.1)—(10.3) we have
II “ ll'2V < (II / ll„, „■ + II <P + It ® ) + e. II«IL • (I®-
\ "• SW «■«
where 1 < q' < q. In the case Tq - T j = 0 this inequality has another form for
q = 3/2. It is proved in exactly the same way as (10.5), with the use of estimate
(9-3), it being assumed that the coefficients of the operator £ satisfy the condi¬
tions of Theorem 9-1 in Q".
356
IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
§ 11. A FUNDAMENTAL SOLUTION OF THE PARABOLIC EQUATION
OF SECOND ORDER
In this section we commence the presentation of another method for investi¬
gating tbe Cauchy problem and the basic boundary value problems for parabolic
equations of second order, namely the method of potential theory. We begin with
the construction of a fundamental solution for a parabolic operator of second order.
Let there be given in the domain = En x (0, T) a parabolic operator
£(*, t, d/dx, d/dt) with coefficients belonging to the class 3“’a//2(D<^>), a < 1.
We will construct a fundamental solution Z(x, (, t, r) that satisfies the equation
-S’[X' w)Z(x• *' ■*)•=*&(* —6)— (11.1)
and is bounded for Jaej —» ~.
The function Z plays the same important role for the operator £ as the func¬
tion r plays for the heat operator. The function Z can be constructed by the
method of E. Levi [69], which consists in looking for Z in the form of a sum of
two terms: a principal term, having the desired singularity at * *= £, t = r, and an
additional term, whete for the principal term one selects a function ZgU - f > ij, t, r),
namely a fundamental solution of the equation with parabolic operator
£0(£, t, d/dx, d/dt), which is obtained from the original operator £ by discarding
the minor terms and ''freezing” the coefficients at the point (£, r). The second
term is looked for in the form of an integral operator with kernel ZQ. The density
of this operator is determined from an integral equation.
The function Zq, which is obviously obtained from the function T by a certain
linear change of the space variables x, has for t > r the form
I-1, t)= ~
[4n (t - r)]T (det A (|, t))'/•
2 A“' T)<je, — W(*y—(11.2)
where /Ilf, r) is the matrix composed of the leading coefficients a.y(£, r) of the
operator S.q, while the > r) are the elementsof the inverse matrix /)*(£, r).
For t < rwe set Zq(x - (, (, I, r) = 0.
The function ZQ(* - £, f ,’t, r) possesses the following properties. As a
function of the arguments * and I it is infinitely differentiable for ( > r and
§ 11. A FUNDAMENTAL SOLUTION 357
|DT«Z,C*-{. I, t, t)|
< c (/ — T)~ y'~exp ( — C 1-y^rJJ8 j (H.3)
(C, c >0). In addition, tbe function ZQ(z, f, t, r) and its derivatives with respect
to z and t satisfy a Hdlder condition in the arguments £ and r. In particular,
|Dr,DUo(z. I. t. x)~D'lDiZ0(z. I', t, t)|
2*
exp
0<p<a. (11.4)
Finally, we will also make use of the following property of the function Zq- for
any fixed <f and for t - t> 0
| Z0(Z, t, x)dz = 1;
Bn
consequently, for 2r + s > 0,
J Dr,DlZ<,(z, £. t. -r)d2 = 0. (11-5)
It will frequently be necessary for us to consider the integral
J D'.Dl Ze (x - /, t) d\ (2r +-« > 0).
E„
By virtue of (11.5)
J D[DSXZ0 (x -1, t. t)rf|= J \Dr,DsxZ0(x — l, I, t. x)
En En
— D'.D'tZoix — l, z, t. T)|,.;Jd|.
from which, with the use of (11.4), we obtain
f Dr,DsxZ0(x-t 1. t, x)d\
ft + gf + S » . ,
<c(/-t)' 2 J |*_||“ex?(-cJ^!i)^ (11.6)
2r+j-o
»C(<—X)
Let us consider the volume potential
358
IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
u(x, t, T)= j dX J Za(x — y, y, t, X) f (y, X)dy, (11.7)
where / is a function satisfying a Holder condition in all its arguments. The
derivatives of this potential are expressed by the formulas
da
tx7
d!u
/ dX j -ZAx /(y. X)dy,
asrsrj- 1dA J *>-/<*• ^
T £n
+ / /<*• W (u-8)
l = /(^ +
t
-j- j dX j dz^x~y- y- >■). [/(y, A.) — / (j:, X)]dj>
1 £n
+ J /(x. X)dX J (11.9)
* £n
It is easily verified with the use of estimates (11.3) and (11.6) that the inte¬
grals in the right sides of these equalities converge.
Equalities (11-8) and (11.9) are the easiest to prove, by considering a se¬
quence of functions
t-h
t, t)= J dX J Z0(x — y. y. t, X)f(y, X)dy
E„
and arguing in exactly ehe same way as in the proof of (1.8) and (1.9). Here one
makes use of the relation
lim f Z0(x— y, y, £. t — k)(p (y) dy *=r q>(jc), (11.10)
&•» 0/
&a
which is valid for any continuous <p(x) that does not increase too rapidly at in¬
finity. Indeed,
§11. A FUNDAMENTAL SOLUTION 359
J Z0(x — y, y, t. t — h)<p(y)dy
*»
= J Z0(x — y, x, t, t — h)<p(y)dy
E«
-f JlZ0(Jf — y. y, t, t—A)—Z0(x—y, x, t, t~-h)}<f(y)dy
Bn
= /,-4-^2-
The first term I^ of the right side tends to <fi(x) for A —» 0 (this is established
in exactly the same way as (1.7)). As to the second terra, by virtue of (11.4) we
have the estimate
|/2|<CA"T J
En
which implies that it tends to zero for A —> 0.
In this way one proves relation (11.10) and, consequently, equalities (11.8)
and (11.9).
It is also not difficult to calculate that
(*• '• -S- £)*<*• *• '• Tx' *)“*<*. t, x)
t
= /(•*. 0+ J dk J K{x, y. t, X)f(y. X)dy, (11.11)
1 en
where
K(x, y, t. X) = ^ {au(y, X)-atJ(x, t)]
i, )~\
+ jr,(*. t. ±)za{x~y, y. t. X),
a
**(*. *>£)-**■& +%(x' <1M2)
i-i
Let us proceed to the construction of the fundamental solution Z. We will
look for it in the form
Z(x. I, t, x)*=Z0(x — l. t. t)
t
4- jdX J Z0(x — y, y, t. X)Q(y, X, x)dy, (ll.IV
360 IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
where Q is a function to be chosen in such a way that the equation (11.1) holds.
Applying to the potential
i
Z'(x, I. t, x) = j dl j Ztt(x — y. y. t, V)Q(y, X, t) dy (II.14)
T Bn
formula (11.11) and noting that
* (*•'• Tx' w) zo(x~l-6.f)=6(*-4) 6(t~x)-+-K(x, I, t. x).
we see that (11.1) will hold when
t
Q(x, I. t,x)+jdXj K(x, y, t, X)Q(y, X, x)dy + K(x, £. t. t) = 0.
* Bn
(11-15)
This equality is a Volterra integral equation for Q, whose kernel K has a weak
singularity since
IMy. *) —«y(x, 01 <c(lx —yja-i-lf — xlT) (11.16)
and consequently
_ £Jt2z2L~ / »n
y, t, t)|<c(* —t) 5 exp(—C'tttt)- (11.17)
Let us also estimate the difference X(x, £, t, r) - K{x‘, f, t, r). Let % be that
one of the two points * and x which is nearest to Suppose for the sake of
definiteness that x" - VPe have
|AT(jc. I. t. x)-K(x\ i. /. x)\
< 2 *>-««(*'• ^i|f-^(a^r-T) 1
n
+ 2 i ai/(£- t)—aij(x'' oi
i, /-i
I d'Z„ (X — 1,1, t, T) &Zn (X‘ -1, S. t, T) I
I dxt dXj dx'j dx’j j
+ \[^(x, t. t.^)]z0(.x-l, I. <. t) |
+ \j?l(x'. t,^Za(x-\. 1.1, t)
-&x{x\ t.^Z^x'-l, I, t. t)|. (U.18)
§ IX. A FUNDAMENTAL SOLUTION 361
One easily verifies that
| fflZ, (x — E. S, t, t) d*Z„ (x' — I, f. t) |
j dX[ dXj dx\ dx'j |
<c|x — xr\(t— x)~~exp(—C -yfpl1-) (11.19)
(the elementary proof of this inequality is omitted).
From (11.19) and (11-3) it follows that for any y € [0, l]
■ (x-j, I, t, x) (X'-I, g, (, t) I ^ d'ZAx-j. 1.1. t)
I dxt dx j dx\ dx'j I dxfix j
d=zB(x'-t 11, T)iv/i d’z, I I 6>za iy-y
dx'[dx'j j y dxi dx j | | dx\ dx'j j J
”+?+Y ,■ I yll . \
<; | JC — x'^(t — x)exp ( ~ ^ JZ-X )■ (11-20)
Using inequalities (11.16) and (11.20) for y = a and (11.3) for an estimate of the
first two terms ot the right side of (11.18), and making use of analogous inequali¬
ties for an estimate of the remaining terms, we obtain
| K(x, I, t, x) — K (x\ £, t, x)|
Ht-’J
exr.
c\x — x'\a(t-x) 2 e*p(-cJ4fi!i). (11.21)
From this estimate and from estimate (11.17) one easily obtains the mote general
inequality (cf. (11.20))
|K{x, I t, x)-K(x', |, t, t)|
a+2-ta-g) .
<cU — Jc'P(/ — X) 2 exp C -J—i— J, (11.22)
in which (8 is any number satisfying the condition 0 < /3 < a.
The integral equation (11.15) is solved by the method of successive approxi¬
mations, and its solution Q is expressed in the form of a series
OC
Q(x, I. t. x) = £(-\TKm{x, t, x), (11.23)
where K is an iterated kernel:
m
362
IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
(
Km(x, J, t, t)= J dX J K(x, y, t. X)Km.x(y. X. x)dy. (n.24)
’ Bn
With the use of induction one can prove the estimate
IKm(x, 4. /. t)|
■(“)
(11,
25)
This estimate holds for m « 1 (see (11.17)), mid in order to prove it for m > 1 it
is necessary to estimate the integral
jdxj K(x. y. t. I) Km (y, |. X, x)dy
nini—H p/n f ilA - a-n-2 ma~n~2
M 2 V27-I(/_X) 2 (X—T) 5 dl
/ma) J 1
Since
and
J
(< — t) 1 (X -x) 2 dX--
q j-wtq-2
it follows that
Am n>rt + l (®A
§11. A FUNDAMENTAL SOLUTION
363
and estimate (11.25) is proved. Inequality (11.25) guarantees the uniform conver¬
gence of the series (11.23) for £ - r > 0 and yields the estimate
|Q(*. S. 1.1)| <e«-T)“iT=£exp(-cl^|il). (ii.26)
for Q. This estimate ensures the convergence of the integrals in (11.14) and
(11.15).
Thus we have constructed a solution of the integral equation (11.15) and
proved estimate (11.26) for it. Let us now prove that the solution Q(*, (, t, r)
satisfies a Haider condition in its first argument.
From inequality (11.26) and inequality (11.22), in which /3 “ a.' < a and the
exponential factor in the right side is replaced by
»p(-cJSl£)+-p(-=iT5ft)
it follows that
t
JrfX J y. t, X)~K{x'. y, t. X)]Q(y. %. X. x)dy
' B" i
C m-2-(a--Q*) »+2~c
—jc'|“ J (t — X)~ 2 (X —T) S-dX
X J [exp(-ci^lH)
+ exp(-Ci££-2J!)]exP (_ C J^t£)dy
n+l-ln+a' Ijr* —il»\
<e|* — x'f (t — x) * exp^— C-L—I1-). (n.27)
Now estimating Q(x, t, r) - Q(x', t, r) with the use of equation (11.15), we
see that
|Q(*. t. x) — Q(x', I, t, T)|
n+2~ia-a’)
<C|JC—x'\ (t~T) 2 expC-L—a'<a. (n.28)
One should now substitute the obtained solution Q into (11.13) and convince
oneself of the fact that the function (11.13) really satisfies equation (ll.l). But
this has in fact already been done above in the derivation of equation (11.15);
one need only note that the application of formula (11.11) to the potential (11.14)
364 IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
is valid by virtue of estimate (11.28).
The construction of the fundamental solution Z is complete.
§12. SOME AUXILIARY INEQUALITIES FOR THE FUNCTION Q
In this section we will obtain certain inequalities for the function Q which
will be needed by us for exact estimates of the fundamental solution Z.
First we will improve estimate (11.28) by proving it for a' - a:
IQ(x. 4. /. x) — Q(x\ 4. x)|
< c I X — x' |a (t — T) ~ exp |— C -*t !.*-); (12.1)
x", as in §11, is the closest to of the points x and Secondly, we will show
that for l > J* > t
|Q(*. 4. t. X)~Q(X, 4. t\ t)|
— itl / - 112 \
<(((_,')!(('_t)- 2 exp(-Cii_|L). (12.2)
Finally, we will consider the function
t
q(x. t)— f dx J Q(x. 4. t. t)d4
« Bn
and show that it belongs to the class ^):
<12-3)
Let us proceed to the proof of estimate (12.1). Let
*,(*. 4. t)
I, /“I
Kt(x, 4. t, x)*=^(x, t, -jL)z0(x-4. 4, t. x).
The kernel K2 has a weaker singularity than Ky, since it does not contain second
derivatives of the function Zg. In particular, in place of (11.21), for the kernel
K ^ we have
|K2C*. 4. t,x)~K2(x', 4. t, T)|.
< c | x — x' I" (f — i)" +i exp [ - C If |2 j (12.4)
§ 1Z SOME AUXILIARY INEQUALITIES 365
and therefore in place of (11.27) we will have
t
f rfX. f [AT2(jc, y, t, %) — Ki{x', y, t, A.)]Q(y. X. x)dy
n
< c I x — x' |° (t — *)'“ exp (— C If—~||8). (12 5)
Let us now estimate
t
j dX J iK,(*, y. t, X) — Ki(x’, y, t, ?„)1 <?()». £• x)dy
' e«
t
= j dX J Kx(x, y, t, X)Q(y, |. X, x)dy
x a
t
- j dX J Kx(x\ y, t, X)Q( y. %, X,x)dy
t a
t
+ JdX J [Af,(jc. y, t. y. /. X)]Q(y, £, X. t)dy. <i2.6>
t Ea\<,
Here o = is the ball with center x” and radius 2\x — x'\, and <r2 is
the ball with center £ and radius \%H - f j/2.
In order to estimate the first two integrals of the right side of (12.6) it is
necessary to take into consideration the fact that if y € a then y € E^\of2* aD(^
consequently
We have
t
J dX J Kt (x, y, t, X)Q(y, X. t) dy
x a
t
<c J (t — Xj
n+2-q n+2-a
2 ^ — t) ^ dX
* J exp( —C-4—-IJl)exp(-Ci£-^li)dy.
We divide the integral with respect to A into two parts: an integral with limits
(r, (( + r)/2) and one with limits (it + r)/2, t). In the first integral we apply the
inequalities
366 IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
t — X t). exp(— cij5^|8)< 1.
and in the second, the inequalities
k — T), exp(-~C <exp[-y
As a result we obtain
t
j dkj K,(x, y. t, k)Q(y, g, X, x)dy
t+X
a x
p i . , ...
+ j*» J (t-k) T-exp(-clfgfl)rfX
0 f+t
-‘■j*-
In completely the same way we prove the inequality
i
JrfX J *,(*', y. /, X)Q(y, |. X, t)rfy
i a
f(,„t)-^exp(-£l^).
It remains to estimate the last integral in (12-6). We assume for the sake of
definiteness that x" - x'. We have
i
JtfX J !*,(*. y, t, X)y, t. X)]Q(y. X, t)dy
: ^ [atJ(x', t) — au(x, 01X
(12.7)
§12. SOME AUXILIARY INEQUALITIES
xjdX J *.t)rfy
t Bn\o
+ .?. / d\ l }a‘f(y' K)-a“(*'' 01
7^/,AH]Q(y> I, X. T)rfy.
367
X)
^.(Ar'-y. y. A A)1
dx',
(12.8)
Let us consider the latter term. We note that £n\cr = (£n\o j)\j (<7j f| a2),
and estimate the integral over £„\<7j.
If y C then it is easily seen that
41-*'—yl< I*—yK-fK—H
and so, by virtue of (11.16), (11.20) and (11.26),
(
J (IX J [au(y, X)-atj(x', /)l[^
f)*z0 (-* — y, y. t. >•)
dx, dx
t Bn\v,
^£»(£—y-AMIq(y, 4. X. t)rfy
dvj dXj J
(OJCj
<C
<+T
2
-*T J (f-
n+2+a
• X) 2 (X —-t) * rfX
X J exp(_cK^)exp(~cl^lJl)[|x'-yr
■+(< — X)TJ<fy-H* — *'| j (/ — X)" 2 (X—t) *
f f T_
x J „p(-C^)„p(-Cl$=i£)
dX
e.N«,
(12.9)
Here we make use of the inequalities
368 IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
K-,r=P(-c.
<></-i)’«P(-c'i£=i£). ic < c>.
<exp[-F^r(l-e'-j’l2 Hy-!l2>]
Denoting the left side of (12.9) by 1^, we obtain
/^rjlje-xTP—O'T2exp(-C, '--=%?-)
(12.10)
f + T
2
f 2-q n+2-a
X J (*•— T) 2 |x—x'|(/ —T)
t
Xexp(-C, 17 £~tS **) J dy J (/
2
-*>
n+3-o
2
«.N^t
2
X exp (—C; 1 y-y |!j rfA.}
">2-o { „ | jr* — £ |» \
Let us now consider the integral
H,= /<t J !•,,».
If y € a2, then |x' - y| £ \x' - (| + |£ - y\ < 3|*' - f |/2 and \x - y\ £
I* - ?l - If - yl > I* - f I - I*' - f |/2 > |*' - £|/2 > I*' - y|/3- Consequently
/• **2 _n+2-a *
J (t- Xf 2 (A.-t) 2 dX J
1 o,n»,
+ {t — *.)T] exp C3 exp (- C3 dy -
§12. SOME AUXILIARY INEQUALITIES
<cexp^-
-C.
'* t—t
t+x
“T
(t—x)~ 2 j dy J (X-xf 2
.n»i
Xexp(— C5 1-|JT'-)^+
n+2—a * i. / »*/ « is\
+(if~T)' * j rfy J (<-*) 2 exp(—CgL—PjrfX
< c | jc - (< -1)'*^ e*p (-C4 .
Thus we have shown that
t
I J d>- J [atj(y, X) — alj(x\ <)]
** c,\o
^c\x-x'\a(t-%)~ 2 exp(-cl^--iii).
Finally, let us show that
J d\l i. *• t)dy
<e(,_Tr^exp(_ci£5|J!).
We have
e_\o
dj:; dx/
(4 X
8
Qrfy
n+2
a+2-a
<c| (t—X)~ 2 (J. — t)"" 2 rfJl
I jc—y f
Further,
(12.11)
(12.12)
370 IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
J
/«j &;<>*
£L En^°
2
J A J lQ(y. £. x. t)-Q(x. IX.,)] dy
t+X
2
+ Jp„.E.x.„»
t+ X
2
X J s/' «,»)<«»,■--.
ffl *
where <a is the boundary of o-. Making use of estimates (11.3) and (11.28), and
repeating calculations that have already been carried out several times above, we
obtain
i «+2 n-t-2-(a-a*J
IJ,|<c J (/ — *.)' s (X-t) 5 dX
t+x
x J |*_^xp(-c<4^)[exp(-ci£^ill)
-f exp (—C dy < c(t—x) exp (-C' ).
|y2|<c J^-Xj'^CX-tf^expj-Cl^MjdX
T+t
2
»+2-2a , ,
;c(t~x)~ * exp(-C|xf~|I).
In
order to estimate / j we use the inequality
dZ„(x — y, x, t, X) I
*y> I n+j
<c|jc —y|(< —A,) "^exp {—C " ) (12.13)
§12. SOME AUXILIARY INEQUALITIES
371
implied by (11.2).
We also note that the set <a coincides with the boundary <» j of the ball a^ if
|as' - £| > 41* - *'|; while if |x' - < 41* - x'\, then a> = w*lere “ j *s
the set of points of &>j the distance of which to £ is not less than \x' - (|/2,
and (o'2 is that part of the boundary o>2 of the ball o2 lying in CTj.
If y € at j, then
I*— *'KI* — yl<3|jt — jc'|;
while if y € (o'2 (in case \x' - < 4|* - *'|), then
Therefore
and
J aZ6 (x — y,x, t, X)
2 exp(-C
a/;— m*>«*
< c | * -11» (t - X)"^ exp (- C
\J3\ < c(t ~ x) 5~ exp (- C '*E|-)
X |x —*T J (' - X)^ exp (- **) d%
t + X
+ IX-41" Ji(#-X)""^exp(-Cl^i£-)t
Ux
2
Ux
372 IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
Thus inequality (12.12) is proved. From (12.8), (12.11) and (12.12) we obtain
t
J dX J [AT, (x, ji, t. X) — Kt (*', y. t, A)] Q (y. X, t) dy
T Bn s«
Combining estimates (12.5), (12.7) and (12.14), we find that
(12.14:
hi
\K (x. y, t, X) — K (xy, t, X)] Q (y. !•. X, x) dy
<e|x — x"C(t — x)~ * exp^— C-L-^—). (12.15)
This inequality and (11.21) imply estimate (12.1).
Let us turn to inequality (12.2). We assume that £ > t' > i and £ - t' < (£-r)/:
First we note that
|K(x, t. x)-K(x. I, t', x)|
_ ?..z2. I I r t 12 \
2 exp(-C (12.16
while for wc have
|K2(x, I. t, x)-Kt(x, I. t',t)i
£ .itIt® / ijc—fin
<c(<—f')2 (<'—*) * exp C-L;—-j.
(12.17
We have
I
J dX f K(x, y, t, X)Q{y, X. x)dy
’ ''
— J dX j K(x, y. f, X)Q(y, X, x)dy
x
t
= f dxjK(x. y, t, X)Q(y, X, x)dy
w-t en
t'
— J dX J K(x, y. t', X)Q(y. |, X, x)dy +-
§12. SOME AUXILIARY INEQUALITIES
373
:i — *
J dX J [K2(x, y, t, k)—Ki{x, y. t\ X)\Q(y. X. x)dy
2f-t
t. /-i
X j" PZtjx — y, y. <■ A.)
i. i-t t zr,
x**>]Q0,4,X,t)rfy. (12.18)
Applying estimates (11-17) and (11.26), we obtain
r r r «+2-o
I <ft j ATQtfy <c J (t — X) (X —tf 2
2t —I En 2t'-t
X !exp(-c-l£5fJi)exp(-cl^i£)rfy
dX
<c(t-ty (t - exp ( - c .
The second term of the right side of (12.18) is estimated analogously. The
remaining terms are estimated with die use of inequalities (12.17), (11.16) and
(11-3)- In the process one proves the inequality
J & j ZhS$=gj^Q(y. 4, X.T)„J>
g+2-c
<c(t — x)~ 2 exp^— C j (12.19)
which is analogous to (12.12).
374 IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
The proof makes use of the representation
J
* ?B
2t’-t
“I dX J aSfe IQ (y' *• X' T) ~ Q(x1 ** X’ T)!
2 f
+ J Q(*. X. x)d\ J *Z>^hJ^dy
•11 En
2
l + T
+ / *• *■
r n
As a result of estimating all of the terms of the right side of (12.18) we
obtain inequality (12.2).
We have assumed that t -1' < (i1 - r)/3- But inequality (12-2) also holds for
t - l' > (t* - r)/3. In this case it follows from (11.26) since
*■*■2-0 a n+g
Finally, let us consider the function
t
q(x, t)~ J dx J Q(jc, y, t, t) dy.
0 En
Let ai be the ball \x* - y\ < 21* - x*\ We have
t
q(x. t) — q(x\ t)— J dy ^ Q (x. y, t, x)dx
Ot o
— j ay j Q(x’- y> *• t)dx
a, 0
/
+ j dy j IQ (x, y. t,x) — Q (*', y, t, t)! dx.
£a\«t o
§ 12. SOME AUXILIARY INEQUALITIES
375
With the use of (11.26) it is very easy to prove that the first two integrals can be
estimated in terms of c |* - *'|a. In order to estimate the third term we represent
Q in the form
where
Qi =*»-+-/ dX J K{x, y, t, X) Q (y, X, x) dy.
* En
From (12.4) and (12.15) we obtain
|Q,(x. £. t. x) — Q1(x', t, t)i
<c|x_x',*(/_t)-TL+4eKp(-Cl^A£).
where /3 = roin(a, 1 - a). Therefore
t
J dr j IQ^x, y, t, t) — Qt(x’, y, t, t)|rfy<c|jc—x’la.
o
Finally,
t
J dy I 1*1 (x. y. t, x)—Kl{x', y. t. t)1 dx
2 [•„(*'. t)~all(x, 01J dt J
U Ba\ o,
I,)-1
» I p
4- X \ dx [a,j(y, x)—a,j(x’, t)]
4, /-I U Bn\v,
v r ***. (x - y. y, o d‘z° <-x' - y- y> <• T> 1
[ dx, dx j dx] dx'j J
from which, by repeating arguments already carried out above, we obtain
t
j*dx J [K\(x. y. t. t)—Kx(x\ y, t.
t)1 dy
u tn so,
s(“)
<c|*—x'f.
Thus < «• Analogously one can prove that
It is necessary in (his regard to use the representation
376 IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
i
g(x. 0—«(*. *')= J rf* J Q(*. y, t. x)dy
mtx(0.2t'-f) En
t*
— J di J Q(*, y. t', t)dy
max (0,2T-*>
max (0,2('-/)
+ f rft J IQ, (jc. y, /. t) — Q, (x, y, t)J dy
•
n
+ £ !«„(*• O —«y(*. 01
J, /-1
X J
* En
„ mtxia.it'-I)
■+ S J rfT / ia‘j(y- *>—**;(■*,
I. y-1 » <
vj- r d»Zo (JT— y, y, f. t) d»Z, (■* — y. y, t\ t) 1 .
I ixidxj dxi dx/ J ”*
Tbe explicit calculations are omitted.
Thus inequalities (12.1M12.3) are established.
§ 13- ESTIMATES OF THE FUNDAMENTAL SOLUTION
In this section we obtain the following estimates of the fundamental solution
Z (x, t, r), which are necessary for establishing exact estimates of a volume
potential with kernel Z:
n+2r+i
\tftD^Z(x,l. t, t)|<c(/-t)' 2 exp(-C-LlillL), a3.i)
where 2r + s < 2, t>r,
\D',DsxZ{x. I. t. x) — D'iDx-Z{x\ I, t, t)|
B + 2+V <142-a+Bl
c f [(JC — x'l'y-T) * 4-|Jt-xY(<-tr 2 J
X exp (-C (13.2)
where 2r + s = 2 (i.e. r = 0, s •» 2 and r = 1, s = 0), 0<y<l, O<0 < a, « > r,
§ 13. ESTIMATES OF THE FUNDAMENTAL SOLUTION 377
and
\d'iDU (x. 6. t. t)-D;.#2(x, I, t', t)|
[a+2r+g+g 2~2r~.y*a £±£1
Xexp(-Clfgl!lj. (13.3)
where 2r + s = 1, 2 and t > t' > r.
When applied to the function ZQ estimates (13-1)—(X3-3) follow from (11.3),
and hence it is sufficient to establish them for the function Z' of (11.14). If
s = 0 or s = 1, then by virtue of (11.3) and (11.26) we have
|D’xZ'(x. I, U t)|
t
< J dX J \D*xZa{x-y, y, t, X)||Q(y. X, t)|<fy
1 En
* n+s «+2-O
<C j (t — Xf 2 (X-x)" 2 dX
X
En
t
<f(<-t)"* J (t - X)~^(X-x)'~^ rfXexp (-C
= c(.~x)-^:iexP(-cl£=M).
In order to estimate D^Z' we take advantage of the formula
t+x
I «! ***%&*» X. T),y
+ J d% J ~ K X)-Qt-
1+ T B, '
n
+ / <?(*. 6. X. T)«tt J (13.4,
2
378
IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
which follows from (11.8). With the use of inequalities (11.3), (11.26), (11.6) and
(12.1) we obtain
d*Z'
\<c
i+x
2
, n-t-2 m-2-a
I (t — xf 2 (X —r)- 2 dX
I dxi dx/
X J eXp(-Cli^li)exp(-Ci^=l!i)dy
r ,
■x) 2 dX J exp f-
Jn+i
(< —X) ~T~(X — X) - «a j exp i — u ■
<+t
X IX — y |° [exp C + exp (— C dy
/• ..o n+2-o , , k ,,,
-t- J (t ~ X)~ 2 (X- xf 2 exp (—C )
<+T
2
— ^ exp (~ C
Precisely the same estimate is also valid for \dZ‘/dl\ Thus for 2r + s < 2 we
have
m-Zr-M-o
\Dr,D*xZ'\^c(t-x)~ * exp (13-5)
and inequality (13-1) is proved.
Let us estimate
u. 11, %) d’z- (x\ g. t, %)
dxt dx j dx\ dx'j
For this purpose we use formula (13.4). By virtue of (11.20) and (11.26)
2
I d% f rd,z° (x ~y’ *•x} diz°(x' -y' ?• *• *■> ]
J J [ dXfdXj dx[ dx) J
X Q(y. i. t)dy
/H-T
2
<c|x — x'|°J (f —X)'
fl + 2+q
* X
§ 13. ESTIMATES OF THE FUNDAMENTAL SOLUTION 379
w-f 2->n
d
e'.
t + x
2
*+2+a f 2-a
<C|x — — t)* 2 J (A.—xf 2 dX
X[exp(~cl£^)4-exp(l-cl^ili)]
(13.6)
We denote by H..(*, f, t, r) the sum of die last two terms in (13-4), and by
, as above, the ball with center x" and radius 2\x- x’\.
It is not difficult to verify that
W1;(*. 4. t, T1.1. T)
j ax j wt*£•x- 5. t)i d?
+_T 0g
2
r
f dx f [0(y, 4, X. i)-Q{x', 4. X, T)1 dy
J J dx, dxt
t-ft o, < /
2
t
f r &ZG (x - y, y, /, A) (*' - y, y, t X) 1
J J I dx.dx, ' dx*i dx'. I
+ t &.\.o. i * / * ' J
t + x En\*
2
X \Q(y. 4. K x)~Q(x', 4. K x)]dy
+ j IQ (jc. 4. X.-z)-Q{x'.l. X, t)) dX J dy
380 IV. LINEAR EQUATIONS WTH SMOOTH COEF FICIENTS
Using (12.10), we obtain
» a+2 *+a
|Af1|<e {t — X)~ 1 (X-t)~5-rfX
l+t
s
X | |,-,rexp(-ci^C.)[«p(-Ci^=it)
+ exp (-C dy<c(t-x) "T* exp (-C' if^UL)
X j I-* — y[°dy J (t — X)~~V~exp C' L*—.jUl)'
Ux
/+t
2
^c\x-x'f(t-xf * exp(-C'
In a completely analogous way
——t) exp[— C Y3~)•
If y € j^0\^| then c ^\xf - y\ < \x - y\ < c 2|a;' - y|. Therefore
i «+3 04-2
|Afa|<tf|jP — Jf'l j <*“ W""1"(X— t)~ 2 tfX
<+*
2
X J |x"-y|nexp(— ci^;^)
X [exp ( - C J^1) + exp (- C ^-if1*)] *y
<c|jc — x'\(t~x) 2”exp
X f \x,,-ylady exp(-C'l^~y
Bn^a I ill
2
< c I * — *' |a (i — T)-~*~ exp (- ~ .
Before estimating we note that
§ 13- ESTIMATES OF THE FUNDAMENTAL SOLUTION
1
dfZ^x-y, y,t,X)
dxi Oxj
dy-
J[
d*Z,(x-y, y,t,X)
dxtdxj
(PZq (jr -
’ f * <■>, f
In consequence of (11.4)
nd»Z,(x-y, y, <, X) d^Z0(x-y. X, t, X) 1 . I
dxidxj dyidyj J y|
2+fl
381
(13.7)
whence
<c(f~X)~ 2 J 1 x — y |° exp C ! * ) dy, (13.8)
f —y, y» <■ M *, *,
i J [ 5573^ dy(ayy J y|
2—0
<«(<-*.)■
If we estimate the second term of the right side of (13.7) with the use of (12.13),
we will have
1J ' ololu-in*
iO, I
—*) ~exp(—C (13.9)
From this inequality and from estimate (12.1) it follows that
*T(<—T)" 2
X exp (— C U”~~TE|i) J (/ — XfT1 ax
t-ft
L 2
J (t- X)~^exp(-C 1 x~Jlp )<
2
<c|* — —r)'^ exP(-c tf~T5|1j.'
\dX
382 IV. UNEAR EQUATIONS WITH SMOOTH COEFFICIENTS
Finally, let us estimate My We have
fpss -U
J | <3jr, <fcr^ <?jrf A*y J
= f ra»Z,(x-y.y.<,X) d*Z,(x' — y, y, f, X)
_ J I d-ir, «U, (Jjrl djr',
d’Ze (x — y. x, t, X) , d2Z0 (x' — y, x\ t, X) 1 .
^y,^y; J
, f fd*Zt(x-y,y,t,k) K,(x-y, tU)l
•+ J [ W^STi \*»
Ol
_ f r PZq (*' — y, y, t, X) d»Z0 (x' - y, y. <, X) -
J [ dx' dx'j dyidyf
The second teem of the right side is estimated by inequality (13-8); for the third
term we have the analogous inequality
| f C d’Z, (x' — y. y, t, X) d>Z0<x' -y. t, X) 1
| J [ dxtdxj dyt dy j j y j
<e(/-X)“5»i J|V-yrexp(-ci4EfJ!)rfy.
®i
We represent the integral / over E\,o^ (the first term) in the form
/s=J f r d‘Z„ (x - y. y, t, X) d’Z, (at' - y. y, t. X)
£n<0,i dx‘dx/ dx'‘dx'i
d*Za (jt — y, x, t, X) , d!Za {x‘ — y. at, t, X)
/[
lyJWj dyt dyj
d2Za (x' — y, x*, t, X)
dy,
dZa (x' — y, x, t, X)
dy
— —] «; O’) da, = /, + /2.
J
By representing the difference of the function d^Zg/dxi dx^ at the points x and
x' in terms of the derivative Zq/dldz.%dz■ where z is a point on the segment
between the points x and %', and I is the direction of the vector x — x', we
obtain, with the use of (11.4),
§13. ESTIMATES OF THE FUNDAMENTAL SOLUTION
.r*
383
IAI<
f dl f y.t.i)
J .J L dldztdzj
en^ai
n+3
x J |y'-y|“exp(-C^^-£-)dy.
(13.10)
In order to estimate 12 we take advantage of the inequality
I dZ„ (x' — y, x', t, X) riZ0 (x' — y. x, t, A) I
I dyl I
< c IX - x' j° I *' - y I (i - Xfexp (- C -L^Ef11) •
which is obtained from (11.2) with the use of elementary estimates. By virtue of
this inequality
n+2
IhKclx-x'f+'it-Xf s exp^— CL~Il£i-j. (13.n)
Combining inequalities (13-8), (13.10) and (13*11), we obtain
t
j dX
<
n+2
c
f Li!5> «t_1rfy
I l»xtdxj dx^Xj]
j\x~yfdy J (<-Xf^exp(-C-^=2il)dX
o, -oo
+ J \x'-yfdy j (t - X)'^ exp (- C Kfplll)
O, -oo
» /» b+3
+ \x-x'\ j Ix"~y\ady J (t-Xf *
Bn No, -® ^
Xexp(-C tfA + j*~*'ra /(*-*)
— 00
X exp(-C < c IX-*T.
dX
•+*
2
(13.12)
From this inequality and from estimate (11.26) of the function Q it follows
that
384 IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
n+2-o
| Ms| < c |* — x> f(t - t) 5 exp(— Cl^if)
<c|* — x'^(t -t) 5 exp|—
Thus all of the M. are estimated and
\H,j(x. I, t. xI. t. t)|
This inequality together with (13-6) proves the estimate
6’Z' (x, j, t. t) d*Z' (x% 1.1. T)
x'i
ff 4-2
dxt dxj dxjdx j
<c\x-~x'\a{t~x)~ * exp(-C 1^-111). (13.13)
In order to obtain an analogous estimate for dZ'/dt we make use of the fact
that, as was determined in §11* the function Z* satisfies the equation
*(*• '• *• *-*>
t
— Q(x. t, x)+ J<tt J Kix. y. /. X.)Q(y, x)<%
x £n
— — K(x, /, t). (13.14)
Transferring to the right side of this equation all of the terms except dZ'/dt and
making use of the already proved estimates (13-5) and <13-13), we obtain
| dZ' (x. j. t, >.) 6Z' jx’, It, r) |
I Ot dt 1
<c| x — x'fif — x) ~ exp C -1 * '* ).
Thus for 2r + s - 2
\D',OsxZ'(x. I. t, t)-D',Dsx-Z'(x. t t, x)|
< c [* - x'f (t - t)~^ exp ( - C ).
From here and from (13-5) it follows that
§ 13. ESTIMATES OF THE FUNDAMENTAL SOLUTION 385
\DrtDsxZ'(x, t, x)~-Dr,Dsx,Z'(x. |. t. t)|
4-2-0+P #
<e|je —*'f</ -t)“ 2 exp(-ci-^ ~TSI ) (13.15)
for 0 < /3 < a. This completes the proof of inequality (13.2).
Let us proceed to an estimate of the difference
D\z'(x, t. x) — D*XZ' (js, t)
for s = 1, 2, t > l‘ > t, I - t' < lA (*’ - r). We make use of the representation
DsxZ'(x, t. t)
r + x
2
= J dX J D^o(JC—y. y. t, X)Q(y. i, X, t)rfy
x £o
t
+ J dX | DjcZo(x — y, y, t. X)
2
XlQ(y. 1. X. x) — Q(x, l. X, x)]dy
t
+ J Q(*. I. X. t)rfX J DjZ0(x—-y, y, t, X)dy (13.16)
f + X
2
and of the same representation for 0* Z' (*, (, t‘, r), obtained from (13.16) by
replacing t everywhere by t‘. Forming the difference of these two functions, we
obtain
DxZ'(x, I. t, t) — DXZ' (x, t', -t)
t’+x
2
= J dX J \DsxZ0(x—y, y, t, X)-DxZ0(x-y, y, t', X)]Qdy
Bn
2t'-f
-f | dx J [o*z0(*—y. y. t. X)— Diz0(x —y, y. t\ x)J
£+? En
2
2V-t
X [Q(y. I. X. x)—Q(x, l, X, x))dy+ J Q(X. X. x)dxX
r+x
386 IV. UNEAR EQUATIONS WITH SMOOTH COEFFICIENTS
X J [DJZo(at — y, y. t, X)~ O^Z0(x — y. y, t\ X)]dy
t
+ J dX J D*Z0(x-y, y, t, X)[Q(y, g. X, x)-Q(x, g, X. x)]dy
— J dX J DsxZq (x—y, y, t', A.)
v-t en
X iQCy. I- *)—Q(x, g, X, t)) dy
t
+ J Q(x. 6. X, x)dX J DxZa(x — y, y, t, X)dy
*’-* Bn
v 1
— j Q(.x, 4. X. x)dX J DsxZ„(x — y, y. t'. X)dy~'%Nl.
2V —t Ett im 1
Each term of the right side of this representation is estimated with the use o!
inequalities (11-3), (11.6), (11.26) and (12.1). The assumptions t > t* > r,
t - t* <{t* ~ r)/4 are also used:
f'+T
2
• a-tst-i n-t-i-n
|W,|<e(f — f) j (t' — X) *~-(X — xf 1 dX
xJ„p(Jcu=it)„p(_cii=it)„
n+s+7~a t g.
<c(t-t')(t’-x)~ 5 exp(-C
<c(t — tf) * (it'-t) ^re»p(-c^3|l ),
2/'—/
i% ff+f+2 it'i’i
|Ar2|<c(< —/') j (f'-X) 5—(X — T)"_T'dX
t‘ + X
X J i«-,r hp(-c^)+.-p(-cJ^EV1)l
£n
2 \
X exp (-C dy < c (< - <0 - tf X
§ li. ESTIMATES OF THE FUNDAMENTAL SOLUTION 387
Xexpf-cJ^iL) J (f'-X) *~d\
i' + t
LL±*
2
2-*+a
The fourth and fifth, and also the sixth and seventh, terms are estimated in die
same way; for example,
<
j* tt + S
|A?4|<C J (*~X)“ 2 (X —if 2 rfX
xlM-'WW-'W)
£«
+exp (- c '-y—)] \*—yf*y
*h2 / ■ 112 \ f s—ct
<c(*~T)" 2 exp(—I — ~ dX
»’*/
2-J+g «+2 , . . „.
= «(<-<') 2 (f-t) 2 exp(_CIff;.*1 ),
and by (11.6)
|WS|<C J IQ(x. |, X, t)| (/ — X)-^ dk
2t-r
2-s+a o+2-a . |f
<c(/-f') * (t — t) s exp^— C ^* )•
It remains for us to estimate the third term N^
J [D*xZa(x-y, y, t, X) — D}Z0(x — y, y, f, X)Jdy
*
— J dt" J Dt-Dx Z0{x — y, y. t . i.)dy
r En
and since, by (11.6),
388
IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
J [DxZq (x — y, y, t, X) — DxZa (jc — y, y, t', X)] dy I
I
i 2+a-a
<ej(f" — X) 4 dt". (13.17)
it follows that
»+2-g
a exp(— C — ■ )
i 2t’-i
f f 2+g-o
X J df J (*'—*)” 2
2-S+g JI+2-g
<«(*-*') a 2 exp(-C-^=|L).
Combining the estimates, we obtain for s = 1, 2
\D’xZ\x, 4. /. t)~D,JCZ'(x. 4, x)|
2-«+q «+2
<C«-0 * «'-T) * cxp(-Cl^iL). (13.18)
This estimate is proved under the assumption that «-«'<(«'- r)/4. But it also
holds for t - t’ > (t* - r)/4, since in this case it follows from (13.5).
Let us estimate the difference
£>,Z'(*. 4. t, x)-D, Z'(x. 4. t).
This is easily done with the use of the already proved estimate (13-18) if one
takes advantage of equation (13-14). From this equation, from (13.18), (13-1) and
from the easily verified inequality
|K{x, 4, t. f)-K(x, 4. t)|
< c (t - t'f2 if - if~ exp (- C
it follows that
\D,Z'(x. 4, t, t)— DfZ'{x, 4, t', t)!<
<c(< — O0/V — exp(—(13.19)
(we assume, as above, that t > t' > r).
Estimates (13-18) and (13.19) together with (11.3) prove inequality (13-3)-
§ 14 SOLUTION OF THE CAUCHY PROBLEM
389
§14. SOLUTION OF THE CAUCHY PROBLEM
(T)
Let us consider in the domain Dnthe Cauchy problem
■31 {*• *' ~5x’ 4i)v{-x' 0.
«Uo = »(x>- (14.1)
We assume chat the function f satisfies a Hdlder condition in all of its argu¬
ments, and that <p is continuous. In this case the solution of problem (14.1) can
be written in the form of a sum of two potentials with kernel Z:
I
v(x, 0— f dx J Z(x, I. t. r)/a. t)d%
« Bn
+ jZ(x. £. /. 0)fO)<. (14.2)
e«
In order to ensure the convergence of the integrals in the right side of (14.2) it is
necessary to require that the functions f and <p do not increase too rapidly as
|*| —• 00; it is sufficient to require that they increase no faster than a function
e°*2, where a is some positive constant depending on T and on the constant C
in inequality (13.1). These restrictions on f and <p are formulated more precisely
in [25a], where the potentials of (14.2) are studied in detail in classes of functions
increasing at infinity. We will consider problem (14.1) only in a class of bounded
functions.
In order to verify that the function (14.2) is really a solution of problem (14.1)
one should consider separately tbe functions
t
*,(*. 0- jd-tf Z(x, I, i, t)/(|. x)dx.
9
V,(x. 0= J Z(X. 4. t. 0)<p(|)d%.
e»
They are solutions of the problems
S”0i =/• *il,_0 = °.
-2^2 = 0, v2 — (p. (14'3)
Of these equalities only the equality requires a verification, since
the construction of a fundamental solution in § 11 was based on the relation
S.v j = /, while the remaining equalities are obvious. Since by virtue of (13.5) for
390 IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
r= s = 0
lim f Z'(x, I t, 0)<p(|)d£ = 0,
we see, upon using relation (11.10), that
lim v2(x, f) = Iim f Z0(.x—\, t, 0)f 0)rf| = <p(.*),
t-+Q *
*8
Q.E.D.
Let us now prove, using estimates of the preceding sections, that if f €
tfa’“/2(D'r+j) and 9e»2+t(FB), then
Kl^n <«I/I5n* (14-4)
<e!?Ca) (14.5)
and, consequently, 8+1 "
We will thereby, in a different way, have proved Theorem 5*1 for I m a < 1, In
order to prove inequality (14.4) one must establish that
H2+;<n <‘\/$n (14.7)
,2-s+a\
W«|); 0<r> '<«!/Cn <* =1 • 2)- (14.8)
* U+l ^ Mt. 1
and also
<0,*,>JTin <«l/lS?n* (14-9)
'• n+1 n+1
<*,>« <c\ft»iT) (*-.0.1.2). (14.10)
fl+1 Jl+l
Since these estimates are proved in approximately the same way as estimates
(2.8), (2.9), we will not verify them in every detail. We make use of the equality
i
DxvJ (X. /)= | dx | DXZ (*. y, t. T)[/(y. *)—/(*. t)] dy
» en
t
.+ J fix, x)dx j DxZ(x. y, t. x)dy. (14.11)
§ 14 SOLUTION OF THE CAUCHY PROBLEM 391
Since by virtue of (11-6) and (13-5)
J J D^Zix.y.t, T)rfy|<e(< —t) r (5=1. 2).
Btt
JZdy|<c, (14.12)
it follows from representation (14.11) that
2-*+q
IDSflfi l<ct~r~ |/|W (s = I. 2).
Dn+l
I®i)<rf|/I(“m (14.13)
°n+l
Using the equation £v f for an estimate of &tv ±y we obtain
°«+1
The latter three inequalities prove estimate (14.10), which, however, is completely
obvious for s = 0, 1.
Let us pass to the proof of estimates (14.7)--(14.9)- We begin with (14.8). In
order to prove it one must estimate the difference
£>i {x. t) - D>, (x, t') (s = 1. 2).
For the sake of definiteness let t' < t < T. If t > 21', i.e., t - t' > l\ then by
virtue of (14.13)
2-J+o
|£>>i(*. ,(*. Ol<c(t~~r~
2-g+c 2-54-0
+*' 2 )i/i(;;n . d4.i4)
»+l °*+.l
The same inequality also holds for t - t' < t', but in this case the proof requires
more subtle arguments. With the use of (14-11) one can prove that
Dlvi (x, t) — Dlvi (x, t')
t
~ J dX J DxZ <‘X’ y' *' T) ^ (y' V ~ ?*•*' dy ~~
392
IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
~ j dx j DxZ(x, y, t', x)\f{y, x) — f(x. x)]dy
v-t en
2t'-t
+ [ dx j [DsxZ(x, y, t, t)
0 en
-D’xZ{x, y, tt)]|/(y. x)-f(x. x)\dy
t
•+ j fix, x) dx j DxZ(x, y, t. x)dy
2V-t En
t'
— J /(•*. *) dx J DSXZ (.x, y, t\ x) dy
2t'-t Em
n
2t'-t
+ J If (.X, x) — f (x, 01 dx J [D!XZ (x, y, t, t)
0 ««
2r-t
-DXZ(x, y. f, t)] dy \ f (x, t) j dx J \D*xZ{x, y. t. x)
1
— DXZ (x, y, t‘, t)J dy = 2 Jj- (14.15)
/-i
Tbe first six terms of the right side are easily estimated with the use of inequali¬
ties (13.1), (13-3), (13.5) and (11.6):
t> 2-5 + 0
g | Jj |< «(* — 1/ fin<14-16)
The /7 teem is transformed in the following manner:
2/'—*
J dt | [DiZ (x. y. t, x) — DXZ (x, y, t'. t)| dy
» **
ar-<
= J eft J fOi-2T0(JC—y, y, t, t) — DsxZu(x-y. y. t', x)]dy
*
t
j dx \lfxZ'(x. y, t, t)dy +
tt’-t En
§14 SOLUTION OF THE CAUCHY PROBLEM 393
V
+
-f- J dx j" DSXZ' (x, y, f, t) dy
c
Jdt J DsxZ'(x.y,t.x)dy-[ dx J Dsxz‘(x. y. t\ x)dy
By virtue of (13-17) and (13-5) the first three integrals of the right side do not
exceed c(t - t‘)^2~s *a^2. The fourth term can be written in the form
t f
J dx J Dxz! (jc. y, t, x) dy— J dt J Dxz' (jc, y, t', t) dy
0 E«
t
jdx J Z0(x — y, y, t, x)q(y, X)dy
! Bn
— J dx J Z0(jc — y, y, t', x)q(y, x)dy , (14.17)
0 £»
where
x
qiy. *)— J d% J Q(y, 4, t. X)dl
is a function that, as was shown in §12, belongs to the class Ha,a'^2{J)^j) of
(12.3).
If from the very beginning we had considered instead of v j the volume poten¬
tial
t
«»**= J dx J Z0(x—y, y, t, t)/(y, x)dy.
0 en
then for this potential we would have had representation (14.13), in which the
kernel Z is replaced everywhere by Zg. For the right side of this representation
we would have had estimate (14.16) and also, by virtue of (13.17), the estimate
2—s+a
(14.18)
| Jr|<c (< — /') * |/|
<o>
Consequently inequality (14.8) holds for the potential v® and the right side of
(14.17) is estimated in terms of c(t - t')*‘2~s *a^2 (we also make use of (12-3)
here). Hence the Jy term of representation (14.IS) is also estimated by inequality
(14.18), and estimate (14.8) is thus proved.
394 IV. LINEAR EQUATIONS TOTH SMOOTH COEFFICIENTS
Inequality (14.7) is obtained by means of the same arguments from the repre¬
sentation
S»vt (X, f) <Pvt (*', Q _ r d% r <PZ{X. y. t, t)
dx, dx, dx', dx'. J J dx, dx,
’ ’ o «, '
X {l/(y. -c) — f(x. T)] -f-(/(jc. t) — /(jc'. oil dy
U 0, *
+ [/(*'. *) — /(*'. 01) rfy 4- J rft J
dxt djty J
which is a consequence of formula (14.11). Here a j is the ball with center x and
radius 2\x - x’\. In estimating the right side one should use inequalities (13-1).
(13.2) and (13.9).
Finally, inequality (14.9) is a consequence of the already proved inequality
(14.S) and the equation & j = /.
Thus we have proved inequality (14.4).
Inequality (14.5) can be proved by writing the function in terms of a
volume potential. Since this function is a solution of problem (14.3), the function
v3(x, t) = v,(x. <) —<PM
is a solution of the problem
_2>v3 == — S’? (X), lr_o ~ °*
Consequently,
<
«3 — — j dx J Z(x. y, t. r)&<t(y)dy
, a e„
and
§ 15. THE SINGLE-LAYER POTENTIAL
595
vt (x, t) — q(x) — J dx J Z(x, y. t, x)S’<t(y)dy.
o e.
If we now apply inequality (14.4) to the tight side, we obtain (14.5).
Estimate (14.6) is proved.
§15. THE SINGLE-LAYER POTENTIAL
Let Q = Q, x (0, T), where 0 is a bounded domain in the space En with
boundary S 6 0 < 1. Consider in a parabolic operator
£(x, t, d/dx, d/dt) with coefficients from tfa,a//2(0^j) and let Z be its funda¬
mental solution.
The potential
t
V(x. t)= f dx J Z(x, I, t, t)q>£. T)rfS«
0 s
is called a single-layer potential. This potential possesses many of the proper¬
ties of an ordinary electrostatic or beat single-layer potential.
By means of estimate (13-1) it is not difficult to show that the function
V (x9 t) is for any bounded <p a continuous function of x and t. Let us calculate
the conormal derivative of the potential V(x, 0 on S:
Let
lim V au(x, t) nt (xa).
*JC,£S ■ axl
V(x, t) = V0 (x, t) + V'(x, t),
where
V0= J dx J ZoVdSt. V'= j dx j Z'ydSt.
0 5
Estimate (13-5) for r = 0, s = 1 implies
Ilm j au(x, t) dVd(X' l) M*o)
dV' (x, t)
dxi
t.J-l
«<(*<>)• (15.1)
jt-jcj
396
IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
Let us calculate tbe conormal derivative of the function Vg. We fitst consider
the directly defined value of the conormal derivative on S
t
W0(xo. /)= J dxj H0(x0. I, t, t)<p(4, t)dSv
where
»/(■*<>)
i,i~ i
tt
hrf 2 a,j(xa, t)Au‘*Hl.x)
i. /.*-1
X — 4*) nt (*0) Zo C*o ~ 4. 4. /. t).
Since
l+B
and
it follows that
= J 2 (*o,~ Si)"'(■*<)) j <*l*o—41
I dl, (jc0. /) ■- a„ (4. T) I < e (I *0 -1 f + (f - #).
I«ol<*(|*o-il,,+l*o-4f-+ (t-xj*)(t-T)~~*L
X exp (-C 1 **2^) <«(/ — t)"2^exp (- C(15.2)
where 8 = min (a, j8). Consequently, the kernel HQ has a weak (integrable) singu¬
larity.
We assume that die point x approaches *q along the normal to Xq. Consider
the difference
n
S °W(JC-”»(*<>)-^o(*o. o.
I./-I
One easily verifies the identity
av, (x, t)
*/
1.7-t
Jrft J X v
u s i,/-i
§15. THE SINGLE-LAYER POTENTIAL
X (<P(|. r) — <p(x0, t) ) dSi — J dx j H0(x0, £, t. r)
X(<P(t <P(*o. T))rfSt
(»<•*». t) — <p(x0, /))dT
‘H/1
<
J (V(x0, %) — <p(x0, t))dx J H0(x0, l, t, x)dSi
+ <P(*o- 0 J dx J.2 atJ(x, t)
x " u,w«,
fcfct t> j
dxt
JrftJ j «,,<*.<)(-
u s i, j= i V
Vi
u)
vJ
dZQ (x — l, z% t, t)
i2«4r9/
jr •».*>/
-f- q> (x0. t)
Lo 5 f, /-I
X («,• (x0) — », (4)) rfi’i — j dt J 2 fl,; (x0, t)
x in,(x0)- mil))dS5
jr-oz.
dZg(X-~ £. /, T)
dxj
+ <P(*o- 0} J* dx f JJ a,j(x, t)
|0 § /./-I
Xo,®^f- | dx ! i aij Oo. 0
x
<» 5 /,/-I
az„(*—I,-?,*, t)i
dxj
z-x0
X-*X0
n,®dSt
397
(15-3)
398
IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
If the function <f> is continuous, then all of the expressions in square brackets
in the tight side tend to zero as * —* xQ. Consider the first expression. We fix an
arbitrary small positive c. Let 5 be a number such that for |£- Xq| <5 we have
<p(£, t) - <p(x0, r) < f . Let 2 be a part of the surface S lying inside the ball with
center and radius 8, and let a be a part of S lying inside the ball with cen¬
ter *0 and radius 2\x - . We assume that S/2 >|x - *0|. We have
rfT J S a‘i('x' ^ 117 n‘ T)~'P(** T>1 dSi
/■
a t,Jm I
-<ce
1* « fl+1
j dSEj (i!~xf 2
0 0
™p(-C-7— t
ce |*
3£ji=T J
since x lies on tbe normal to *0 and, consequently, *0 is the nearest point to x
of the surface S.
By virtue of (15.2)
t
J dx f H0(x0, t, t)l9(g, x) —<p(x0, t)] dSt
0 ff
ft i* /l+l-d . . . g
< ce j dS$ J (t — xf 2 exp ^— C !-■— ii-j dx < ce.
a o
In ordet to estimate the remaining integrals we make use of the inequality
2 a,j(x. t) t'JL n, (xa) - H0 (xa. t. x)
.l-t 1
< e 11 x~ x0|(f — T)_-r“+ jx — jc0|a(< — t)--7- }
X { exp (-C il^) + exp (-C-tigil) },
which follows from (11.3) and from
I«</(*. <) — Ai/(X0. OK«l* — X„f.
Since for £ £ S\a we have
yl*b-fcl<l*-&l<f |*b —61.
(15.4)
(15-5)
§15. THE SINGLE-LAYER POTENTIAL
it follows that
M[s*
I 1
XfoKI. t) — 9(*o. T)ldSi
t
<ce J ||jc — x0\(t —*)
0
+ 1jc —xor(< —tr~}d'c f exp(-C 1 dSl
S\9
<ce H* —X0! J T^^+U~*ol“
L 2\ff J
|jf — x0f\log\x — *oll]<«
and
0 4^7 n< ^o) — «0
1 »r *
J * j s «,,(*•d
XI<P(|. -t) —<P(-K0. T)1 ^55|<c(6)|x —x0i“.
Combining all of these estimates, we see that for small \x - *0| tbe first
term of (15.3) does not exceed ce, and consequently it vanishes for |x - *q| -
Let us consider the second term. Let jj be a number such that for |( - r |
we have
|q>(j:. t)~<p(x. T*|<e.
Repeating die calculations just carried out, we obtain
J l<P(*o. t) —<p(jc0- Old* J Jj —W0l</sJ
<e Jdt[J
La y«t a J
400
IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
<ce J dSl J (t — xf 2 exp (—
o L/-ti
ft fl+1-6 # , __ _
+ J (#—f)“ 2 exp(—
dx
i-n
<ce
f dSt , f asi
J | J
<ce.
and, further,
J [<P(*b. 0—<P(*o. t)\dx J [ 2 ,
-tl S\o L t, 1
t f'r «+» *+n
<ce j rfS5 j [|jc_x0|(/—x)~ * 4-|x—x0l“(<—t)“~J
S\o l-ti
X [exp (- C 1 tE |J)+exp (- C J)] dx
<cell +|*~ Arol“|log|x — je0||]<ce
dSt
and, finally,
. n
J !<P(*o. D-<P (x0. 01 dx H aVTJ7~
o s L/, y—i
<cfTl)|x— JC0|a.
Combining these estimates, we see that the second term of the right side also
tends to zero for * —»
The third and fourth terms also vanish for * —» *q, as is easily established
with the use of estimates (11-3), (15-5) and
lM*o) —n;(UI < c | *0 — | P. (15.6)
In these terms one can pass to the limit under the integral sign for x —*» x^,
§ 15. THE SINGLE-LAYER POTENTIAL 401
since the kernels of the potentials in these terms have a weak singularity.
Let us consider the expression in braces in (15.3). We have
j t
— «»J £ M*-*> dZa(x~dl x' t' x)Ml)dSl
•**% $ l.j-1 '
is J
”•“.15f *! ^
+ |2 J !«*/<*. *>-«„(*•
The function Z0 depends on t through the difference t - r and through the
functions r) and det/4(£, r). Let
Z(x~ x, <— x, x) — Z0(x — l, x, f, t).
The last argument r of Z coincides with the argument of A^’1^ and det A.
It is obvious that
d2T0 dZ (jr — t, jr. t — t, X) I
TT= ^ L
— ' - 0 +Z(*. 1. <. T).
where
, _ ( d2(x — £, x, t — i, t) dZ (x—I, X. t~ x, k) \ \
’ \ dx dx LJ*
One easily sees that
|Z(x, £.
and consequently
402
IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
.„n f rft f *Z&=±*J,JL41
= lim I Z(x —jc, e. /)<*!— f Z(x — \, x, t, t) dl
i
- J dt J %(x, 5. /. x)d\.
0 u
By transforming in a similar way the second term of the expression in braces
in the right side of (15*3)* we see that ail of this expression is equal to
lim f 2<x — x, e, — f Z{x0 — x0t c, f)d|
e*>0^ e~*°«
— |"J Z(x-|, x. t, t)dI- J Z(*o-|, <0.
- I J dx f %(x, fc. t, T) d\- f dt j S(JC». 6. <. T)rf||
Lo U o a J
+ [ EM
Li,/-is o
* r t
JJ J <ft J (.a,j(xB, t)—a,j (x0, 0)
i,1 o
X
d'ZAxH — X, xa<, t)
alTdfj
4
It is obvious that the expressions in square brackets here also tend to zero
for x —> *o' Therefore
lim
U /-I
S a‘l <*• '> 1ZJ *i <*») ~ roOo. #)
<P(*o. 0 I lin> f Z(* —X. e.
I'-*0 5
J Z(■*<,— J, *0, e,
2 j
-lim
«-*0 J
§ 15. THE SINGLE-LAYER POTENTIAL 403
One easily calculates, introducing the new variables of integration z = (x- f )/\/t~,
that
lim [ Z(jc—x, e, i)dl = ( *' ** X^°' -
•■+« J lo, if x£Ea\Q,
and
lim f Z(x0~l x0. e, t)dl =
1
n 1
(4n) 2 (del A, (je-o, t) )J
| dz,
where §> is a certain half space whose boundary plane passes through the origin.
It is easily seen that the latter integral is equal to one half the integral of the
same integrand taken over the whole space En, since this function is even with
respect to all of the variables z. Thus
lim f Z(Jf0—6, x0t e, t)dl~~ f Z(z, jc0, 3. €)dz^\
and
n
lim 2 Ilf “*(•*<>) == 0 ± y'K*0* 0. (15.7)
/ jm\
where the plus sign in the second term corresponds to the case when the point x
tends to xg from within Q, while the minus sign corresponds to the case when
x —> *0 from without.
In proving relation (15-7) we assumed that the point x tends to Xq while
remaining on the normal to S at the point xg, but, just as in classical potential
theory, this relation is also valid under more general assumptions. Suppose x is
in a neighborhood of xg and x is the nearest point to x of the surface S, so that
x lies on the normal to S at the point 5. For definiteness let * € O- We have
j Z(zt *0# I, t)dz
2»
40 4
IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
i. /-I
<
a
S “1/(*. 0 —ffi-9 »/ <*o) - W'o (*o. 0 - j<P(x0. t)
£ «„(*• t)^~—~nt(x)—Wa(x. t)-jq,(x. t)
1-1
+ |r0(J, *)-r0(*0. Ol + yltfx. 0-1>(*o. 0|
+
2 *<;(*• Q•g^Le%W —«I<«)1
It is easy to see that the first three terms of the right side tend to zero as
x —Xq. In order to estimate the fourth term we make use of inequalities (15.6)
and the fact that
I dV„(x,t) |
1 ixl '
<cmax|<p| j J expC-!~^~Lj(#—t) 2 dx
S (I
■< c, max |
dS,
jrr<C2|log|j: — x\
Thus the fourth term also tends to zero if x tends to xQ in such a way that
1*0 — x|9log|jc — *|-»0. (15.8)
This is always the case if x remains within a certain cone with apex at Xg and
axis directed along n (xQ).
From (15.1) and (15-7) it follows that under condition (15.8) we have
dV(x, t) 1
lim JJ alj(x, t)°l£dlnl(x0)*~W(x0, t)± j<t(xB, t), (15-S»
j- I
where
W
(.Xq, f) = J dx J H(x0, I, t, T)<P(|. *) dS%,
«,(ar0).
(15.10)
t, /-i
§ 15. THE SINGLE-L AYER POTENTIAL
405
This permits us to use a single-layer potential to solve the second boundary
value problem for the equation £k = 0, both in the interior domain Qj. and in the
exterior domain D^+^Qj- • Consider, for example, the interior problem
*
y. «„£«,<*> I =®(*. o. «i,=o=o. (i5.il)
ifjli '
Its solution is sought in the form of a single-layer potential
t
u{x, 0= JrfT J Z(.x, I- t. T)d|. (15.12)
0 s
The potential is continuous in the closure of the domain and satisfies inside Q
the homogeneous equation of (15.11) and also the zero initial condition. Requiring
that its conormal derivative be equal to we arrive by virtue of (15*9) at an inte¬
gral equation for the derisity <px
t
<p(g, t)~ — 2 J dx J H(I, T), t. T)f(t). t)rfS^ + 2<D(i, f).
0 s
Since the kernel H has a weak singularity, this equation is solved by the method
of successive approximations. Its solution will be a continuous function if $ is
continuous. It is also easy to show that if $ satisfies a Holder condition in its
arguments, then <p will also satisfy a Hdlder condition.
The solution of the exterior problem
S’n — 0 (x£E„\n),
au-~~~nt(x) I = <£, a|/ssO=:0 (15.13)
i, ; sr
can also be sought in the form (15.12), and the equation for the density <p will be
t
<p«. 0 = 2 J dx J H{6, f|, t, T)<p(ti, T)dS„—2®(|. f).
0 s
Thus we have proved
Theorem 15.1. Let the coefficients of the operator £ be defined in 0*
and belong to the class Let 0 be a bounded domain with boundary
406
IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
(16.1)
S £ For any continuous function on Sy. problems (IS.11) and (15.13)
have classical solutions that (i) are continuous in the closed domains Qj, and
^n+\\@T respectively, (ii) have continuous derivatives, namely, those entering
into the equations, inside these domains and (iii) satisfy the boundary conditions
when the boundary is approached along any curve on tehich relation (15.8) is ful¬
filled. The solution of the exterior problem is bounded at infinity.
§16. SOLUTION OF THE FIRST BOUNDARY VALUE PROBLEM
In this section we obtain a solution of the first boundary value problem
*(*- t’-h'-wh-f-
u\s = ®(x, t), a|(=0 = q)(jr).
For f=0,ip = 0 the solution of this problem can be sought in the form of a
double-layer potential
a(x.t)
r r "
=jax j s «</«• t>az(^y~ ""'(&)>*(&• T>d5t- a6-2)
0 S t.l-l ’
However, it is necessary in this connection to impose additional restrictions on
the leading coefficients a- of the operator £; these restrictions are connected
with the need to differentiate the function Z with respect to the variables (.. If
one assumes that the functions a., satisfy a Holder condition in I with exponent
*/
a/2 and have derivatives da-./dx^ satisfying a Hdlder condition with exponent a
in the variables x, then one can show that the function 7. has derivatives dZ/3^,
d21/dxk dt; j, Z/dXj dx^ ■, d2Z/dt ■ that are continuous functions for
x t£ £ and are subject to the inequalities
^TnJ dZ (*,&,<, t) 1
D‘D* W) 1
n+lr+s+l
<c(f-x) * exp^-C-'-j^-l1- ) (16.3)
for 2r + s < 2, so that the double-layer potential has continuous derivatives
D\D'^u (%, t) (2r + s < 2) inside Qj, and + j-• F°r the function Z' one has
tbe estimates
* exp(~ci^|!l) <s<2). (16.4)
§16. SOLUTION OF THE FIRST BOUNDARY PROBLEM
407
The proof of estimates (16.3), (16.4) and the continuity of all of the above-men-
tioned derivatives of the function Z will not be carried out; it can be found in
[25a] or [31b],
The limiting value of the potential (16.2) for * —* xQ € S is calculated in the
same way as the limiting value of the conormal derivative of the single-layer po¬
tential in § 15. It is expressed by the formula
lim u(x, t)
I
= J dx f M (x0, t, t)n(l. 0If -i H(x0. t), (16.5)
o i
where
M{Xo. I, /. *) = 2 «„(6. T) d7S^±V-„di)
i. J-1
is the kernel, which satisfies for Xg E S the inequality
|iW0(*0, I, t, T)|
exp(— 6 > 0, (16.6)
and hence has an integrable singularity at Xg = * = r. The minus sign in formula
(16.5) is taken when x —> from within Q, while the plus sign is taken when
x —» *0 from without. Relation (16.5) is first established in the case when *
tends to along the normal to S at the point Xg. But after that, injexactly the
same way as in classical potential theory (see, for example, [36], Russian p. 71,
and also the end of §15), one can show that (16.5) holds no matter how * tends
to xg, even without condition (15-8). From this it follows, in particular, that for a
continuous JU the function u (x, t) is uniformly continuous both in Qj. and in
dZ\\Qt-
Relation (16.5) permits us to reduce problem (16.1) with f = 0 to an integral
equation both in the interior (with respect to S) domain Qj< and in the exterior
domain D^+\\Qj- The solution of the interior problem is expressed in the form
of the potential (16.2), the density fi(£, 0 of which satisfies the integral equation
I*(I. 0 = 2 JrfTj M(l, t|. t, t)^. T)</S„— 2©(|. t),
while in the case of the exterior problem the integral equation has the form
408
IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
I
**(!. 0 = — 2 J dx | iM{|. n, t, x)dS,, + 2®(|, t).
0 5
Both equations are Voiterra equations with a weakly polar kernel and sue solved
by the method of successive approximations. Their solutions are as smooth as the
function $(£, t).
Thus we have
Theorem 16.1. If the coefficients of the operator £ belong to the class
/yS«/2(D(n)> and
in addition the leading coefficients a-j have derivatives
daif/3xk satisfying a HSlder condition in the variables x with exponent a, while
the surface S satisfies Ljapunov’s conditions, then problem (16.1) for f = 0,p - 0
and any continuous <J> has a classical solution both in the interior domain Qj and
in the exterior domain This solution has continuous derivatives,
namely those entering into the equation, at interior points of the corresponding
domain, and it itself is continuous up to the boundary.
We now proceed to solve the first boundary value problem (16.1) with the use
of a Green’s function.
The Green’s function for problem (16.1) in an interior domain is a function
G(x, y, t, r) possessing the following properties for all (y, r) 6 Qj.:
&\x, t, •^•)o(jc, y, t, x) = 6(* — y)6(< — t).
0(x, y, t. x) = 0, 0(x. y, t, t)|,(4 = 0.
It is connected with the fundamental solution 2 by the relation
0(x, y, t, x) = Z(*. y. t, x) — g(x. y, t, x), (16.7)
where the function g (*, y, t, r) is the solution of the problem
^(x- *' ~Sx' ■£)*<*• * <’T) = 0’
g(x, y. X. x) —0,
g(x, y. t. x)|JtCJ = Z(Jf. y. I. x)!^s. (16.8)
If one requires that the coefficients of the operator S. satisfy the conditions
of Theorem 16.1, then one can express the function g in the form of a double¬
layer potential. W. Pogorzelski [94d ] proposed the ingenious method set forth
below, which permits one to obtain the solution of problem (16.8) in the form of a
single-layer potential for a- € //“’
§ 16. SOLUTION OF THE FIRST BOUNDARY PROBLEM 409
Let (y, r) 6 Qj, and for x € £„\0 and r < ( < T let
g{x, y, t, x)~ Z (x, y, t, t). (16.9)
This function is obviously the solution of the following second boundary value
problem in the exterior domain D^+\\Q f '■
J?g = 0,
g (x, y, x. t) = 0,
|;4"iWLrH(i-y-<iT)-
where H (x, y, t, r) is the function (15.10).
From the results of §15 it follows that the function g is representable in the
form of a single-layer potential
t
g(x. y, t, t)= J rfX J Z(x, t, Jv)co(4, y, t, x)dSz. (16.10)
T s
where the density a> is determined from the integral equation
<#(4. y, t, X)
t
=2 j dX J H&, tj, t, X)®(ri, y, X, x)dSi — 2H(\, y, t, t). (16.11)
t s
which is solved by the method of successive approximations.
Let us now define the function g in Qj, by formula (16.10) and show that it
satisfies the relations of (16.8). Of these relations only the boundary condition
requires verification. It follows from the continuity of the single-layer potential
in the domain + j • Indeed, if the point * tends to a point *0 € S while remain¬
ing in the domain then by virtue of (16.9)
lim g(x, y, t, x) — Z(x, y, t. x). (16.12)
But since the limiting value of the single-layer potential does not depend on from
which side of the surface S the point * approaches *0, it follows that relation
(16.12) implies the necessary boundary condition.
Thus the function g is expressed by formula (16.10).
It is seen from this formula that g (x, y, t, r) does not have any singularities
410 IV, LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
when the points x and y are not on the surface S.
Let us estimate g with the use of relations (16.10) and (16.11). We first note
that for y 6 fl the function H (f, y, t, r) possesses the following properties:
l«ft. y. T)|<c(f-T)"^exp(-Cii5i£).
t
J dX J (//ft. y. X, T)|rfS|< c. (16.13)
t s
In order to prove the latter estimate we represent H in the form
where
H'-Zye.t)1*'*’; ‘%11
is a function for which the estimate
B-ft-Q .* .2.
|W'(|, V. * exp|—C L|—y-j
holds by virtue of (13-5), and hence
/ a
J dX J|/f'(i. y. X. T)|rfSs <c(t -t)t<c,.
T S
For the function Hq we have the formula
H0(l. y. /. i(D
- «—</<*
- 2(r^oS^-^)^ft)2o(l-y. y. *>.
J~1
so that
§ 16. SOLUTION OF THE FIRST BOUNDARY PROBLEM
411
t
jdxj ]H0a, y. X, x)|d5i
The boundedness of the latter integral is proved in classical potential theory
(see, for example, [36], Russian pages 275—276, for n = 3).
Thus , y, t, r) is subject to inequalities (16.13). By using the represen¬
tation of the function co in the form of the series obtained in the solution of equa¬
tion (16.11) by the method of successive approximations, it is not difficult to show
that the same estimates hold for o>:
We now estimate the function g with the use of (16.10). Let />(*) be the
distance from the point * to the surface S; then for 0 < 2r + s < 2 we have
t
x s
\DrtDsxg(x. y. /, t)|
t
< sup \D\DsxZ(x, I, t, X)| f dX I J<a<|, y, X, t)|dSt
In addition, we can obtain an estimate for g that does not depend on p(x).
Let o- be a part of the surface S lying in the ball with center x and radius
\x-y\/2. We have
412
IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
|£C*. y, t, T)| <C
n+l *
(/_,)" 2 f —X)~
z<a
t+x
T"
x / e*p (-c -^bF) exp (-c
Ux
2
+(/-tf ^expl-C-iyf) j dX J|®(|. K T)j</S{
t A*\a
4- (< — xfT exp (— y 'I'E'f"')
t+X
xj (X-xr~exp(-4-4^£)rfxJ,
t <»
< c (/ - t)"T exp (-C, LL~^|i!).
It follows from this estimate that the Green’s function has the same singu¬
larity at the point x - y, t = r as the fundamental solution:
|0(jf. y, t. T)|<c(f — t) Texp(—(16.16)
Let us consider the function
i
u(x, t)= J dx | 0(x. y. t, X)f{y, x)dy
0 Q
-4- J 0 (x. y. t. 0) <p (y) dy = b, (x, t) + at(x. t) (16.17)
a
for feH^a/2(Qr) and ip € C (Q), and show that it is the solution of problem
(16.1) for $ = 0. On account of (16.7) die equation £u = f is fulfilled in the
domain Qj., Further, by virtue of (16.16) and tbe results of §14 we have
limuO, <)= limut(x, f) = <p(x).
i-* o <-»o
Finally, let us verify that the function u satisfies the zero boundary condition.
It is obvious that for t > 0
§ 16. SOLUTION OF THE FIRST BOUNDARY PROBLEM
413
We fix a small S > 0. On account of (16.16) we hare
t
J dx J 0(x, y, t, x)f(y, x)dy <cmax|/|6.
1-6 Q
Since the function G (x, y, t, r) does not have any singularities fot I > r, for any
preassigned i/ > 0 we can find a d > 0 such that, when the distance from the
point x to the boundary is less than d,
i-i
I dx J 0(x, y, t, t)/(y, x)dy <t|.
o a
Thus for a point * that is sufficiently close to the boundary we have |iij(x, t)| <
c8 max|/| + ij, i.e. = 0.
We have thus proved that the function (16.10) is the classical solution of
problem (16.1) for $ <* 0.
We have considered problem (16.1) in an interior domain, but all of the argu¬
ments with obvious changes carry over to the exterior domain D^Q\Qj.. Thus
we have
Theorem 16.2. Suppose the coefficients of the operator £ belong to the class
ffa, a/2(£j<r)) Then problem (16.1) with $ = 0 has a classical solution in both
interior and exterior domains for any bounded f and <p if f satisfies a HSlder con¬
dition and <p is continuous.
Theorem 10.1 and inequality (16.16) permit us to obtain exact estimates for
the derivatives of the Green’s function. We present them without proof.
Theorem 16-3. If the coefficients of the operator £ belong to the class
//“>“/2^ thcn the following inequalities are valid'.
\&d;0(x, y, t, i)|<e(f—tf * exp(—cH-31-).
IDrtD*0{x, y. t. x) — Drt,D*Q(x. y, t'. t)|
o • t «+2+q . . „.
<£(/—ty T) a exp(-oH—ziy
where 2r + s ■= 1, 2 and r <t' < t, and
|d;dJ0(*, y, t. t) — D'iD%'0 (x\ y, t. t)|
a+2+a , , -
— x'\a{t — t) 2 exp (— C l~r:Zi. j,
414
IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
where 2r + s = 2 and x " is that one of the points x and x‘ which is closest to y.
§ 17. ON THE ESTIMATES OF S. N- BERNSTEIN
In the paper [8a] S. N. Bernstein proposed a method for estimating tbe maxi-
mums of the moduli of derivatives of any order of solutions of linear parabolic
equations under the assumption that the solution itself and all of the known func¬
tions in the equation are sufficiently smooth. Local (interior) estimates are given
in [8a], In combination with the estimates of derivatives on a boundary given by
him earlier (see [8b] and §3 of Chapter VI) they form a sufficiently complete col¬
lection of estimates even in the closure of a domain (with the use, of course, of
the boundary and initial conditions). The fact that the estimates on a boundary
are carried out in [8b] only for elliptic equations, and in two-dimensional domains,
is nonessential: they go through without any changes for equations of both types,
elliptic and parabolic, in any multidimensional domain. These estimates, as we
will show below, are simple and beautiful. But they cannot be considered as exact
estimates (concerning this see §§3 and 4 of Chapter I) and therefore cannot be as
effectively used in proving theorems on the solvability of various boundary value
problems as the estimates in Hdlder spaces and in the spaces given above.
Let us present Bernstein’s method for |aj. Suppose u is a classical solu¬
tion of the equation
with continuous coefficients and free term f and with the satisfying condition
(2.5) of Chapter 1.
If its values are known on Fj.: »|pr = with S and t/> being sufficiently
smooth, then maxj-y |ux| is estimated in tetms of known quantities in the way
done in Lemma 3.1 of Chapter VI (with natural simplifications for the given linear
and all of the coefficients and the free term of equation (17.1) are differentiable
with respect to x at each point of Qj- Suppose that max^|u| = jIf. Consider the
function v (x, t) - «*(*, t) + \u2(x, t) - fit, where A and n are two numbers which
will be chosen sufficiently large later on. Let us calculate S.V.
case). In order to estimate u I one must know that the derivatives
' X*
§ 17. BERNSTEIN’S ESTIMATES 415
3"v — 2a (u 4-^“ . 'j
** l* xk*ixj *k*t)
+ 2X« (ut - + «,«„) - 2aij\*,u V/
— 2 Xa,.ara_ —p..
*/ *l *j
By virtue of equation (17.1)
Jgrv = 2ur (—aur — a u + f_ + <*.,, a,, —a u )
**\ x* ** k ‘i*k xl*l iJ!t V
4-2Xb(—aa-\-f) — 2a.,ur —2Xa.,a,o, —j».
11 ikil xkxj IJ X, Xj
We estimate £i> from above, using Cauchy’s inequality (1.2) of Chapter U and
condition (2-5) of Chapter I, as follows:
+!)«» + (! +J,)JW* + /*]
4- &u\x -+- If1 — 2v«^. — 2vXaJ — |i,
where the constant c depends only on maxQ^a^, a., aljxfc> aixk’ a’ “*4!' ^ake
(=7x/, A =* (l/2i/)c(l + l/2i<) and p = c(l + A)M2 + c ma*QTf^ + AmaxQTf2+ !•
Then £v < - 1 < 0 at all points of Qj, and hence the maximum of v is on I*1 i.e.
maxv — -Xa2 — maxfa^-4- Xu2 — u/V (17.2)
Qr Qt ; rr w ;
And this is the desired estimate for max^ ju^j in terms of M and the already
estimated max j* lttx| •
In order to obtain a local (interior) estimate of we consider the auxiliary
function
•ai (x, <) = £2(x, t) ifi' (x, t)-j- Xu?(x. t)— jit,
where £ (x, t) is a cutting function that is equal to zero in the vicinity of Fj-. It
is easily seen that £w will be negative for sufficiently large A and p, and hence
max^j^ w will occur on Tj,, i.e.
w(x. t)^.maxw-^XM2.
rr
This inequality also gives an estimate of max^ |«,J for any domain Q' that
is separated from Tj- by a positive distance.
In order to estimate second-order derivatives one must consider the function
416
IV. LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS
v(x, t) = u\x(X, t) + Xu\(x, t) — \lt
or the function
w(x, t) = t?(x. t) ulxx (X, t)+Xu2x(x, t) — (If,
depending on whether a total or a local estimate of |uxx| is desired, and choose A
and ft so large that £v or £ui will be negative in Qj.. (A detailed derivation of
this can be found in [Wi] and in [123a].) The derivatives of u of any order are
estimated in essentially the same way.
CHAPTER V
QUASI-LINEAR EQUATIONS WITH PRINCIPAL PART
IN DIVERGENCE FORM
This chapter is devoted to quasi-lineat equations with principal part in diver¬
gence form, i.e. to equations of the form
jg>u^ai — -^a,(x, t. u, ax) + a(x, t, u, ax) = 0. (0.1)
They do not include all of the quasi-lineat equations. But all of the widely
known equations of mechanics that are of parabolic type belong to this class. Cer¬
tainly the equations
a,~alf(x. t, u)u„ +«(*. t. u. «,) = (>
and
",~au{xv *< *• «• “,,) = °
can be reduced to it in an obvious way. We will investigate the generalized and
classical solutions of the equations (0.1) and prove the unique solvability of the
basic boundary value problems. The restrictions under which this is done are
essential and, as the examples of Chapter I show, they cannot be weakened (in
the terms of the adopted spaces) for the whole class of equations (0.1).
The solvability of the boundary value problems, as was noted in §4 of Chap¬
ter I, is proved by us on tbe basis of the Leray-Schauder theorem and a priori es¬
timates of the norms |u °f aH possible solutions of them. We begin the
present chapter by obtaining these estimates. In §§1 and 2 we give estimates of
the norms max^ |uI, |u|^) and |b|^,, then (in §3) of the norms
and after this of ma%Q^\ux\ (§4) and (§5) for the whole domain Qj. In
this connection we essentially use the results of §§7 and 8 of Chapter II on func¬
tions of the classes which contain the solution u themselves and their deriva¬
tives u . Estimates in the vicinity of Sj,, and thereby in tbe whole domain, are
417
418
V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
carried out in these sections for the first boundary conditions. They and Theorem
5.2 of Chapter IV on linear equations with smooth coefficients permit one to verify
the fulfillability of all of the conditions of tbe Leray-Schauder theorem and to es¬
tablish with its use the solvability of the first boundary value problem. This is
done in §6. In §7 analogous arguments ve carried out for other (nonlinear)
boundary conditions. In §8 the Cauchy problem is briefly analyzed. In §9 the
Stefan problem is considered. Its specific peculiarity is the fact that the common
boundary of different phases (or the boundary of die domain of definition in the
case of a one-phase problem) is unknown. Finally, in §10 we give another method
for estimating the oscillations of the solutions of equations (0.1), which does not
require one to turn to the classes 82 but uses essentially the same analytic
facts as the first method.
At the end of §6 we give a proof, which is based only on the energy estimate,
of the generalized solvability of tbe boundary value problems.
§1. BOUNDED GENERALIZED SOLUTIONS.
HOLDER CONTINUITY
Suppose the functions at(x, t, u, p) and a(x, t, u, p) making up equation
(0.1) are defined for (*, t) 6 Qj. and arbitrary values of the arguments u and
p = (pj, • - -, pn), are continuous in u and p, and satisfy the conditions
a,(x, t, u, p)Pi>v{\u\)p* ~^(x, t), (1>1)
la,(x, t, a, p)!<|x(|«|)|/>|-f-«p,(Jc, t), a-2)
\a(x. t, a. p)|<|i,(|«|)p* + »,(jt. 0. (1.3)
in which v(£) and the fii (<f) are, as elsewhere, positive continuous functions of
(> 0, with t'(f) monotonically decreasing, the monotonically increasing,
and the functions (x, t) nonnegative and having the finite norms
II ‘Pallor, Qf’ II fl ^2g,2r,QT ^(*2- (1.4)
where q and r are arbitrary positive numbers satisfying the coadition
§1. HOLDER CONTINUITY 419
fot n 2,
f€U. <»]. r€[j~j7' i~2kt]’ °<*i<T
for n=l.
(1.5)
By a bounded generalized solution (bo. gen. solution) of equation (0.1) we
will mean a function uix, j) from Fj’0 (Qj-) with vrai ma*^y,|u j < « that satis¬
fies the integral identity
t
J uix, t)mx. <)<**£ +J J [— + «,(*. t, u.
fi 0 8
-j-a(x> tt u, 0 (1.6)
°. _
fot any function ijix, t) from Jf'j. (Qy) with vrai m'AXQ^r\rj\ <00 and any i from
lo, n
As a consequence of assumptions (1.1)—(1.5) all of the teems of (1.6) are
finite for arbitrary u and »/ from the indicated classes, and from (1.6) it follows
that
| u(xt t)f\(xt t) dx
q
t
+ J J [—B1»t+ai(x> l- “• ux)wdxdt
t6 a
i
< J J(w,«*+<pa).l’i|rf*^. o<f0<*<r, (1-7)
/„ C
where (vrai max^ |u|). Let vrai max^ |uj a* U9 ^ p{M)t v - u{H)
and = ftj(M). Tbe following theorem holds:
Theorem 1.1. Suppose the functions a^(x9 t, u, p) and a(x, t, u, p) possess
properties (1.1)—(1.5)> while u{x, t) is a bo. gen. solution of equation (0.1) with
vrai max^j,|tt| = M. Then u{xt t} belongs to #a,a^2(().p) with some positive a
depending only on nt v, /*, q, r and The quantity for any Q* C Qy
separated from V ^ by a positive distance d, is estimated from above by a con~
stant depending only on n, M, v, n, /i2, q, r and the distance d.
If u is Holder continuous {of exponent /3 with respect to x and fi/2 with
respect to t) on some part V of the boundary Fy. satisfying condition (A), then
420 V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
uix, t) will belong to Ha,a^2iQT (J P') with some a > 0 depending only on n, v,
H, q, r, Mp^/v and /9. The norm |«|^, for any Q' belonging to Qj. and separ¬
ated from rT\r‘ by a positive distance d, is estimated from above by a constant
determined by n, M, v, n, fly fij, q, r, j3, the constants 60 and of con¬
dition (A) and the distance d. In particular, when P' = Tj., S satisfies condition
(A), “|rjr 6 (rr), the solution uix, t) 6 Ha,a^2(Qj-) and is es¬
timated from above by a constant depending only on n, M, v, p, /ij, 9> r> /3, the
constants and of condition (A) and |u |
Remark 1.1. The parameters q and r in (1.4) can be different for different
<£(., and in place of the t) in the right sides of (1.1)—(1.3) there can be finite
sums of different rj>, it being only necessary that the parameters q and r corre¬
sponding to them satisfy the requirements (1.5).
In order to prove the theorem it is sufficient to demonstrate that u (x, t) be¬
longs to the class S^iQ^., if, y, m, 8, k) with parameters y, m, 8 and k deter¬
mined only by M and the numbers entering into conditions (1.1)—(1.5).
To this end we first note that identity (1.6) implies the identities
J J [Vi+“<(*• *• “•
i, a
-f- a (x, (, a, «x) r^] dx dt — 0 (1.8)
for h<i0<tl<T-h with any function t)[x, t) from V^'0 iQj._^). The symbols
( )h and ( )£- in them denote the Steklov averagings over future and past t de¬
fined by formulas (2.7) and (2.10) of Chapter III. Relations (1.8) are deduced from
(1.6) in the same way as (2.11) was deduced from (1.16) in Chapter III. In (1.8) we
put t)(x, t) « £2(*> f)max {uA (x, t) - k; Ol s £2u^\ where fix, t) is an arbitrary
nonnegative continuous piecewise-smooth function that is equal to zero on Sj.,
and we transform the first term as follows:
11
f f u^Vdxdt
* *
= i I (“**’<*• 0^1,';- J J {afj&'dxdt.
fi /• a
We then pass to the limit io all of the terms as h —► 0. As a result we obtain
HOLDER CONTINUITY 421
h
t)Ux, Oft af+ J J [-(«<*>)*££,
* U a
+ a( (u<*V)Xi-+-OB**1?2] dx dt = 0. (1.9)
The validity of the passage to the limit in all of the terms as ft —* 0 follows from
Lemmas 4.5—4.7 of Chapter II and from assumptions (1.1)—(1.5) on the functions
a-(x, t, u, p) and a(*, t, u, p), and also from the membership of it in V^’0 (Qr).
In (1.9) tg and tj can be taken arbitrarily from (0, T) and, since all of the terms
of (1.9) are continuous functions of tg and !j, also from [0, 7"].
We will assume £(*> t) to be different from zero only for * € Kfi. By virtue
of assumptions (1.1)—(1.3) it follows from (1.9) that
i
')?(*. Oil!,* f+v f J [uWf^dxdt
“ ' ‘I *P
t
*» At.p
+(n,«*+<p2)(8—] dx dt. (1.10)
Estimating the terms of tbe right side by Cauchy’s inequality (1.2) of Chapter II,
we obtain the inequality
j || «“> (x. t)i(x. t) f2 K |' + v J [aWfS2 dx dt
t
A*. p
-f (u — kf S|+4- q>2) I2 ^ maIX ^ u {x, t) — kj
+(“ — (HI)
in which Q(p, t- tg) = Kp x (tg, t) and tj is an arbitrary positive number.
This inequality has tbe same character as inequality (10.1) of Chapter III. The
only difference consists in tbe fact that the right side of (1.11) contains the term
(mai^jA t-trf u(a, t) - the coefficient p.j(maxAt-1())“(*> *) - 4) °f
which is in general not small and which accordingly cannot be cancelled by the
422
V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
second term of the left side. In order for this to be possible we impose a restric¬
tion on the choice of the level k- Namely, we take only those k for which
max u(x, t)—A6 = -r~. (1.12)
<?ip. t)
Then (I.IX) with fj = v/4p for t 6 [»0, + r] implies
|| «<*>(*. OUx, 0|gi/r -t-v J [af]n2dxdt
“ Q(P, t-W
<[!«<*>(*, «„)£(*,/0)||^ +c J [«WF(£+S|£,|)rf*<tf
p e<P. t-h)
t
-)- J J &x(x,t)dxdt, (1.13)
where c = 2(4pt/v + 1) and 2)j(*, t) = 2 (<f>n + + 8<f>^)£2- This inequality co¬
incides with inequality (10.2) of Chapter III for the case M = 0. Moreover, the
assumptions (1.4)—(1.5) on the </>i are such that the 3)j from (1.13) and the 3)j
from (10.2) belong to one and the same space L In view of this (1.13) im¬
plies inequalities (10.4) and (10.6) of Chapter III, in which M is put equal to zero,
or equivalently, inequalities (7.5) of Chapter II. Analogous inequalities hold fot
- u(x, t). They imply inequalities (7.1) and (7.2) of Chapter II and thereby the
fact that the solution u is an element of
ma*!*: ll^ilW.Or}
©2(<?r. V. r, 6. k) with y— iSKTFvi ’
x = ~L, 6=-—-,
and an element of
3 t n_
%(<3(Po- Po5)- M- V. r, 6, k) with 9o m. [c || 3t |t^V. Qr] “ *» 7 " 4.
V~ mln Jl; v} ’ r ~ r^T *1 + 2H^ 6=4j--
But, as was proved in §7 of Chapter II, this guarantees the membership of u in
Ha,a^2(Qj.) with some positive a, which is a continuous function of the argu¬
ments y = (2/min il; vi) (4fx2/v + 1), q, r and M/S = 4Mfi -Ju in an admissible do¬
main of variation of these arguments (for M/S this domain is the semiaxis [0, ~)),
and the estimate of (u ) ^q).
§2. BOUNDEDNESS OF GENERALIZED SOLUTIONS
423
The other two assertions of the theorem follow from Theorem 8.1 of Chapter
II and the fact that the solution u belongs to 82(Qp M, y, r, 5, k) and '£>£Q(p0, pg),
U, y, r, 8, k) with the same values of the parameters M, y, r, 8 and k as above.
Remark 1.2. It is easy to see that the membership of the function u(x, t) to
the classes was actually deduced by as not from the fact that a (x, () satis¬
fies the identity (1.6) but from the fact that inequalities (1.7) are valid for it.
Therefore all of the assertions of Theorem 1.1 are valid for an arbitrary function
u(x, e) from K j’0 {Qj) with finite vrai max^|u| that satisfies inequalities (1.7)
for all of the indicated 1), and t.
Inequalities (1.13) for u (but not for - a!), and hence inequalities (7.1) and
(7.2) of Chapter II for u, underlying the definitions of the classes Bj, were de¬
duced by us from inequalities (1.7), in which tj takes only nonnegative values.
§2. ON THE BOUNDEDNESS OF GENERALIZED SOLUTIONS
We consider here arbitrary (not necessarily bounded) generalized solutions of
equation (0.1) that belong to the space iQj>)- It is natural to define such
solutions u(x, t) as elements of (Qj.) satisfying the identity (1.6) for any
function t/(x, t) from IT^’1 (Py). In order for this definition to make sense it is
necessary to impose restrictions on the functions a^(x, t, u, p) and a(x, t, u, p)
that are somewhat more rigid than (1.2)—(1.5). If one formulates them in the form
|o((x, t, u, /0|<c|j»|4 e!“f 4 <M*. <)• (21)
|o(jc, t, u, /?)]<£|J»f_p+c|«|v + q)2(«. t),
then the function ^(x, t), <£2(*> *) and the parameters a, (8, y must satisfy the
conditions
0<a<iL±i:
0<v<i+4; *i€*,(«r).
i+^r=‘+f- *>>• *>'•
(2-2)
Conditions (2.1) and (2.2), as can easily be demonstrated by means of Holder’s
inequality and inequality (3.4) of Chapter II, guarantee the finiteness of the inte-
grals jg^ a({x, t, u, ux) rjx, dx dt and Jg^a(x, 1, a, ux)t]dxdt for any functions
u and -q from V\’®{Qr). Let as show that any generalized solution u(x, t) from
424
V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
Vj’0 iQf) of equation (0.1) will be bounded and that it is possible to estimate
vtai max^lul from above in terms of ll“ll91 rj qt and known quantities if for
I“ t > *0 t*le ^unct*ons aip) and a(x, t, u, p) also satisfy the conditions
“(P)Pi>v\P\ — lM“| —aJq>(x, t), v> 0,
and
in which
(6 — 2) ^ J- + -7--j =1 _ jt,, where 6 > 2,
A^w^r*00]’
0 < x, < 1. for n^>2and
<*1. 1 -x, 1 I - 2x, ]'
1
(2.3)
(2.4)
0<xt<2-, for n= t,
vn.
i2-n
(2.5)
while cfi (x, t) is an element of r2 with q^ and satisfying the con¬
ditions
1 . n ,
_ + „==l_X2>
% 6 [2(1 —
Xj)
?2€I1. °°1*
•«]. 0 < Xj < 1.
fot n 2,
^[t^* 1—2k,]* °<x2<‘2
fot «*1.
(2.6)
The numbers <?j and can be any positive numbers, even less than unity,
and fi0 < v. The number 5 is assumed to be greater than 2; for 5 = 2 the term
|b| 8 can be assumed to be included in u2<f>(x, t). We confine ourselves here to a
total estimate of |u|, i.e. to an estimate of vrai |u |, assuming in this con¬
nection, of course, that vrai ma*j. |u | < <*>. From the present section and §8 of
Chapter III it can be seen how one varies the reasoning to obtain local estimates
of vrai max {u | (without assuming that vrai max^ ju | is bounded). If fi = 0 in
(2-3) and (2.4), then vrai max^^juj is estimated only in terms of known quantities.
§2. BOUNDEDNESS OF GENERALIZED SOLUTIONS
425
If it is known a priori that the solution u(x, l) is bounded, then the condition
/ig < v can be dropped. We will prove all of these assertions.
Theorem 2.1. Suppose the functions a^x, t, u, p) and a(x, t, u, p) satisfy
conditions (2.1)-{2.6) and < v. Then my generalized solution uix, t) from
V2of equation (0.1) with vrai max^ |u| e Mg < •» is a bounded function
and vrai max^ |u| is estimated from above by a constant c determined only by
the quantities v - n0, p, \\<l>\\q2,r1,QT< ?2> rV ?1« rV S> *0’ T and
mes fi. The parameters ql and can be any positive numbers, when /i * 0 the
constant c does not depend on Mljj.rj Qj.-
In order to prove Theorem 2.1 we consider relation (1.9). For k>Ug the
function £(x, t) in it can be put equal to 1 since the function rj m £2 will
be equal to zero on Fj.. From (1.9) with £ s 1, tg = 0 and k > max jjK0; £g| it
follows by virtue of assumptions (2.3) and (2.4) that
i||«<S>||’ ()+ J K-|iI uf-u*v)dxdt
< — f au~dxdt^ J U\)dxdt, (2.7)
where, as elsewhere, (*, t) = maxJuOt, t) - k; Ol, Ak(t) is the set of points
x 6 0 at which o(x, t) > k, and Qt^ (A) is die set of points (*, t) 6 Qt =
fl x (0, tj) at which u(x, t) > k. .Since by assumption fig < u, from (2.7) we have
the inequality
J 0*1*1*+-^)^. (2-8)
in which c depends only on v - fig from conditions (2-3) and (2.4). Let us esti¬
mate the integrals /'j = Sq( (ty l“i dxdt and j2 = u24‘dxdt, assuming
that s > 2 and the quantity is known. By virtue of conditions (2.5)
and Holder’s inequality we get
426
V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
.11
II9-2
V V <?r
(2.9)
where q l~ 2 ql/ql - (S - 2) and = 2rl/{rl - (8 - 2)). Conditions (2.5) imply
l/r J + 7i/2<?J = + Kj)/4. where Kj = 2<Cj/b and
i.€[2. ?.€[*• £] ** «>s.
?,gI2. oo], r,6[i-qfrssr* I"] for B=r'1-
We take the numbers 9j = 9j (l + «j), r"j = r j(l + Kj). They satisfy relations (3-3)
of Chapter II; thus the embedding theorem (3.4) of Chapter II is valid. In view of
this and inequality (3-5) of Chapter II,
_2 2_
Jl < 21! u ||“-2V Qr p21 «<*> (S) ^ J raesT, A„ (t) dtj
? < + K|
+ *a ^ J mes *> Ak (t) dt j
<21| a"6-2
"v <?r
I 4-- — V
P21 a<*’ le (wUi 7' mes?.Q/
h
2x,
+ A* I j mes «* (/) <#
(2.10)
For q j = S - 2 (this is possible only in the case n m 1) qj goes to 00 and in
place of (2.10) we will have
M*. f.)
+it. I
?1 (*. Vj .
(2.11)
§2. BOUNDEDNESS OF GENERALIZED SOLUTIONS
427
where fi(k, ij) is the measure of the set of those points t in the interval (0, ij)
at which mes A^it) > 0.
The integral j2 is estimated analogously:
<211 <P
v V Q,t m Ir >v Q,t
<*)
«s. r,, Qr
<2toK.'r*r
a— k\l . 4 *2||1||- - 1
%<■> -v*' V'rV*>J
I ± J- V*
4»PQI m W,* mes «» 0/
m
•bk1
l+Xi
r>
(2.12)
where ?2 = 2q2Aq2 -1 ),r2~ - 1), *2 = 2k2/h, q2 = q2(l + «2),
r~2 = r 2(1 4- *2), with the numbers q2 and ?2 satisfying relations (3-3) of Chapter
IL For <?2 - 1 the index <f2 “ 00 instead of (2.12) we get
e2^,^f* (*. *,>
h< 2
r,. Of
1-frMa
+ k2(l '<
(*. <l)] .
(2.13)
where fi(k, J j) has the same meaning as in (2.10). Let us substitute our esti¬
mates for /'j and j2 into inequality (2.8). Then for values of t j so small that
4cp2
nll<
/ _L j.
“lC.Tr,, QrVt7' mes ?1 ®)
+ l*H,r r„ Qr(v’ m«^o) ] <1,
we will have
l+*l
I “(J) !<?/, (*) < c‘* [(I mes ?' A i, (t) dtj
0 4 V
4" mes (<) dtj
l + Xj '
r*
(2.14)
428
V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
for k > max IUQ, where for f(- = ■» it is necessary to take pik, tj) in place of
the integral /p1 mesr,/,,‘ A^il'ldt. Theconstant cj is determined only by known
quantities. For p = 0 (i.e. for 8 - 2) it does not depend on From
inequality (2.14) we conclude on the basis of Theorem 6.1 of Chapter II (more pre¬
cisely, Remark 6.1 to it) that uix, t) is bounded in from above and
vrai u does not exceed a certain number determined only by kQ, the
constants cj, r£, k£, and mes 0 and T, i.e. by the quantities indicated in
Theorem 2.1. By considering uix, t) successively in the cylinders £1 x (tj, 2tj),
il x (2/j, 3tj) and so on, we estimate in a finite number of steps vrai max u from
above in all of Qy. In order to estimate uix, t) from below it is necessary to
apply the result just obtained to the function u (x, t) = - uix, t), which satisfies
an equation of the same type as uix, t), namely
*• “*) + «(*• *• “■ «,) = 0
with
a(ix, t, u. ux) — — a,(x, (, —u, — ux)
and
a (jc, t, a, ux) = — a (x, t, — a, — ux),
subject to conditions (2.3)—(2.6). Theorem 2.1 is proved.
Remark 2.1. We note that the estimate of vrai mas^it from above is actually
deduced not from the identities (1.6), which the generalized solution uix, t) of
equation (0.1) satisfies, but from the corresponding inequalities
^ uix, t) lfix, t)dx |* + f (j [- ui)t + a(.(x, t, u, ux)ijx.
"a 0 n
+ aix, t, u, ux)if] dx dl < 0 (1.6')
01 i
with a nonnegative function i) from {Qj). Analogously, only inequalities
(1.6') with a nonpositive function ij are needed for an estimate of vrai min^ u
from below.
As indicated, conditions (2.1)—(2.2) are imposed only in order that the inte¬
grals in (1.6) will have a meaning for any of the solutions in question. If one
assumes that the solution u is better behaved than in Theorem 2.1, then the re¬
quirements (2.1)—(2.2) can be weakened. In addition, it is possible to get rid of
the restriction fiQ < v, which was imposed above. Namely, the following theorem
bolds.
§2. BOUNDEDNESS OF GENERALIZED SOLUTIONS 429
Theorem 2.2. If the functions Oj(as, t, u, p) and a(x, t, it, p) for (x, t) € QT,
|it|, |p | < “> are subject to restrictions (1.2)—(1.3) with 4>2 from iQ-p),
and satisfy far k>kg and the same x, t, u, p conditions (2.3)—(2.6) with an arbi¬
trary constant ftQ in (2.4), then for any bounded generalized solution it 6
the quantity vrai maxg^, |b| is estimated from above by a constant M depending
only on k0, v, fx, ft0, 8, qv rv Il0l?2,r2,<jr> ?2 md r2 from conditions (2.3)-
(2.6), and on Mq = vrai maxry |it|, the norm ||it|? ^ q^,, mes Q and T.
We consider the function v ■= |n|m, m > 1, and the function v^ <• Ittjl”1,
corresponding to it, and in identity (1.8) we put
n(jr, o = I “a (X, t) r_1 «J»J (x. t) sign uh (*. /),
Here is the Steklov averaging of u over t defined in §2 of Chapter HI,
while (*, t) - max t) - k; 0|.
This leads us to the inequality
sj [•©(*.«fdxt
8
+ J J \at(x, t, a. Bx)O“*r",*lSn“*t'S0Sr<
+ a(x, t. a, «x) (i T~ ‘ sign dxdt=0.
from which, passing to the limit as h —» 0 and setting tQ - 0, we deduce
~ J [«<*>(.*, t)fdx
Q
ti
+ J J [*,(*• t. u. ax) (| a l^uv™+(m_ 1) |„ |”-2
+ a (x. t, a, ux) | a I”1'2 avw\ dx dt — 0; (2.15)
for h > max \ '1q ; A™I we obtain from (2.15), (2.3) and (2.4) the inequality
430
V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
whete is the set of points (x, t) from Qt^ at which v(x, t) > k.
Hence, choosing m > Hq/v and taking into account that |u|m = v and
|i> |, we see that
J [*H*>(*. /,)),dx+ J itdxdt
a Qt, (*)
-<c J m -f-ifiyjdxdt. (2.16)
<?!,<*>
*v
Here the function «£(*, t) is the same as in (2.8), the exponent 8 » (5 - 2)/m + 2
is subject to the restriction
and in regard to v(x, t) it is known that
®*rri = 11 “ M*i’ rr qt'
Thus inequalities (2.16) have exactly the same form as (2.8), with the same re¬
strictions on tft, 8 - (5- 2)/m + 2 and v that were imposed on <j>, 8 and u re¬
spectively in inequalities (2.8). This permits us to draw the desired conclusion
concerning the estimate of vrai max^.f and hence of vrai max^|uj from above
in terms of the constants indicated in the theorem.
In connection with the above discussion we mention the paper [53], in which
the validity of the strong maximum principle is established fot generalized
solutions from (Qf) of equation (0.1) if in it (x, t)ux. and
a = <j(b, ux) > M\uJ.
§3- ESTIMATES OF max^, |<g AND
We assume that the properties of the functions a^x, i, u, p) and a(x, t, it, p)
are somewhat better than in §1, namely, that the partial derivatives
da.(x, t, a, p)/ dpj, dajda and da^dx- exist and that for (*, t) 6 QT, {u [ < M
and arbitrary p the following estimates are valid:
i-i i i-i
(3-1)
in which
with
§3. ESTIMATES OF ma Xq, ) u%\ AND 431
t. u. p)| + j da‘^ “,£-)-|<|ilp|+<P,(Jr, 0. (3.2)
l0P2 + <ft.(*. t). (3.3)
a(jc, t. u, p)|<»t/>J +<&(•*. t), (3.4)
ll<Pi. <Pj. 1>3ll2?,s,,0_<M1. (3-5)
— + -S~ = 1 — X| and
r ' 2q
0 < x, < 1 fot n ^ 2,
2
(3.6)
0<K,<y for n=l.
We will prove that under the fulfilment of these conditions the quantities
max^,, \ux | and f°r £he solutions «(*, <) pf equations (0.1) and some
a > 0 are estimated ftom above by a constant depending only on the numerical
parameters in conditions (3.1)-(3.6), max^ |u| and the distance from Q' to Fy.
For the sake of simplicity we will assume that u € O2’^(Q^.). As to how to obtain
these estimates for generalized solutions we refer the reader to Chapter IV of the
book [65q], in which this is sho#n for equations of elliptic type. The indices q
and r in (3.5) can be different for different functions. Thus, suppose u(x, t) €
o2,1 {Qf) and ma*Qr j ii j = M. We will first estimate IIux£\\ L2(Qt)’ where dx> *)
is a smooth function with values between 0 and 1 that is equal to zero on Tj..
To this end we consider the equality
i
J j Jg’aeiu!?dxdt== 0, X — const ~^> I.
o a
and transform it by means of an integration by parts to the form
I
X J •U(* VI*. t)dx U + J f [- x-Stt,
C 0 S2
-f- aj (x, t, u, ux) (Xux,+2£Cr() + o£2J dx dt = 0. (3.7)
432
V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
By virtue of assumptions (3-1) and (3.2) we have
at(x. t, #, p)pt
l
f da,(x..t, a. p)
==PiPj j
di+Pi<*i(x. t. u, 0)
p-ip
dp,
>v/’2~]£IlPil<Pi(*. 0>yP2-^T<P?-
Therefore
i i
f f ^uy-t?dxdt < f J e^[||^| + |5.<p?
0 0 0 0
+((*| «, | ■+ %) *| c, | ■+ (K + %)$]dx dt
t
< J’ J •u(t»«‘I + &«*+ KC2+^+2q»,q5,|
o a
-f-HirSp+cPjP)dxdt. (3.8)
We choose A = 8pjv and take into account the fact that <f>\, <£3I < c(p^\
then from (3.B) it is seen that
J u^dxdt^C. (3.9)
Or
where c is a constant depending only on a, l/u, p., py U, max^ (|£x|, |£,|) and
mes Qj. (c is an increasing function of the indicated arguments).
Let us now show that it is possible to estimate the integrals dx
for any s (this need only be done for s not exceeding a certain fixed number sQ
determined by the parameters q and r of (3.6)) in terms of known parameters, as
long as Kfi and the concentric ball K2/3 belong to 0 and p does not exceed a
certain number pQ. The number pQ is also determined only by known parameters
and Sq. It is easy to verify that the assumptions of Theorem 1.1 follow from con¬
ditions (3.1)—(3-6). We arbitrarily fix an interior cylinder Q' and carry out all
considerations within it. By virtue of Theorem 1.1, fot any KpC0' we have
OSC2 [a, /<■„} < cp».. (3.10)
Let us take a cylinder K2p x (0, tj) and a smooth function £(x, t) with
values between 0 and 1 that is equal to zero in the vicinity of tbe lower base
§3. ESTIMATES OF max^, |i»J AND (ux)(q) 433
and lateral surface of this cylinder.
We consider the equality
0 K2p*-1
t<iv s = 0, 1 (3.11)
and transform it by means of a double integration by parts to the form
t
2s + 2 /+ PC*-J[2VX2-|“,r22%
*jp 0 *Sp
(|«^rf*dt = 0. (3.12)
It is not hard to see that the passage from (3-11) to (3-12) is legitimate in spite
of the fact that the derivatives a and D^u were used in the intervening trans¬
formations (see page 211).
We leave the trivially nonnegative terms in the left side, while the remaining
terms are carried over to the right side and estimated from above, using inequality
(1.2) of Chapter II and (3-1)—(3-6):
sdj-2 J !“,(*. t)dx
^2p
+ l“xl2s~!‘iXltuXtX uXlaXlXp\dxdt
*I J
t
— J f rg-j. „ | Ux |2j uXk%tt,
0 Klp I Xi
+(It+ s*- - a6*) -^(1“***Ux^
i
j [Ei«,f«u2+^|a,r^+
u *■*>
434
V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
n
+ (K -*■ fi Kl+K +f).+K +%) 2
i,k-\
2C|S,|] dxdt.
-i-i. i*•+*»
Hence
t
2s^f2 J ol2i+2?2(-#. + J J
**p 0 N>
*-i \*«1 /
V , As 2 *2
■jj | “■» I ***£
dxdt
< J J I“jt |2j£2 + ~ I «^|2l+2 E* 4 £«L i Uj f* £J
(3.13)
4"
/-i \ft-t /
+ ^Vk P*+4 f + 3«V I Us f*4 e + SteV I «x r+2 fi
+ |"xla,+2^l^ |“K<Pl l^xl + Va + ^s) 5] |(lB*l*,<ljrJ^JF|| K* ^
/,a-i J
We choose e » y/8 and t ^ » i//2, and we transfer the terms containing second
derivatives of u with respect to x to tbe left side and reduce similar terms. This
gives us
/
/<«(«. v- <*)/ J [(s+i)Kr<£’
S ATjp
+1 “,|*+2(£ + W) + (». | «,| +¥*+»*)
x 2
dxdt.
(3.14)
In order to estimate the first term of the right side we make use of the inequality
§3. ESTIMATES OF maxg, \uj AND (u^gi 435
< e,pa' J [(s + l)11 «, J2* u\p +1«, |*+5 JJ] <**. (3-15)
which by virtue of (3.10) follows bom inequality (5.8) of Chapter II. The constant Cj in it
depends only on n and c from (3.10). Let us take pQ such that
c,c(«, n](«o + I)3 Pg* = - (5-l6>
Then for p < p0 aod i = sQ it follows from (3.14) and (3-15) that
J ki*+*p**+J J ii\'.rw
ifix*
t
<«(*. v-11)/ j(M'(*+j)+i]i^r+^
° **>
+ 1 “t |2i+2S/ + (<Pl+<P2 + <P3)(l +|“xl)
xSld^lX^H^^. (3.17)
/, fe
Inequalities (3.17) have the same character as inequalities (11.7) of Chapter
III. The differences in the constants and domains of integration are nonessential.
The role of the function S) is played by the sum + <t>2* $y which, as can be
seen from a comparison of conditions (3.2)—(3.6) with conditions (11.1) of Chapter
III, has tbe same finite norm ||3)j|2? 2r Qf"
In the same way as in Chapter III, we obtain from (3.17) and (3.15) for s = 0
and from inequality (3-9) the estimate
+j J [“L+Kr]^*rf'<c-
20 o /r2p
self the estimate
7
| $apdX(U<c,
°<‘<T 0 *2P
and from it and equation (1.1) itself the estimate
7
2p
436
V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
where the c's are constants determined only by known quantities.
In addition, the following integral estimates were deduced from inequalities
(11.7): r
max f |«x(*. f)|*’+***+ f f\uxfs+*dxdt4ZcSl, (3-18)
*p « K0
e > 0, P<Po,
with the constants cSQ depending only on n, M, sQ, tbe parameters from condi¬
tions (3.1)—(3-6) and e > 0. Consequently, estimates (3.1B) ate also valid in the
present case as long as pQ and Sq are subject to condition (3.16). In the case
when u(x, 0) is summed over K2p with power 2s^ + 2 one can take the function
£ to be independent of i and thereby obtain estimates (3-18) also for i»0.
Let us now proceed to an estimate of max^, |uj and Each of the
derivatives uX£ can be considered as a generalized solution of a certain linear
equation whose coefficients satisfy all of tbe conditions used in §10 of Chapter
III for estimating the Hdlder norm of any generalized solution of such a linear
equation in terms of its norm in L2(Qf)- To this end we transform the identity
T
J j ^7ur\^dxdt = Q (3-19)
by means of an integration by parts as follows:
f «xt(x. t)dx
t
+ ^ + -g) \ ^ M = 0: (3-20>
here we assume that rj(x, l) is equal to zero on the lower base and lateral sur¬
face of the cylinder Qj = Kp x (0, tj). Tbe passage from (3.19) to (3.20) is legiti¬
mate, although the intervening calculations contain the derivatives u , and l)\u
(see die explanation with regard to a similar situation on page 211). By virtue of
(3-20) the function can be considered as a generalized solution from
Fj’® (9j) of the linear equation
e)u*'j+/‘(x' <))==0’ (3 21)
§3. ESTIMATES OF maX(?, |uj AND (ux)(q! 437
in which
. /.. ^ dat(x,i,u(x, l),ux<.x, i))
t) = g— ,
*1
while
/?<*. Q = da‘{x' *' U,t(X, t)
, da, (x, t, u (*, t), u,(x, t)) _6(fl(x< „(Xi /}> , (x> 0).
»•**
The functions o^.(*, t) and /j(x, t) making up this equation, by virtue of oar
assumptions on the solution u(x, f) and the functions o({*, t, it, p) and
a(x, t, u, p), possess the following properties:
v(/Mf)|»<o(/(r.
and
/f(*. *)|<(H| “*<■*• f)|+<P,(*. 0)|*,(*. 0|
0 + 0+<P3(*. 0
1 »
<3(i«J(x, f)+<P2+<P3+Yqr?-(Pi+6+T+TI
where 5 is an arbitrary positive number. We take $ so small that the numbers
** ?/U + d) and = r/( 1 + 8) satisfy a condition of type (3.6); namely,
JL_i_ " =1±±4-11±*]L£ = (1_»,)(! 4-6)< 1. (3.22)
rt ' 2?i
1+ Si .
For such a 8 the function <f>^ + <t>$ + (1/(1 + i will have a finite norm
in £ . In order for 3fiu*+ (S/(l + * to belong to it is suf¬
ficient, in consequence of (3-18), to take where m = max fl,(l + 8)/28],
and then /* will belong to £?J rjWPo- r)) in «** arbitrary cylinder Q(p0, r) C
(Jj, that is separated from Fj. (or from Sj. if u(x, 0) € Z,230 + 2(Q)) by a positive
distance, with pQ determined by condition (3.16).
Ic view of this Theorems 8.1 and 10.1 of Chapter III apply to it^ - uXJc as
the solution of the linear equation (3.21). They and considerations analogous to
those developed in §6 of Chapter UI guarantee the validity of the following propo¬
sition.
Theorem 3.1. Suppose u(x, t) is a solution from 0*' * (Qp) of equation (0.1)
438
V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
with maxg |u| = M. If the functions ai (x, t, u, p) and a(x, t, u, p) in equation
(0.1) satisfy conditions (3-D—(3-6), then for any domain ()' C Qj~ separated from
Fj. by a positive distance i the quantity | ux j^) with some a > 0 and the norms
\\uxx, ttjHj Qi are estimated from above by a constant depending only on n, M,
the parameters u, ft, f(j, q and r from conditions (3-D—(3-6), and the distance d.
The number cl > 0 is determined by n, v, /x, q and r. If furthermore it is known
that max^gjj !“*(*> 0)1 < “ (or \ux (x, 0)|^ < ■»), then maxg, |uj (respectively
|ux |^, ) and the norms ||bx;1., b(|| 2 Q’> for anY domain Q' C Qj separated from
Sj. by a positive distance d, are finite and are estimated from above by a con¬
stant determined only by n, M, v, fi, ftj, q, r, d and tnax^g D | ux (x, 0)j (respec¬
tively \ux(x, 0)\<P and fi).
§4. AN ESTIMATE OF max [ttj IN THE WHOLE DOMAIN
Let us now concern ourselves with an estimate of max |;jJ in the vicinity of
Sj.. Suppose tbe functions a; (x, t, u, p) and a (x, t, u, p) satisfy conditions
(3.1)—(3-6), while the solution u(x, t) of equation (0.1) is an element of 02'HQj).
We first assume that = 0 and Sj. satisfies condition (A). From tbe equality
t
J J \)dxdt — Q, *<7\
0 Q
where A is a sufficiently large number, we obtain the estimate
J a*dxdt^c, (4.1)
Qt
where c is a constant depending only on n, U - vrai maxq^\u\, u, fi, jij from
conditions (3-l)-(3.6), T and mes Q (c is an increasing function of these argu¬
ments). This is deduced in literally the same way as inequality (3.9) was deduced
2 + 2
from (3.7). We will now obtain estimates of the integrals }\ux\ 5 dx for any s
and for domains adjacent to 5. This is done in basically the same way as in §11
of Chapter III and in §3 of the present chapter.
Let us assume that
max I “jc 1 jsr — ^2-
We consider the function
(4.2)
b (x, 0=
§4. AN ESTIMATE OF max |ttx| 439
0 fot | Ux(x, f)lS<'W2=='M.
\ux? — M for +
1 for | ax P >• M + 1,
which is equal to zero on Sj..
We form with its help the following integral identity:
- J J S *^1W-3)
0 aap*-l
where £2^^ = K2p fl ^2p *s an arbitrary ball with center on S and p not ex¬
ceeding some number p0, while £(*) is a cutting function for K2p-
Let us denote a2 by v. We transform the term containing u£ as follows:
I" J J V‘^° dxdt~'% j v!)P dx j —i j* f v(bt?)tdxdt
0 s2p ° 0 %
t '
= y f J (i -\-M)b,&dxdt
S2p 0 C2p
f «*ee</jc| —-4- f (* + ;»)*£*</*I
X < ,o
°2P
The remaining terms are transformed by means of a double integration by parts.
As a result, from (4.3) we obtain
J
O, 0 °jp ^
+ ■5?% + 3^) + 2«,/&r4)
4- a (A ub& -f- = 0. (4-4)
The passage from (4-3) to (4.4) is legitimate in spite of the fact that the deriva¬
tives uxc and D^u were used in the intervening calculations. In tbe left side of
equality (4.4) we leave the positive integrals
440
V, QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
r
t
and carry out estimates of the moduli of the remaining terms from above, using con¬
ditions (3-D—(3-4) and Cauchy’s inequality. As a result of these estimates and an
estimate of the positive terms of the left side from below we obtain
One can easily see that the left side of (4.5) is estimated from below in terms of
The first four integrals in the right side of (4.5) are bounded by known quan¬
tities, since 0 < b(x, i) < 1 and estimate (4.1) is already known for the integral
fg^vdxdt. The latter integrals can be estimated in the following manner by
means of inequality (3-6) of Chapter II, taking into account conditions (3.5) and
(3.6), as well as the boundary condition it|s = 0 and tbe Hdlder continuity of u
in x:
t
t t
n(q— 1)X|
q(2x,4-n) '
Choosing
§4. AN ESTIMATE OF max |bJ 441
n
J 2dx
— — J o [&«&£* -(- 2«XJ|BX<
%
+ **kvb*P + dx
<e,p’ j {a\xbP+-LbV^dx-\-c^ J vWdx +«,. (4.6)
p»+p«< —L—minfl; v; c'\.
from (4.6) and (4.5) we get
r
max J vb&dx+v J J (u^Jj -f- ^ dxdt
)<t<T" • % r
+ J J dx <*/ ■< const.
0«<7
*p
These estimates together with equation (1.1) and analogous estimates fot the in¬
terior balls K2p obtained in §3, guarantee the inequalities
max f u‘i(x, t)dx + f j ux |4 dx dt < es,
»«**£ <jT
T
J J {U\x +U^dxdt^c'v (4.7)
o U'
in which the constant c2 depends on c from (4.1), M2, !!“,(*> 0)||2>u, a°d
a, while c j also depends on the distance from Q' to 5.
The above arguments are insufficient for obtaining an estimate of
Jq fa *lx dx dt since 6 vanishes on the boundary of the domain. To this end
special considerations are required in the vicinity of the boundary, which we will
now develop under the assumption that 5 6 02. Suppose Sj is a not large portion
of the boundary S, while £(*) is a cutting function for a ball Kp with center on
Sj and not intersecting S\Sj. Without loss in generality one can assume that the
equation for Sj has the form xn = 0. In the same way as in the interior estimates
442 V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
we consider equality (3.11) with £=0 and transform it by means of an integration
by parts to a form analogous to (3.12). Since £(*) does not equal zero on S, it
follows from a comparison with the case of interior estimates in (3-12) that there
will exist additional integrals over Sy
5 S Cos(b, x.) - u^.CosU, xk)]£2 dxdt
'Si
t
( { «i*xkM£xkCosin, x.) - 2^. Cos (b, zk)]dxdt = /; + 1T
8 S,
OS i
<
Si
The second of these integrals is obviously estimated in terms of U2 and known
constants. The first, on account of the boundary condition u | = 0, takes the
form
<imi 5 -•!«*.«{>£2**
0 Si
n~ 1
2 aiuxnxii2dxdl-
'S! i~l
Introducing the notation b^ix, l, u, p) - jJft a-{x7 l, u, p^, * * * » pn_ j, r) dr, one
can transform /j as follows:
il
I
ft- 1
I5S (2b.aXi,b.x.^)dxdt,
Sji-l
Hence we see that the integral is estimated in the same way as I2 in terms of
and known constants. In other respects the estimates of f^fg u2x£2dxdt in
the vicinity of the boundaty do not diffei from the interior estimates. In this way
we estimate fj fB dxdt. Using equation (1.1) again and the second of the
inequalities (4.7) we arrive at the estimate
T
5 ^ ^uxx + dx dt - cy
o li
where c j depends only on the quantities determining c2 in (4.7) and on the norms
io 02 of the functions defining the boundary S.
§4. AN ESTIMATE OF maitluj
443
In order to obtain estimates of the integrals of higher powers of u% it is nec-
essary to employ the relations
i
~ J J -sruik\iv ~~dx
0 % M«>
s— 1. 2
where, as above, v= u*, M = M\> A2f>,M^ '* thc set of Poiats * 6 °2p~ ^2p 0
at which o{x, t) > M, while is a ball with center on S, and with them to de¬
velop arguments analogous to the derivation of estimates (3-18) from (3.11)- This
leads to the inequalities
T
max f \ux?**2dx + f (\ujs"dxdt<cs (4.9)
o«<rBJ o Q;
for 1 < s < $q and p < pQ» where and are connected by a condition of type
(3.16). Inequalities (4.7) and (4.9) together with the same estimates (3.18) for in¬
terior cylinders permit one to conclude that for any s > 0
T
max
o</<r
J Krwj J Iux f**dx dt« c's, (4.10)
u a
where c's is some constant depending only on s, M n, die constants v, /t, ft j,
q, r from conditions (3-1)—(3-6)» mesfl, T, ||u%(x, 0)||2j +2iaa“d By vir¬
tue of Theorem 1.1 the norm |u |^, is estimated in terms of \u f},tnax.Q^\u\
and the constants in conditions (A) and (3.1)—(3.6)- Further, by arguing in the
same way as at die end of die preceding section, we show that tif, - ux^ is a solu¬
tion of the linear equation (3.21) with the o.. and f. satisfying the conditions of
Theorems 8.1 and 10.1 of Chapter III; and hence their conclusions hold for uXfi’
Thus the following theorem is proved.
Theorem 4.1. Suppose u(*, t) € 0^’1 (Qf) satisfies equation (0.1) and van¬
ishes on Sv which is assumed to satisfy condition (A). If the functions
a.(x, t, u, p) and a(x, t, u, p) satisfy conditions (3.1)—(3-6), then max^|i»J is
estimated from above by a constant depending only on the parameters in conditions
(3.1)—(3-6), n, max^lul, maXy^luJ, mesfi, T and ag and in condition (A),
while luj^ is estimated from above by a constant depending on the same param¬
eters, and fi. The number a is determined here by n, n, is, fiy q, r and
444 V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
fi. If furthermore s 6 02, then |\uxx, “(l| 2 Qy *’* estimated from above by a con¬
stant depending only on the parameters from conditions (3.1)—(3.6), n, |it|,
raaij.j, \ux |, mes 0, T and the boundary S.
This theorem given an estimate of boxqt\u |, ||bxx, “tll2,<?r aod
in terms of the quantities m«urg. |«x| and \ux\ j-j.. which are not known from die
conditions of the boundary value problems. In Chapter VI (see Lemma 3.1)
maxr juj will be estimated for solutions of the first boundary value problem in
* I *
terms of known quantities only, and this will be done directly fot solutions of
general quasi-linear equations without referring to any results of the present chap¬
ter. An estimate of |ux|^|, will be given in die following sectioa in terms of only
known quantities and max^ luj-
Theorem 4.1 is easily generalized to die case of a nonhomogeneous boundary
condition. To this end it is necessary to introduce in place of u(x, t) a new un¬
known function v(x, t) = u(x, t) - tfr(x, t), where tj/[x, t) is some function from
02-HQt) that is equal to u on Sj., and to note that Theorem 4.1 is applicable
to it. This leads to the following proposition.
Theorem 4-2. Theorem 4.1 is also valid for the case when u coincides on
Sj. with the values of some function *jr(x, t) from 02’^(Qj). The estimates of
ma*Qj, !“*l. 11“**. u^l) 2 Qj. <*nd will depend in this connection on the
norm of \j> in 02,l(@r), i.e. on max^, ]i/r, ipx, <pxx< tfr(|.
§5. ESTIMATE OF {uxY^, AND HIGHER DERIVATIVES
IN AN ARBITRARY SUBDOMAIN OF THE DOMAIN Qf
In this section we give estimates of tbe quantities and |u(, uxx\^q)
for solutions u(x, t) in an arbitrary (not necessarily strictly interior) subdomain
Q' of the domain Qy To this end we first prove a theorem guaranteeing an esti¬
mate of max^, |u(|.
Theorem 5.1. Suppose that a solution u(x, t) of equation (0.1) belongs to
C2’HQt) and has max^ |u | » M, na3lQT l“J “ «»<* suppose that for (x, t) 6
Qf, |o| <M, |p I < My the functions a, (*, t, u, p) and a(x, t, u, p) satisfy a
Lipschitz condition in t, are differentiable with respect to u and p and are sub-
feel to the inequalities
v|2 < daiix^-&-, < nls.
(5.1)
§5. ESTIMATES OF AND HIGHER DERIVATIVES 44 5
1 daj (x, t. u, p) a, (x, t + h, a, p) — at (x, t. a, p)
I du h
%L. 2jL, f&hSsJL |<f(* t>, (5.2)
wAere ||<£|? r q^, < Mp c^e numbers q and r satisfy condition (3.6). Then maxg^juj
i« estimated from above by a constant depending only on n, v, /ip <?, r oni
maSp^,|ut|. If r' is an arbitrary portion of the surface F, then aaxg, where
O' is any pan of the cylinder Qj. that does not intersect with Fp\r', is esti¬
mated in terms of n, v, (l, (ij, q, r, maxr, |«£| and the distance from Q’ to Tj.\F'.
For a proof we consider in the cylinder Qf_/t the function
«*(*, 0 = [“(*■ r-j-A?) — u(x, 01. Af = *>o,
and show that it is a generalized solution of a certain linear equation. We take
the divided difference in I of both sides of equation (0.1):
AJ?’« __ do* n
tt dxi A/ ' A<
and represent Aat/At, Ao/Aj in the following form:
■jr—jfia/ix- “(*■ *+Ao. “X(x. f+Aoi
— a,\x, t, u(x, t), ux(x, /)])
I
“"S? j * + ta(x. *-f-A0 + (1 — r)u(x. t),
xax(x, <+A/)-f-(l—x)ux(x, t)\dx
-f-i- (ajlJC, t -f Af. u(x. t). ux(x, t)] —
— at\x, t, u(x, f), ttx(x, Oil
‘ o 3 o
+ (5-3)
and analogously
~3- = clv*i-+cvk + d.
This gives tbe following relation for v1:
446 V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
difl d.
~§r ~ HFt[aUvxI+V* + *! + -+• CV> + d = 0, (5.4)
which can be considered as a linear parabolic equation for v* with coefficients
and free teems satisfying the conditions of Theorems 7.1 and 10.1 of Chapter HI»
By virtue of Theorem 7.1
max | v'11 2e max [max | ti*|; 1), (5.5)
Or-A rT-h
where c is determined by n, t', nv q and r and does not depend on h, and
hence an analogous estimate is valid for lim^gtf* = u(. The first part of Theorem
5.1 is proved.
For a proof of the second of its assertions, on local estimates of maXq, [u{|,
we make use of Theorem 8.1 of Chapter III, which implies the inequality
max | v>' | <! cf
Q'
for any part Q' of the domain Qj.. If Q' is an interior subdomain of Qj., then
the quantity c' is determined by the constants determining c in (5.5) and (be dis¬
tance from Q' to rT-h, while for cylinders adjacent to a portion T' of tbe sur¬
face ry, the quantity c' depends only on n, v, fi, jtj, q, r, maxj , | v^\ and the
distance from Q' to Fj.\r‘. It is clear that analogous estimates will also hold
for u(.
Remark 5.1. On the basis of Theorem 10.1 of Chapter III, for J1, and hence
also for ut, it is possible to estimate in terms of the same quantities as
maxg, and the distance from Q' to VT if the latter is positive. If S satis¬
fies condition (A) and then it is also possible to estimate •
Remark 5-2. It is not difficult to see that the assertion of Theorem 5.1 remains
in force for solutions u(x, l) from C2,1(^r\S0) f| C(Q^) with vrai max r?,|tt£| <°c
where SQ = i* € S, t = Ol, it being only necessary to understand maxr |u(| as
tin sup
Af-*+o (x.oerr-*1 1
Let us now proceed to an estimate of assuming that S belongs to
O2, while the functions o. (*, t, u, p), a(x, t, u, p) satisfy in addition to condi¬
tions (5.1). (5.2) the following conditions for |u| < M, |p| < My
§5- ESTIMATES OF (?x)fQ> AND HIGHER DERIVATIVES 447
sup J <Mt. *>... (5.6)
(6(o, n11 axi ii*. a
To this end we consider (0.1) for fixed t as an elliptic equation
— -gjp (<*, (x. t, a. nj ) f / (x, t) = 0 (5i7>
with free term
/(*, t) — u,(x, t) + a(x, t, u(x, t), ux(x, 0).
Conditions (5.6) and tbe estimates of maxg, |u{| established in Theorem 5.1
imply tbe boundedness of the norms
!i/(*. OIU^M0'). <?!>*■ (5.8)
For such equations the following assertion is valid.
Theorem 5.2. Suppose u(x) belongs to VjCO), has vrai max0|ux| < and
satisfies almost everywhere in Cl equation (5.7) (in which t is fixed: t = !q). If
the conditions
v>0.
are fulfilled, then for any subdomain £J‘ CQ separated from S\S' by a positive
distance d the quantity \ux(x)/ ^is estimated from above by a constant de¬
pending only on n, v, ft, n2, <l2> ^1’ > a> & an(^ t^le distance d.
The constant a > 0 is determined by the same quantities except for d. The piece of
boundary S' (it can be the empty set) must be a surface of class 02 (or at least
This theorem is proved in §6 of Chapter IV of the book [65q] (see Theorem
6.1 and the remark to it), only there the dependence on the boundary values of u
is expressed in terms of the norms ii<A !ll^ c* where ifr(x) is some function coin-
" 3*
ciding with u on S. But this difference is nonessential since by virtue of Theo¬
rem 2.3 of Chapter II the norms ||iji and inf ||\[i j| ^ ff, where the infimum
is taken over all functions tfr from (fi) that are equal to \ji on S, are equiva¬
lent. 93
On the basis of Theorem 5.2 and the assumptions stated above, for the solution
448
V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
u(x, t) in question, we obtain an estimate of {uz(x, t) for arbitrary do¬
mains O' C 0 in terms of known quantities.
Suppose ma*0< (<r {“*(*> «))*“* < c. By virtue of Lemma 3.1 of Chapter II,
from this inequality and the inequality | iz(| < Cj there follows the inequal¬
ity + < c2 w*th c2 depending only on c, Cj, a and the boundary
S. Analogously, using local estimates of maxg, |uf| and „ we obtain
an estimate of + for any Q’ C Qj.. In this way the following proposi¬
tion is proved.
Theorem 5-3- Suppose the functions a.(x, t, u, p) and a(x, t, u, p) satisfy
conditions (5.1). (5.2) and (5.6), while S€ 02. Then for any solution
it € C2’^(.Qj) of equation (0.1) with maxq^,\u\ = M and max^^Ju^l = Afj the
norm j u%\is estimated from above by a constant depending only on n, u, ft,
Mj. 9. r, V2, 92’ U’ MV maxrrW’ S’ “ar0< t< T M*» ^ /9}> m,i ?3> with
> n. For any pan Q' of the cylinder Qj. intersecting with a portion F* of the
surface F J and separated from P J \F' by a positive distance d, the norm
jax is estimated from above by a constant depending only on n, v, [L, fj.j, q, r,
H2, q2, U, My d, P', maatp, ju{| and the quantities max^^^ ||u(x, t)||^2
and > n if T' = S' x [tj, t^ and tj > 0, and max0<J< ^ ||b(*, £)||^?
-/n\
||u(z, 0)|j<2) a, and q^> n if F’ is the union of the surface S' x [0, t2] and the
part \(x, 0): * 6 0* 1 of the lower base of Qj.. The index a is determined in both
cases by the same quantities as 1^*1^ except for d.
Remark 5-3- The assertion of Theorem 5*3 remains valid for the solutions de¬
scribed in Remark 5-2.
After estimating the norms and |uj^) one can consider equation
(0.1) as a linear equation with smooth coefficients (if, of course, a.(x, t, u, p)
and a(x, t, u, p) are smooth functions of (heir arguments) and make use ot the
corresponding theorems of Chapter IV on linear equations for estimates and other
qualitative investigations of its solutions.
Let us formulate a proposition that follows from Theorem 5-2 of Chapter IV
and Theorem 5-3 of this section.
Theorem 5-4. Suppose that on the set 28 - ((*, t, u, p): (x, t) 6 Qy, |u| < M,
Ip I £ My\ the functions a.(x, t, u, p) and a(x, t, u, p) satisfy conditions (5.1)
and (5.2), while the functions a(., da./dp-, da./du, da-/dx. and a are Wilder con¬
tinuous in the arguments x, t, u, p with exponents fi, p/2, j8 and /9 respectively.
If S belongs to the class H2+^, then for any solution u from H2*& 1+ &2{QT)
§6. THE FIRST BOUNDARY PROBLEM 449
of equation (0.1) having maxg7.1“i < M and <Ml the norm |u
is estimated from above by a constant depending only on U, Vj, jB, the quantities
v, ii, q, r, hi from conditions (5.1), (5.2), the norms of a^ da^dp., dajdu, dajdx.
and a in JjfA #2, A 3H), S and the norm
Let Py be an arbitrary portion of the surface Fy. For any subdomain O' of
the cylinder Qj, that is separated from Fj.\F' by a distance d > 0, the norm
is estimated from above in terms of d and the same quantities as the
norm except for the quantity which is replaced by the norm
|*|<^>T
Theorem 5.4 together with the theorems of §§1-4 of this chapter and Theorem
5.2 of Chapter IV for linear equations permits one to prove the solvability in the
large of the first boundary value problem for equations (0.1).
§6- THE SOLVABILITY OF THE FIRST BOUNDARY VALUE PROBLEM
Let us consider in Qj. the first boundary value problem for equation (0.1),
i.e. the problem of determining in Qj. a function u(x, t) satisfying in Qj. the
equation
«,-(j:, t, u, uj-i a(x, t, u, «J = 0, (6.1)
and coinciding on with some known function xjj:
«lrr=1>lrr (6.2)
Equality (6.2) contains in itself the initial and boundary conditions. We will as¬
sume that ijr is the value on Fj. of a function t/r(x, t) defined in Qj. and possess¬
ing a certain smoothness, which will be specified below. If tfr{x, t) 6 C(Qj),
then the compatibility condition of zero order (for the initial and boundary condi¬
tions) is said to be fulfilled. If t/r(x, t) 6 * (Qy) and on the manifold
S„ *\x 6 S, t = 0|
t% +• w + a(x> *• i>. <w=o, (6.3)
then the compatibility condition of first order (for the initial and boundary condi¬
tions and the equation) is said to be fulfilled. It is clear that the compatibility
condition of zero order must be fulfilled for a solution u(x, t) of problem (6.1),
(6.2) belonging to C(Qj), while the compatibility condition of first order must be
fulfilled for a solution from C2,1(QT) that satisfies (6.1) in Qr
450
V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
The solvability of problem (6.1), (6.2) can be proved by makiag use of theo¬
rems from the theory of abstract nonlinear equations, bearing in mind that it has a
uniqueness theorem for the classical solution. We will apply the Leray-Schauder
principle (its formulation can be found in [68b] or in §8 of Chapter IV of [65q]).
We write equation (6.1) in the form
on determining the function v; the function tu(x, t) in this connection is assumed
to be given. We introduce a linear Banach space B g of functions w (x, t) that are
Under certain restrictions on the functions a{, a, tfi and S, problem (6.5) defines
an operator $ in B 5 which associates each function to of B s with a solution v
of the linear problem (6.5):
This operator is nonlinear and depends on r. Its fixed points for t - 1 are solu¬
tions of problem (6.1), (6.2). In searching for the fixed points of $(w; r) we will
apply the Leray-Schauder principle.
Let ur be one of the fixed points of the transformation <P(w, r), i.e. let
+ A (x, t, u, ux) = 0,
(6.4)
where
A(x, t, u, ux)
= o(x, t, it, ux)
dai (x, t, u, ux) dty (jc. I, a, ux)
and consider the family of linear problems
-$-xA(x, t, w, wx) — (I —T)[$t — Ai|)] = 0,
(6.5)
iT ir
continuous together with their derivatives with respect to x in Qj and have the
finite norm
l®li6=M5r + !™J
v = $(10; r).
(6.6)
§6. THE FIRST BOUNDARY PROBLEM
ssxjS’u -f~(l — i)(at — A u — ij>( -f- Aij>)
535 — ^ *' “• “*> -H1 — T) “*,]
-f ta(jc, t, a. Uj,) — (1 —i)!^ — A\|>] = 0,
*lr- —♦Ir-*
451
(6.7)
The operators £r possess the property that if the conditions of some theorem of
the preceding sections, in which an estimate is obtained of one of the norms con*
sidered in the book, ate valid for the operator £, then the same conditions ate
also valid for £r when r 6 [0, l]. In fact, suppose conditions (2.29), (2.30) or
(2.29), (2.32), (2.33) of Chapter I are fulfilled for £. Then, as is easily seen, anal¬
ogous conditions are fulfilled for £r, and estimate (2.31) or (2.34) of Chapter I
holds for all possible classical solutions ur of problems (6.7), so that
niax|«T|-<.Mt
Qt
t£I0. I].
(6.8)
We assume further that for (*, t) 6 Qy, |u| < M and arbitrary p the functions
a.(x, l, u, p) and a(x, t, u, p) satisfy the inequalities
dai (x. t, u, p)
dp
(6.9)
i i, i~i ’
<nd + |p|)2.
Obviously, the coefficients of the operators will also satisfy conditions
of the same type, where tbe constants vf and p.f for them can be chosen indepen¬
dently of all r from [Q» l]. We retain the notation v and fi for them. From condi¬
tions (6.9) follow the conditions of Theorems 1.1 and 4.1 of the present chapter
and the conditions of Lemma 3-1 of Chapter VI, and hence the following estimates
hold for all solutions uT from C2’1 (Qy) of problems (6.7):
owx|0<T<1, (6-10)
with the constants M' and My being the same for all uT and depending only on U,
the constants v and fi from conditions (6.9), \<Px(x, 0)| and tbe
boundary S, which we assume belongs to the class 02.
Further restrictions on the «,(*, t, u, p) and a(x, t, u, p) will be imposed
only fot (*, c) € Qy, |a| < M and |p j < Kj. Thus, in order to obtain a priori
452
V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
estimates of aaxgT\uTt\ and it is necessary to add the requirements (5.2)
to (6.9). They will be fulfilled fot all with the same fuactioo <fi (*, t) and con¬
sequently, for the whole class of solutions ur,
max|«J|, 0<t<1, (6.11)
Or t
with one and the same a and one and same constant depending only on die
known parameters from oar conditions, S € 0^ and |^| Finally, the estimate
(6.12)
is given by Theorem 5.2 of Chapter IV on the solutions of linear equations with
smooth coefficients if one assumes in addition that tbe functions a( (*, t, it, p),
daj/dpj, da^du, da^dx- and a ate continuous functions of (x, I, u, p) satisfying
a Holder condition in x, t, u and p with exponents /3, (3/2, /3 and /3, that 5 €
H2*P, i/i 6 + 1 +&^2(Qj) and that <ji satisfies the compatibility condition
(6-3). In tbis way one establishes the uniform boundedness in the norm of
H2*P, l*P/2 0( all possible solutions ur of problems (6.7) or, wbat is the
same, of all fixed points of the transformation <Mw; r). The uniform boundedness
of the norms of the ur (in the space in which the transformation 4>(u>; r) is con¬
sidered) is one of the conditions of applicability of die Leray-Schauder Theorem,
and the most difficult to verify.
Let us now formulate the first existence theorem for the classical solution.
Theorem 6.1. Suppose that the following conditions hold.
a) For (*, t) 6 Qj, and arbitrary u conditions (2.29), (2.30) or (2.29), (2.32),
(2.33) of Chapter I are fulfilled, which when applied to equation (0.1) have the form
da,iVp P) o>0. A(x. t, u, 0)«> —V2 —*2
OT / P 0 m
A(x. t. a, 0)«> — a>(|«|)|«| — J ^I_ = oo, <P>0.
u
b) For (*, t) G Qj, |u | <M {M is taken from (6.8)) and arbitrary p the func¬
tions at(.x, t, u, p) and a(x, t, u, p) are continuous, the at-(x, t, u, p) are differ¬
entiable with respect to x, u and p and inequalities (6.9) are fulfilled.
c) For (x, t) 6 Qj, |»| < M and |p| < J/j (itf j is taken from (6.10)) the
§6. THE FIRST BOUNDARY PROBLEM
453
functions a,., a, da./dpj, da^du, and dai/dxi are continuous functions satisfying
a Holder condition in x, t, u and p with exponents fi, f}/2, /8 and j8 respectively.
d) For (x, t) € Qj, |u| < M and (p| <Mj the function a(x, t, a, p) has the
partial derivatives da/dpi and da/du and the functions a(- and a satisfy condi¬
tion (5.2).
e) t) 6 + ^2{Qj) and satisfies condition (6.3).
f) seff2+/3
Under these conditions there exists a unique solution of problem (6.1), (6.2)
from the class H2* & 1 + ^2(<?j-)- Moreover, this solution has derivatives uxl
from L2(Qt).
Proof. As the basic space in which we will consider the transformation $
defined by (6.6), we take tbe space B 8 with sufficiently small § {8 will be stipu¬
lated below). As was shown above, the theorems of tbe preceding sections guar*
antee the uniform boundedness
with some 0 < a < I for all solutions uT from C2’1 (?j>) of problems (6.7). But
each such solution uT is obviously a fixed point of the transformation 4>(w; r)
and, conversely, any fixed point uT 6 of the transformation 3>(k>; r) is a
classical solution of problem (6.7), more precisely, a solution from
ff 2 + ^8, l+/38/2(^)_ jn {act, from the membership of uT in Bs and condition c)
it follows that the coefficients of the v%i%- and the free term in (6.5) are ele¬
ments of H^' ^2(Qy) tor w =* ur> and therefore by virtue of Theorem 5*2
of Chapter IV the solution 'v = 4>(«r; r) of problem (6.5) will belong
to H2*pt'l*pi/2(QT).
We modify, if necessary, the functions a;(x, t, u, p) and a(x, t, u, p) outside
of the domain of variation of the arguments \(x, t) 6 Qr, |u | < M, |p | < Mtl so
that they satisfy properties a)-d) of Theorem 6.1 on the set \(x, t) 6 Qj,,
|it | < M + e, M <Mj + fl, where e is some positive number. We take 8 = a < 1
from (6.12*) and consider tbe operator $(w; r) on a convex bounded set 1 of the
space B g, consisting of all elements w 6 B g satisfying the inequalities
454
V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
By virtue of (6.12*) all of the fixed points ur of the transformation 4>(uj; r) Ue
strictly in the interior of this set W.
Let us verify that f) on 31 x [0, 1] is a family of operators that are
equicontinuous in w aad -r and uniformly compact. For all w with Iw ]g ? + (
and r 6 [0, I] the functions v = 4>(k>; r), as solutions of problem (6.5). have, by
virtue of what has been said above, the uniformly bounded norms
\v. vx, v„
v !<oB
Uxx lQ
(6.14)
By the uniform boundedness of
and as was proved in Lemma 3-1 of
Chapter II, the norms for the functions v are uniformly bounded, while
the set of such v is compact in Ba, since a < 1 and, consequently, the opera¬
tors <t>, 0 < t < 1, take the whole set 3! into a set that is compact in Ba. In order
to verify the uniform continuity in w of 4*(«); r) on 3)1 x [0, l] we take two close
elements ui‘ and tv" from SI and, corresponding to them, v'= 4>(u/; r) and
v" = 4> («/'; r). We subtract equation (6.5) for v" from equation (6.5) for v and
consider the result as a linear equation for «•»«/- v“ of the form
ii
dat (x, t, w', w’jj dat (x, t, w", a>“)
dw.
*l*j
-1[/4 (x, t, w', — A (x, t, w", «Qj,
Also,
rr~
= 0.
(6.15)
(6.16)
We consider tbe right side of (6.15) as a free term. It is not difficult to dem¬
onstrate that when the quantity | w' — u/' j ^ is small the right side of (6.15) is
small uniformly in t 6 [0, l] as a function of (*, t) in the norm of the space
aP/2(Qj.). But then by Theorem 5-2 of Chapter IV the quantities
|t), vx, vxx, , and hence also the norm Mjga> will he small.
The uniform continuity in r of #(tu; r) on ffl x [0, l], and thereby the uni¬
form continuity of 4>(w; r) in (w; r) on 58 x [0, 1), is proved analogously. The
applicability of the Leray-Schauder Theorem will follow if we establish that the
transformation 4>(u>; 0) for r = 0 has a unique fixed point inside ffi and that the
§6. THE FIRST BOUNDARY PROBLEM
455
transformation w - 4>(ui; 0) is invertible in a neighborhood of this point. But this
is indeed the case since $(w; 0) takes all of ® into the single element v(x, t) s
t) which is tbe solution of the problem
v,— Ai = <|>,— A4>. w lrr = <t> |rr.
Thus for each r from [0, 1] there exists at least one fixed point ur(*, t) for
cj>, which, as explained above, will be a solution of problem (6.7) from
j/2 + afi, 1 + o.fi/2 {Qj) (by virtue of (6.12) the variation of the functions ai and a
beyond the limits of the set \{%, t) 6 Q j, |tt| <M, |p| < Mj! has no influence on
the ur). Let us verify that the u will actually belong to H2*& (QT). In¬
deed, the membership of the if in I]2*a^’1 + a^^2(QT) implies by virtue of
Lemma 3.1 of Chapter II the membership of the uTx in H1,l^2(QT), and from this
and condition c) of the theorem it follows that the coefficients of equation (6.5),
if one substitutes ur{x, i) in them for w(x, t), will as functions of (*, t) be ele¬
ments of H&' But a solution uT of problem (6.7) can be regarded as a
solution v of problem (6.5) with w - ur, and therefore (see Theorem 5.2 of Chap¬
ter IV) v = ur is an element of H2*^' Thus it is proved that prob¬
lems (6.7) for all r from [0, l], and in particular problem (6.1), (6.2), have solu¬
tions from H2 + P’ 1 +
The uniqueness of such a solution is proved in the usual way. Let u' and u"
be two solutions of problem (6.1), (6.2). They satisfy the integral identities
t
J | a,(x, t, u, iix)T\Xl + a(x, t. u, ojr|]ixtf/ = 0 (6.17)
o a
with an arbitrary continuously differentiable function tjGc, t) that is equal to zero
on Sj.. We subtract one identity from the other and write the increments of the
and a, following (5.3), in the form
ts.al^a,{x, t, o', «')—at{x, f, a")
r dat [jc, t. to' + (1 — t) u", xu!x + (1 — t) b'1
J —1 wx dx
0 *f
I
4-« J ^-dx^a,jv1C] -f biV, &a~~clvJt. + cv
0
where v » u - u . As a resuit we obtain the identity
456 V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
t
J J + + «®)r&dxdt=*0.
0 0
from which by virtue of the theorems in §3 °f Chapter 10 on linear equations it
follows that v s 0.
For a proof of the existence of the derivatives d\/dtdx from LjiQf) ol the
solution u we revert to the proof of Theorem. 5.1- By virtue of our assumptions on
the a. and a, relation ($.4) can be regarded as a linear equation for
= ~pj [n(x, t ~f- t\t)—u (x, tyi
with such properties of the coefficients as ensure the uniform (in Ai) boundedness
of the integrals. /q ~A< /b (Au^At)2 dx dt. From this latter fact we obtain the ex¬
istence of utx and the finiteness of ||u(J.||2 The theorem is proved.
If one assumes that the data (possess a greater smoothness, then the solu¬
tions will also be smoothet. In fact, if it is known that the solution u(x, t) of
problem (6.1), (6.2) belongs, fot example, to H2*& then, considering
u(x, t) as a solution of the linear problem (6.5) with tv = it, we obtain on the basis
ot Theorem 5.2 of Chapter IV additional information on it, for example, its member¬
ship in tfi*A(3+If concerning the a-(x, t, u, p) and a{x, t, u, p) it is
known that they, the partial derivatives of the a; of first and second order and the
partial derivatives of a of first order satisfy a Holder condition with exponents
/3, /S/2, fi and fi respectively. In general, this improvement is already cleat from
the theorems of Chapter IV on linear equations and does not require an analysis of
the nonlinear problem itself. We will not enumerate here all of the implications
which one can extract from Theorem 6.1 and tbe theorems of Chapter IV, assuming
that the reader can do this independently.
From a comparison of the conditions of Theorem 6.1 of the present chapter
with Theorem 5.2 of Chapter IV it would seem that condition d) in Theorem 6.1 on
the differentiability of a with respect to u and p and tbe Lipschitz continuity of
the «. and a in t is redundant (overstated). From the point of view of the unique¬
ness theorem the differentiability of a with respect to u and p (or, what is roughly
the same, the Lipschitz continuity of a in a and p) is necessary. But it is pos¬
sible to prove the existence theorem without these assumptions, using the just
proved Theorem 6.1 and various theorems of Chapters III—V on a priori estimates
for quasi-linear and linear equations. Hie differential properties of the solutions
guaranteed in this connection will be determined by tbe smoothness conditions for
§6. THE FIRST BOUNDARY PROBLEM
457
the functions o(- and a, and also by the smoothness of i/H r y, and S and by the
order of compatibility of the boundary and initial conditions, while the solutions
themselves will be obtained as limits of solutions um ix, t) of boundary problems
for equations £mu = 0 with smoothed functions a" and ora and smoothed S" and
t/rm. Among the possible consequences of Theorem 6.1 and the a priori estimates
of the preceding sections we establish, for example, the following propositions.
Theorem 6.2. Suppose conditions a)—c) of Theorem 6.1 are fulfilled. Suppose
further that the following conditions are fulfilled:
d) >J>L £ 0*1 (STy, max | f, ix, 0) | < oo; * £ tfY’ ^ (Qr);
T x£Q
e) seo2.
Then there exists at least one solution uix, t) of problem (6.1), (6.2) belong¬
ing to Ha> a^2iQy) and having ux that are bounded in Qy and derivatives u(,
u that belong to H^’^^2(Qy). For the uniqueness of such a solution it is suf¬
ficient that the function aix, I, u, p) satisfy a Lipschitz condition in u mid p ,
uniformly on any compaction of the form }(*, *) €■ Qy, |u| < c, |p | < cL
We take an infinite sequence of embedded cylinders Qy =» flm x (0, T),
m = 1, 2, —, such that 3 Q^.*1 D Qy for all m and such that the boundaries
ilm (and thereby also the lateral surfaces S” of the cylinders (?”) belong to
H2*^ and have uniformly bounded norms in Q2, and we will assume that the func¬
tion tfiix, t) is extended onto the domain Qy, while the functions ai ix, t, u, p)
and a(x, t, u, p) are extended onto the domain ((*, t) 6 Qy, |it| < |p| <oo), so
that </' € O2'1 (Q x [0, T]), while for the a. and a the smoothness proper¬
ties and inequalities enumerated in conditions a)-c) of Theorem 6.1 are fulfilled.
We construct with respect to the at-(x, t, u, p), a(x, t, u, p) and i/tix, t)
averagings o” (*, t, u, p), am {x, t, u, p) and <jjm ix, t) over x, I, u and p with
averaging radius pm - d/m, where d is the distance from Sj. to Sy, with a Stek-
lov averaging to the right being taken over e.
We consider now the family of boundary value problems
S”nu^~ul—~ af(x, t, u. ux) + Am(x, t, a, a_t) = 0.
•L-fU * = 2. 3- (<U8)
1T *r
in the domains in which the af (x} i, u, p) and t//m {x, t) are the averagings
458
V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
constructed above of the o; and ifr respectively, while Am (x, t, u, p) equals
am(x, t, u, p) fot ((*, t) € Qj, |u | <00, jp| < oo), equals
—*?•<.*. 0) + ^at(X. 0, «|)M (*. 0), *?(*. 0))
for {(*, t) 6 SjJ, u » ifim (x, 0), p = U, 0)(, 11 and for the remaining values of
the arguments is defined so that Am (x, t, u, p) is a smooth function of its argu¬
ments and the functions Am (x, t, u, p), m = 1, 2, ■ ■ •, satisfy conditions (6.9)
uniformly in m, i. e.
\Am(x, t. B./OKudttiKi+lpI)*.
Problem (6.18) satisfies all of the conditions of Theorem 6.1 in Qj and
therefore has a solution um (x, t) possessing the properties indicated in Theorem
6.1. On the basis of Theorems 1.1 and 4.1 of the present chapter, inequalities
(2.31) and (2.34) of Chapter I and Lemma 3-1 of Chapter IV, we have the uniform
(in m) estimates
I um & < c. max I | « c. (6.19)
Qt Q?
with some a < y, and
|«"|(!*m.<e(rf). (6.20)
where Q' is a subdomain of Qj that is separated from Tj by a positive distance
d > 0, while c{d) is a constant depending only on the constants of conditions
a)-c), max 1 |</r | and the distance d. In view of (6.19) it is possible to choose a
Qj _
subsequence from the sequence lam! that uniformly converges in Qj to u(x, t),
which by virtue of (6.19) and (6.20) is a solution of problem (6.1), (6.2). The
uniqueness is proved in the same way as in Theorem 6.1.
If condition c) in Theorem 6.2 is discarded, then problem (6.1), (6.2) has a
solution u belonging to “> a^2(Qj), having derivatives ux{ that are bounded in
Qt and Holder continuous inside QT, and having ut and uxx belonging to
Q't — Si’ x [0, / ], 12f C SI. This assertion is proved in the same way as
1) We recall that according to our notation Fy is the totality of points (x, t) belong¬
ing to the lateral surface and lower base of the cylinder (3p while Sg is the totality of
points (*, 0) with x belonging to the boundary Sm of the domain Qm.
§6. THE FIRST BOUNDARY PROBLEM
459
Theorem 6.2, only in place of (6.20) we will have according to Theorems 3.1 aod
4.2 the uniform boundedness of weaker norms of um, namely
where Q' is any subdomain of Qj that is separated from Tj. by a distance d > 0.
Thus we have
Theorem 6.3- Suppose conditions a) and b) of Theorem 6.1 and conditions d)
and e) of Theorem 6.2 are fulfilled. Then problem (6.1), (6.2) has a solution
u(x, t) from Ha'a/7(Qj) (I W\’1 (6r) ux from Hy'y^(Qy) and with finite
superl«x| . If furthermore a(x, t, u, p) satisfies a Lipschitz condition in u and
p (uniformly on any compact set), then the solution is unique in the indicated class.
The uniqueness is proved in the same way as in Theorem 6.1, but it is better
to verify the fact that u(x, t) satisfies equation (6.1) for almost all (x, t) € Qj
in the following way: write in place of equations (6.18) the identities
| [8?tl + <*f(*. '■ «"• “?) \
Ot
-(-am(x, t, um, «") rijdxdt — Q (6.22)
with a smooth function 7) that is finite in Qj. and pass to the limit 'in them with
respect to some subsequence (a”4!, k = 1, 2, • • •, converging to u uniformly in
Qj, and such that the converge uniformly to ux in each interior subdomain
Q1 C Qr while the a”4 and a** converge to it( and uxx weakly in L2 (Qt) ■
Since the functions 0; (x, t, u, p) and a(x, t, a, p) are continuous in (x, t, u, p),
their averagings converge to them uniformly.
By what we have said, it is possible in (6.22) to pass to the limit with respect
to k — 00 and obtain an analogous identity for a, which is equivalent to equation
(6.X) since u has, besides continuous ux, generalized derivatives u( and uxx.
Under a further weakening of the conditions on ifj and S we lose the bounded¬
ness of Thus, if in Theorem 6.3 we require instead of condition b)
that the at(*, t, u, p) and a(x, t, a, p) be continuous in u and p and satisfy in¬
equalities (3.1)-(3.6), and we replace conditions d) and e) by the membership of
\ r j- tn //'^(Pj.) and condition (A), then instead of (6.19)—(6.21) we will have the
lesser information
460
V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
S'.
|«
and T
(6.23)
I “7 ff + I U?i, + 1 aXX%, Q. < C (d).
These uniform estimates permit us to conclude that there exists a limit function u
for the um that belongs to Ha’ a^(Qy) f| 1 ). has a% from Hy> (Qy),
satisfies (6.2) at each point of Ty, and equation (6.1) almost everywhere.
In this way, with the use of the approximation described above, we prove
Theorem 6-4-
Theorem 6.4. Suppose that the following conditions hold.
a) Condition a) of Theorem 6.1 is fulfilled.
b) The a((x, t, u, p) and a(x, t, u, p) are continuous in (a, p), the
ai (x, t, u, p) are differentiable with respect to x, u and p, and inequalities (3.1)-
(3-6) are valid for them.
c) <2r|rreff%v>.
d) 5 satisfies condition (A).
Then problem (6.1), (6.2) has a solution u from Ha> a^2(Qy) f| 1 (Q’)
with ux from y^2{Qy), where Q' is any interior subdomain of Qy.
Condition a) in Theorems 6.1—6.4 allows one to obtain an a priori estimate of
1“ I all possible classical solutions of problems (6.1) by making use of
the so-called “maximum principle”. It can be replaced by conditions of integral
type, which permit one to estimate niax^ju| on the basis of Theorems 2.1 or 2.2.
Since these estimates are applied to "good” (classical) solutions, the restrictions
on the a^x, t, u, p) and a(x, t, u, p) reduce to the fact that these functions are
integrable over Qj. for any finite u and p and conditions (2.3)—(2.6) are fulfilled.
Thus, for example, from Theorem 2-2 and Theorem 6.1 follows'
Theorem 6-5- The assertions of Theorem 6.1 remain in force if its condition
a) is replaced by the following condition.
a*) For (x, t) € Qy and arbitrary u and p the inequality
2lM*. t. a. p)\ + \a(x, t, a. p)|<(i(|«|. \p |)q>, (x, t)
is valid with any continuous function (i(£, r) and <f>^(x, l) € L j (.Qy), while for
|u| exceeding some number kg the assumptions (2.3)—(2.6) are fulfilled, where,
if /I >0 in them, then the uniform boundedness of Jur| ^ must he known
§6. THE FIRST BOUNDARY PROBLEM
461
for all possible solutions ur (in Theorem 6.1 this must be known simply for the
solutions from of problem (6.7)).
Condition a’) guarantees by virtue of Theorem 2.2 the uniform boundedness of
*»xQt\u% i.e. what we had extracted from condition a) of Theorem 6.1. We recall
that the numbers q^ and in conditions (2.3)—(2.6) can be any numbers greater
than zero and that for p « 0 none of the a priori requirements on the boundedness
of any of the norms of the solutions ur are imposed.
It is possible to give different sufficient conditions when any weak norms of
the solutions are estimated in terms of known quantities. For example, if for
(x, t) € Qj and arbitrary u and p
a,(x, t, ii. p)pt + a(x, t. a, p)u
> v/>2 — % (x, 0 M“ — <Pi (*, I). (6.24)
where v > 0, 0j(*, «) € L1(Qj), a € (0, 2), <f>2ix, t) € and the num¬
bers q2 and r2 are subject to the conditions \/r2 + n/2q2 = 1 + (r/4) (2 - a), r2,
q2 > 1, then the norms \uT\q^, for the if when = 'f’lpj ~ 0 are uniformly
bounded. Indeed, we take the equality
J -f- xai (x, t. u\ «J) aj; + (l — t)(“J)2
-f- to (x, t, ax, dx dt = 0, (6.2 5)
which is obviously valid for any solution uT of problem (6.7), and, making use of
assumption (6.24), we derive from it the inequality
j f (a'fdx yH*v-f i — t) J (u^fdxdt
a lo 0T
<x J (q>s|«'‘|0 + 9I)rf^£f<
<T II % l«r V <?r ll“T »JBj.. J5SL. flr+T 11 *■ >«. Or
<rfa) ll<P*ll,s.,2, QTt«'rQT + TllV.II,, v (6-26)
The desired estimate of |ur|^^ follows from (6.26) since
•tv-f 1 — T>.min[v; 1} and a < 2. (6.27)
462
V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
If for (x, t) € Qj and arbitrary u and p
at (x, t, a, p) p, + a (x, t, it, p) u > v | a |T‘ — (p, (x, t) (6.28)
with some v > 0, ^j > 0, <£j 6 L^(Qj), then from the same relation (6.25) we ob¬
tain a uniform estimate of
lluTll!,~0r+ llutH«, 0r- ? = 2 0±_2|.
It is possible to obtain criteria similar to die criteria just indicated by start- .
ing not from (6.25) but from tbe equality
J + t. «t, «J)^ («*)»-.
■]
-j-q(l-~X)(^)1(rxf~1+xa(x, t, ux, dx it = Q, (6.29)
which is obviously valid for the solutions aT of problem (6.7) when ^|r =0.
The replacement of condition a) by a condition of type a') can also be done in
other theorems of the present section. Another admissible modification of tbe con¬
ditions of Theorem 6.1 consists in the replacement of condition e) on tft by the
condition of simple continuity of \ji on Fj.. We will do this in Theorem 4.4 of
Chapter VI directly for equations of general form. It is possible to go further in
weakening the conditions of tbe theorems and arrive at generalized solutions of
equations (6.1) having only derivatives of first order. Let us cite one of the prop¬
ositions analogous to Theorem 8.8 of Chapter IV of the book [65q],
Theorem 6.6. Suppose the following conditions are fulfilled,
a) For (x, t) € Q J and arbitrary u and p inequality (6.24) is valid and
\a,(x. t, u. P)|(1+|P|)<H(1+!P|)S
+(l+|«|«)<h(*. 0 (6.JO)
and
|a(*. t, a. j»)| <H(1-HpI)!! + (1+I“P)<1>3(*. 0 (6.31)
with
a < 2 and <p$ £ r, (Qr), — H—§57 <' * ’ r3* ?3 S5' I ■
b) For (x, t) € Qj, arbitrary p, and |it| exceeding some constant R
a,(x, t, u. p)pt>v\pf—ix\uf~a\(x, t)
§6. THE FIRST BOUNDARY PROBLEM 463
and
— a(x, t, u. + + «,<j>3(x, 0
with f3 < 2 + 4/n and with function being of the same type as in (6.30)-
c) For (x, t) and (x‘, t') € Qj. and arbitrary u, v, p and q
(Pt — t, u, p) — at(x. t, o, ?)]>v(|u|)j/>— q\\ (6.32)
K(*. <. «. P) — a,(x. (, a, ?)|<n(|ai)|p —?|, (6-33)
|at(x, t, u. p) — a,(x\ tf, v, p)|<e(lx — x'\
+ \t — t'\+\u — «|)t|/’|4-l±(M + M)
+ <P4(*. *) +<&.(*'. t’)\
and
l«(je, /, a, p) — a(x\ t', v, ^)|-^8(|jc — x’\
-fi/——t>i + ip—fi)(it(|0i4*ivi
+ l/’l + M) + <Pi(*. 0 + <Pi(*'. Ol. (6-34)
where v[t) and /i(r) are continuous positive functions of r > 0, e(r) is a contin¬
uous function of r > 0 that is equal to zero for f = 0, L2(Q f), and
^j(*, t) e Lt(Qr).
Then problem (6.1), (6.2) with vHsj- “ ® ^as at least one solution from
Hy’y^2(QT) fl with some y> 0 for any function ifr € H*’ (Fy).
The boundary S in this connection must satisfy condition (A). If furthermore the
functions a; (*, t, u, p) are differentiable with respect to u and p and
| iSti&jL+J* |<ti(ja|)H,p|-H-<|>4(*. 01. <P4€ W). (6.35)
while the function a{xt t, u, p) is independent of u and p, then the solution is
unique in the class of bounded functions from
This theorem is proved with the use of aa approximation of the same type as
in the proof of Theorem 8.8 in Chapter IV of [65q]. Namely, the solutions uix, j)
of problem (6.1), (6.2) are obtained as limit points in the strong topology of the
spaces Hy*y^(Qj) and for the set of solutions ufi{xr t), p —♦ + 0,
of the auxiliary problems
“» ” 2J“ ai (*’ *> “*) + °P(X’ £’ “> ux'> m °* “lSr “ °> ul,=0 =
(6.36)
464
V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
in which the af(x, t, it, p) and ap(x, t, a, p) are averagings of the functions
af(x, t, it, p) and a(x, t, u, p) with an infinitely differentiable nonnegative kernel
<y (r) of tbe form
aP{x, t. a, p)= J 9('"7~I)t-
IX-r'Kp
(in this connection it is necessary as a preliminary to extend the functions
Ojix, t, u, p) and a(x, t, u, p) from Qj, onto a somewhat larger domain with pres¬
ervation of the properties of them enumerated in the conditions of the theorem).
Assumptions a), b) and (6.32) guarantee the fuifillability of all of the Condi¬
tions of Theorem 6.4 (with condition a) replaced by condition a'); see Theorem
6.5) for problems (6.36) and consequently also tbe solvability of these problems.
Moreover, they guarantee the uniform boundedness of the norms and
\uP\Qt some y > O) °f all solutions of (6.36). This fact, conditions a)-c)
and relations (6.36) permit one to choose a subsequence from {a^l that converges
in y < y, and in IF*’® (@7,), and prove that the limit functions u
for them will satisfy the integral identity
J [—“11, + <*,(*• “• «r)n« & (x, t, u, ux) T|j dx dt ~ U (6-37)
Qt
for all smooth ij that are equal to zero on the boundary of Q^, and consequently
will be the desired generalized solutions of problem (6.1), (6.2). Tbe question of
uniqueness of the solution guaranteed by Theorem 6.6 when to its conditions are
added-only conditions of the type
|tf(x. t, u, p) — a(x. t, v, p)|<(i|« —■oH|^l2+<PJ(-«. 01
and
|fl(x, t. u, p)— a(x, t. u. ?)!
<|i|P -?I[|/’|+|?|+<P4(JC' 01 >
where <£j 6 Ly(Qj), while 6 Z.jCPy), has not been resolved in the general
case. But for the case noted in Theorem 6.6 uniqueness follows from Theorem 3.4
§6. THE FIRST BOUNDARY PROBLEM
465
of Chapter III. In fact, for the difference vix, t) = u (x, t) ~ «"(*, t) of two pos¬
sible solutioas a' and u of problem (6.1), (6.2) there follows from (6.37) the
identity
while o;(*, t) = fg (da(-[ • • -]/<9u) dr. The conditions of Theorem 3-4 of Chapter III, and
hence its assertion on the fact that vsO, are realized by virtue of conditions
(6-32), (6-33) and (6.35). Theorem 3.4 makes it possible to strengthen somewhat
the theorem of uniqueness just proved.
The existence of generalized solutions u of the boundary value problems fot
equations (6.1) in the class of functions having only derivatives can be proved
in another way without using Theorems 6.1—6.5 on their classical solvability.
Namely, such solutions can be obtained as limits of approximate solutions
computed by Galerkin’s method. The u.^ are defined in the same way as for linear
equations (see §4 of Chapter III). The special character of the quasi-lineat case
consists in the need to carry out passages to the limit in functions biz, t, uP, t/*)
having in general a nonlinear dependence on the t/* and u^. We do this below by
making use of an idea of Minty [84a], concerning weak passages to the limit
through convex functions and permitting one to pass to the limit in the leading
terms a-(x, (, u, u%) of an equation, and also one of the techniques in the paper
[68a] of Leray and Lions, which is useful for passing to the limit in the minor
term a (%, £, u, u ) (in connection with the topic see also [84, 11, 134,22]).
o
We introduce the space 2^Qt^ m - consisting of all measurable func¬
tions uix, t) that are equal to zero on Sj and have the finite norm
J (a,j(x, t)vXj -f a,(x, t)v)T\x^dxdt — 0,
in which
1
+ (1 — t)u"(x, t), Xu'x(X, t)-j-(l —T) h"(*, t)Jrft,
| VfflJ m«v if » II -1. II it It
and die space V ® (Qf) obtained by a completion in this norm of all smooth
functions that are equal to zero on Sy.
466
V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
They are embedded in tbe spaces L riQf) with q and r satisfying the
conditions
1
2 n
1
where
r ' mn -)-2/n —2n q mn-\-2m —2n '
9^[2, TT^m]1 r^lm> °°]- fot a>m>l^T’
q £ 12, oo), r £ {tn — 2 + 2 —, ooj, for 1 < it •< m,
9612. oo], r£|3m — 2, oo), for m ra — 1,
(6.38)
so that under such <; and r the inequality
(6.39)
is valid for any function u from Vm 2iQ]•)■ One can easily verify that if
m> 2»/(» + 2) for a > 2, and m > 1 for n = 1, then the indices r -- q - min + 2)/b
obey conditions (6-38).
Inequality (6.39) is deduced from inequality (2.9) of Chapter II for arbitrary
m > 1 in the same way as for m = 2, that is, in-the same way as inequality (3.4)
ol Chapter II. We will establish the following theorem.
Theorem 6.7. The problem
ut - a.ix. t, u, tix) + aix, I, u, uj = 0, «|s?, =• 0, = </rQ(x), (6.40)
for any 4’q € L2(Jl) has at least one generalized solution u from
with f£h'71| Aau|| j qth dh < “>, where &hu(x, t) = u(x, t + ft)- u(x, t) , if the
following conditions are fulfilled.
1) For (*, t, u, p) 6 10 x [0, T] x Exx En} the [unctions a^ (#, tt u, p) and
a(x, t, u, p) are measurable in (x, t3 u, p) and continuous in (ut p) for almost all
ix, t) from Qj.; the functions a.(xf t, u, p) satisfy the inequalities
jl*
t, u, p)| t) + c\u\m' + c|p|m“l, <f>y 6 Lm, {Qj)f (6.41)
where q* < q = min + 2)/n, m = tn/im - l), with m > 2n/in + 2) for n> 2 and
m > 1 for n = 1; and the function ai%> t, u, p) satisfies the inequality
§6. THE FIRST BOUNDARY PROBLEM 467
9
|o(*, t, u, p)| «) + c|*|® + c|p|»', tf>2 6 Lqi {Qj), (6.42)
where q' » 9/(9 - l) and m* < m.
o.
2) For any function u(x) from IT^J(£2)
J [a, (jc, ii, ttx)itxt + o(X /. «, «*) u] rfjc
u
T
>v J | I” rfjc -t(f)J(l i-tfydx, v>0, |c (/) dt <!c. (6.43)
a u 0
3) /I monotonicity condition of the form
la fit, t, u, p) - a fit, t, u, q)] (Pi - <?,) > v(\p - q\, u, q, x, t), (6.44)
where v{t, u, q, x, r) is continuous on E* x Et x En x {I x [0, T-], positive for t>0
and equal to zero for r = 0.
By a generalized solution u of problem (6.40) we mean a function of the
class indicated in the theorem that satisfies the integral identity
^ [ - utf>t + at(x, t, u, ux)<f>%. + a(x, t, u, ux)t/>] dxdt
Qt ‘
+ 1 uix, t)<fi(x, t)dx - J t/r0ix, 0)dx = 0
0 ii
for any smooth <f>{x, t) that is equal to zero on Sy.
In order to prove the solvability of problem (6.40) we make use of Galerkin’s
method. We take a fundamental system \^tkix)\ in the space (Q) such that
and maxa |^, </>icx\ = < ~. An approximate solution ix, t)
will be sought in the usual form uF = j (1)1//^ (*) where the cj^ (t) are de¬
termined from the system of ordinary differential equations
(uf, i>k) + (a-ix, t, uN, ux), ipkx.) + (aix, t, uN, ux), if/k) = 0, (6.45)
k - 1, 2, • • •, N and initial conditions
ck (0) = (^o> k = l,2,---,N. (6.46)
From assumptions (6.41), (6.42) and the indicated properties of the functions <pk
468
V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
it follows that the second and third terras in (6.45) are summable functions of t on
[0, I] with sununability power q' for \c^ | < const, and are continuous in the c^,
k = 1, • • •, N (we note that q‘ < m'). Thus for the existence of at least one solu¬
tion of problem (6.45). (6.46) on the whole interval [0, T\ it is sufficient to know
that all of its possible solutions are uniformly bounded on [0, T}.
This type of boundedness follows from the a priori estimate
\uN\m,2,QT<o, (6-47)
in which c does not depend on N, since
0T kW(°)2~0*<¥•
Estimate (6.47) is derived from (6.45) in the following manner. We multiply the
4tb equation of (6.45) by and then sum all of the equations over k from 1 to
N. From tbe resultant equality integrated with respect to t we obtain by virtue of
assumption (6.43)
-j #«"(*. OI|Q + v JXfrfjeiff < Jc(.f)jll + (u")2)dxdt.
Q, 0 8
from which, as we proved in Lemma 5.5 of Chapter II, follows (6.47). From (6.47)
and (6.39) we also have
l*"IUr<** q = m^±l. (6-48)
We now let N tend to oo. Owing to the uniform estimates (6.47) and (6.48) it
is possible to choose a subsequence from (u^i that is weakly convergent in
lq(QT) co some function u and is such that the derivatives of its elements
converge weakly in to * . Let us agree for the sake of conciseness not
to take note of the necessity of distinguishing a convergent subsequence from a
compact (in the strong or weak sense) sequence but to speak of the convergence
of the sequence itself. In accordance with this we assume that the i/*, N ~ I,
2, • • -, converge weakly in ) to u, and the u^., N <* I, 2, , converge
weakly in LmiQj) to n .
Because of assumptions (6.41) and (6.42) the following estimates are valid
for any u and v from Vm,2^T^:
§6- THE FIRST BOUNDARY PROBLEM 469
t, u(x, t), vx(x, t»Im, Q, <
£ Id-!)
+ ci-iyfQ. (• 'sQ'r * +«i*,isr<?1'
s^jCmes Q\ <6-49)
where Q' is any measurable subset of Qj>, and
||o(*, t, u(x, t), Px(x,
i.{1_ij * .Ad--)
+ (mes Q"1^ 9 + “
- V-2(mes Q ' I “ s Q'. || Q>), (6.50)
in which the function /zj(rj, —* 0 f°r r( + —► 0, and ftjtfj, r2, —,0
for r j —► 0, uniformly on any bounded set of the space (fj, r2, r^).
Let us consider the functions ^ (t) = (b^ (x, t), tf>^ (*)), JV, 4=1, 2, • • ■ -
They are continuous in ( and cquicontinuous for any fixed k. To see this we inte¬
grate (6.45) from * to t + h and exploit estimates (6.47)—(6.50) in the following
manner:
|(«*(x, t+h) - U»(x. t), <|>*(*))i
I+ h
< J I(*«(*• *• “?)’ ^r.)+(a(^. *. «*, <).
t
t+h a
< c» J J (21a<(Xi *'aN■ “?)l+1“(*■ *•“N- uTi\)dxdt
t Q i-1
'• OIU«,,,+A {h ■mesQ)^
(A -mesQ)T<c;A7.
The constants and c ^ depend on fc but not on W for N > k. In view of
this the families of functions I^ 4 («), N ~ I, 2, • ■ ■, are indeed equicontinuous
in f 6 [0, T] for any fixed 4. Furthermore, from (6.47) it follows that
maXg<g<7,l^y ^ 0) | ^ o for all /V, A = 1, 2, —. This, as was shown in the proof
of Theorem 4.1 of Chapter III, guarantees the weak convergence in (SI), which
is uniform in i C [0, T], of the to u. The limit function u will thereby be an
470 V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
0
element of Vm j), and estimates (6.47) and (6.48) are valid fot it.
Let us prove that u is a generalized solution of problem (6.40). From (6.45)
follows the identity
Jf—uN4>, + (*. t, uN, +a(jr, t. uN. a") 01 dx dt
‘
! i'"'
-f \uN<fdx\ =0. (6.51)
which is valid for any function tf> = j ^(«)^j(*), where the dk(t) are con¬
tinuous functions of t having generalized derivatives d'^ (t) from L^i [0, T ] on
[0, T]. We denote the totality of such <f> by 9^. The functioa belongs to 9
Let us attempt to pass to the limit as N —* <*in (6.51) for a fixed 4>. Hie func¬
tions A?(x, t) = at(x, t, uN, u^f) by virtue of estimates (6.47) and (6.49) have uniformly
bounded norms in Lm.(QT) and hence converge weakly in Lm.(QT) to certain functions
Aj(x, r)e Lm,(QT) (Lemma 2.2 of Chapter II). The functions AN(x, t) = a(x, t, uN, u^ ),
(x, f) € Qt, are uniformly bounded in Lq\QT) by virtue of (6.47) and (6.50). There¬
fore we can assume that they converge weakly in Lq.(QT) to some function A(x, t)
from Lq.(QT) (see Lemma 2.2 of Chapter II).
This together with the above-described convergence of che iF to u gives as
a limit relation for (6.51) the equality
J [-wj>t + . + A<f>]dxdt + J«(*, T)<f>(x, T)dx-j i/i^(x)<f>{x, 0)dx- 0.
Qt . Q o (6.52)
It is valid for any <j> from 9 k * 1, 2, • • •, i.e. for any <j> from 9 = Uj°=i5V
But the set 9 is dense in the space J, (Qj) consisting of the functions
<f>{x, t) equal to zero on Sr for which <j>t £ Lq, (Qj) and <j>x 6 Ln (Qj-). In view
of this it is easily seen that identity (6.52) holds for any <f> € fT^’ ^, (Qj). From
this identity one can deduce that the function u(x, l) is strongly continuous in
i in L2 (0) and h"1 ||Aftu||j Qr /i —♦ 0 where A„u(x, t) = u(x, t + ft) - u(x, t), i.e.
ti 6 Moreover, the norm will be finite for u (see page 161),
i.e. u € (Qf)- This is proved in the same way as the corresponding asser¬
tions in §4 of Chapter HI (see Lemmas 4.1 and 4.2 of Chapter HI)- Also from
(6.52), in the same way as in §2 of Chapter III, one can prove the validity of the
relation
§6. THE FIRST BOUNDARY PROBLEM
471
|t=or + J + A'^dxdt = °- (6.53)
Let us now prove that for any <j> 6 9
/ Ajtf>x.dxdt =■ J a;(*, <, u, ux)cf>x.dxdt. (6.54)
Qr ‘ Qr ’
To this end we make use of the monotonicity condition (6.44). Because of it
f [<*•(*, t, uN, u?) - a.(x, t, uN, 77 )] (ux. - jj .) dx dt > 0 (6.55)
J t * * » *J
vr
for any r; 6 3*. We replace the integral Jq^, ai {x, t, uN, ux) (ux. - rjx) dx dt in
(6.55) by the expression equal to it from (6.51) with <j> ~ ufl - jj, where rj € ?N
This gives the inequality
J J— uNT\t — a(x, t, ttN, u^(aN — n)
Qt
— a^x. t. u", X){u» — x\x^dxdt
4-(«'v. DlJZo >0-
In it with t — T one can pass to the limit as N - > and as a result obtain
I [- uyt - A (it - rfi- at(x, t, u, rix)(ux. - r/x .)] dxdt
Qr
-5»-is.rr*u.*)r%o- (6-57>
2 ■£’“ l( = 0 l( = 0
In fact, the ii'"'* (x, 0) converge strongly in /- ^ (0) to *//^ - u (x, 0), while the
{x, t) converge weakly in L2(Cl) to u(x, t), so that >oo(u, rj) = (it, r])
and linijy ||u^ (x. «)|2 o < ||tt(*, l)||2 u- Because of estimate (6.47) and the
weak convergence of the itN to u in Lj(Cl), which is uniform in t 6 [0, the
functions ift converge to u strongly in the norm of L y m as is seen from
inequality (2.27) in Lemma 2.4 of Chapter II applied to the difference - it *.
And from the convergence of the a^ to it in Lj m (Qj) follows the convergence
of the uN to it in £<2(12) for almost all £ from [0, 7 ] and almost everywhere in
Qj. Furthermore, from it and the uniform boundedness of ||||^ g (see (6.48))
follows the strong convergence of the it^ to u in (Qj) with any ‘q < q and the
weak convergence in Lq(Qj) (see Lemma 2.2 of Chapter II). From this, the
-IK
''2, B |
(6.56)
472
V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
continuity of the a; (x, t, u, p) in (it, p) and estimate (6.49) i* follows that the
functions a(.U, t, uN, t]x) converge strongly in to a;U, t, it, t)x)
(Lemma 2.3 of Chapter II). The functions fk(x, t) = Ak(x, t) and uk(x, t) =
uk(x, t) - V(x, t)> & ~ 1,2,..., satisfy ali conditions of Lemma 2.2 of Chapter II
(see (6.50)) and therefore hrfkuk dxdt ~* h.TA(u ~ r>')dxdt when * “* ”
So (6.57) is proved.
We now combine inequality (6.57) with (6.53) and with (6.52), in which <f> is
taken equal to - j; € 9^. As a result we obtain the inequality
J lAj(x, t) - «;(*, t, it, ifj] (ux. - i)x,)dxdl > 0. (6.58)
Qt
It is deduced under the assumption that rj € 9^, but since N is arbitrary, while
Ai - a- (x, i, it, j)x) 6 Lm, (Qf) for u, r) € (Qj), it follows that (6.58) is
valid for any ji from V 1.0 ((JT). We put in (6.58) n = it - e£, where £{x, t) 6 f,
m, 2 1
while € is an arbitrary positive number, and multiply both sides of the inequality
by € This gives
f [/!;(*, I) - “,(*, t, ux - e£x)](x.dxdt > 0,
Qr 1
from which, letting c tend to zero, we obtain the inequality
J [A^x, t) - a(.(x, t, u, 1*^)] Cx .dxdt> 0,
Qt
in which tbe equality sign actually holds, since together with £ the function -C
is also contained in S’. Thus (6.54) is proved and hence (6.52) can be written in
the form
J+ a(.(x, t, u, ux)(j>x■ + Atj)}dxdt + J urf> <fc| = 0. (6.59)
QT 1 « lt=0
It remains to be proved that for almost ail (ar, t) 6 Qf
A (x, () = a(x, t, a(x, t), ux(x, ()). (6.60)
We take a smooth function oi S(i) on (- oo < * < oo), which is equal to zero for
- oa < t < 8 and for l> T - 8, where <5 is some small positive number, and equal to
1 for I € [25, T - 281. Let
(uN{x, t)(uSW for <e[0, T] f it(x, t)m(t) for e€[0, T]
(x, t) = J , and v(x, t) ~ <
( 0 for I < 0 and t>T [ 0 f«l<0 and t > T.
§6. THE FIRST BOUNDARY PROBLEM 473
From (6.45) and (6.59) with <f> ~ <o <f> follow the relatioas
+ <uS(«)aj(*, t, J>, u^) rj>x. + fc>S(l)a(*, t, uN, ux) - a*uNl<j> \dxdt = 0
Q ‘ (6.61)
and analogously
r ^ ^
J i- v<j>t + coSai(x, t, u, ux)tf>x. + [&>S /I - <y( ti]tf>ldxdt = 0 (6.62)
Q
where all of the factors under the integral signs are assumed to be equal to zero
for t 6 [0, 7 ], and <f>(x, t) in (6.61) is an arbitrary element of the form
£^=idk(t)\l/k(x) with continuous dk(t) having compact support, while in (6.62) it is
any element of Q-Ux (-•», +«>) equal to zero for large |f|. From these
identities and inequalities (6.47)—(6.50) the estimate
(6-63)
is deduced in the way it was done in §4 of Chapter III (see pages 160—161).
We take a sequence {ityj, N = 1, 2, ■ • ■, 6 9N, such that the vN converge
to v strongly in L?(0 and in the norm | ■ |||q, while the vNx, converge to vx.
strongly in Lm(Q) (in other words the converge to v in the norms of the
spaces L?(0 and IT^ (Q)). Because of (6.61) with $ = VN - uN
J iwf (vN - v) + coSU)aAx, t, uN, uN) (vx. - v )
•r * • % X * % •
Q
+ [<uS(«)a(*, t, uN, u") - a>*uN] (vN - v)\dxdt
= - J iff (v - Vtf) + <u5(«)o((», I, uN, ux) (vx. - vN )
Q 1 **
+ [<u8(<)a(x, t, uN, u%) - 0JtSnW1 (w - vN)\ dxdt z (6.64)
By the convergence of the vN to t> and estimates. (6.47)-(6.50) and (6.63),
the Rn —> 0 for N —► oo since
474
V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
On the other hand, from (6.62) follows the equality
^[*>*0^*, t, u, ux) vx. + (a>S A - a>*u)v] dxdt = 0 (6.65)
<?
which is deduced in the same way as (6.53) ft0™ (6.52), or as (2.13) from (2.15)
it
ol Chapter HI. We combine it with (6.44) and with (6.62) for = - v . As a result,
after elementary transformations of the left side of (6.64)
C l<uS[oiU, t, if1, ux) - aAx, t, uN, a)] (vx. - v ) + a>SU) [<*:(*, t, uN, u )
j t X » A I t
Q
- aAx, t, u, u )] (vx. - v ) + &)S [a(x, t, uN, u%)~ A] (vN - v)
» X A>] 3C
- <u(S(it^-u) (v - u^)! dxdt = R^. (6.66)
The second, third, and fourth members of the left side of (6.66) tend to zero as
N —* oo (it is proved as above, when we derived (6.57) from (6.56)) and RN also tends
to zero. Therefore
iN s fQo>& [a£x, t, uN, wf) - afic, t, uN, ux)j(tgt - uXj)dxdt -* 0, N -*• oo
and because of (6.44)
fQ(us)2v(\u% - uxI, uN, ux, x, t)dxdt 0, N<»
This gives (o)t(tyf'v{\ux(x, t) - ux(x, 01. uN(x, t), ux(x, t)) —► 0 almost everywhere
in Q. According to the properties of p(* * *) and arbitrariness of 6 > 0, we can conclude
that I ux (x, t) - ux(x, f)| —* 0 almost everywhere in QT. From this follows that
a(x, t, uN(x, t), ux(x, t)) nave as their limit a(x, t, u(x, t), ux(x, /)). Equality (6.60)
and Theorem 6.7 are thereby proved. (We had to choose in some places convergent sub¬
sequences from compact sequences, but we did not do this for the sake of conciseness.)
An assertion similar to Theorem 6.7 was proved by Lions in [*34] wjth the
use of an approximation of equation (6.40) by equations of elliptic type.
Theorem 6.7 admits different generalizations. For example, its assertions
remain valid if the a. and a have the form
<*,(*, *, u, p) = at(x, t, u, p) + fT(x, t, it, p),
a(x, t, u, p) = a (x, t, u, p) + a(x, t, it, p) 4 a(x, I, u, p),
where the functions 3f(. are subject to inequality (6.41), the function a is sub¬
ject to inequality (6.42), the functions a. are subject to inequality (6.41), in
§7. OTHER BOUNDARY PROBLEMS
475
which q* is replaced by ?, a is subject to inequality (6-42), in which q* is re-
placed by q, and a is subject to inequality (6.42), in which q* is replaced by
q and m* by m, provided that in place of condition 3) it is known that for any u,
« e vUQ)
m
J [a- (x, I, v, vx) - a i (x, t, v, ux) + 5^ (x, t, v, vx) - (*, t, u, ux)}
0
x (t> - a ) + [o(*. t, v, v) - a(x, t, v, uj + a(x, t, v, v)
Xj A| XXX
- a (x, t, u,uj]{v- u)dx>j l«xt) 1 (6.67)
a
where v(rj, r2) is a continuous positive function for rj >0, r2 > 0.
Theorems analogous to Theorem 6.7 are also valid for quasi-linear parabolic
equations of high order, as well as for quasi-linear strongly parabolic systems
with principal part in divergence form, and they are proved in die same way as was
just done for equations (6.40) of second order, it being possible in this connection
to take as the boundary condition not only tbe first but also analogues of the
second and third boundary conditions (see §5 of Chapter IIJ).
The papers [Ha—d; I34;.124d,e; 22] are devoted to a proof of the existence of
generalized solutions of such problems. A construction is given in [124d,e] 0f
generalized solutions having certain differential properties that are better than in
[lla-d;134] (in the case of equation (6.40) they have the derivatives a, and
which requires greater restrictions than a condition of type I)—3) but in return dis¬
penses with the weak passages to the limit through nonlinear functions. Such so¬
lutions are obtained as limits of approximate solutions constructed by a modi¬
fication of Galerkin’s method, which is similar to that given in
§7. OTHER BOUNDARY VALUE PROBLEMS
In this section we consider the second and third boundary value problems
for equations of the form
JFussu,—atJ(x. t, a)ax^x +b(x, t, u, u^ = <3, (7.1)
which are a special case of equations (0.1). Namely, we will assign on the lateral
surface of a cylinder Qj. the boundary condition
jr<5)« = [ai; (x, t, u) cos(n, jc() -j- <1>(x, t, u)J ^ = 0, (7.2)
476
V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
and on the lower base of Qj. the cootUcioa
(7'3)
Here o = n(*) is a vector of the outward normal to the surface S, which we assume
to be sufficiently smooth-
We will investigate the solvability of problem (7.1)—(7.3) in the class
f]2+f),l+/3/2 (Qp)_ As a preliminary we establish all of the estimates that are nec¬
essary for this purpose. By tbe results of §§1—3, in obtaining the estimates we
may restrict ourselves to small subdomains (/ that are adjacent to tbe lateral
surface of the cylinder We fix, for example, some part S' of the boundary S
and denote by 0' an arbitrary subdomain of Cl that is adjacent to S', and by Q'f
cylinders of the form 0* x (0, T). Without loss of generality one can assume that
S' lies in the plane = 0 and 0 is in the halfspace *n < 0 (this situation can
always be achieved by means of a transformation of independent variables).
Keeping in mind the basic aim of this section, viz. a proof of the existence
of a classical solution of problem (7.1)-(7.3), we will carry out all of the estimates
for solutions uix, t) from C2,1(@j,), although many of them can also be obtained
under weaker assumptions concerning uix, (), as was done in §§1—3-
We begin with a derivation of an estimate of the norm assuming that
max^|tt(*, «)| < U is already estimated. We will assume that for (*, t) 6 Qj. and |b| < U
the functions a--ix, t, u) and ijrix, t, u) are differentiable with respect to x and u and
satisfy the inequalities
t, «)!,!,< Hi* v > 0, (7.4)
I dan (x, (, a) dau dtb dti I
—Si • OF- *■ 75T- 371(7-5>
while bix, t, u, p) for the same (*, t, u) and arbitrary p satisfies the inequality
!*(*, t, a. p)|<u(l + P2) <7-6>
Suppose uix, t) 6 C2,1 is a solution of equation (7.1) satisfying condi¬
tion (7.2). The following identity is valid for any function rjix, t) from V^iQy):
§7. OTHER BOUNDARY PROBLEMS 477
t
+ J J <i>(*. t, u)t\dxdt = 0, 0<f0</<7\ (7.7)
Fot functions rj{x9 t) that are equal to zero oa Sy\S j., wbere S'j - S' x [0, 71,
tbe latter integral can be transformed to the volume integral
1 1
J J ii>n dxdt = J J (TOdx dt
U s
1. a
in which case (7.7) takes the form
t
1 J ["^ + aijuxj\ + fltl + dx dt = 0, (7.8)
U 2
where
, , . 6atj (x, t. it)
a(x. t. u. ax) = gj
da/i
+-^uXf+b(x. t. a, Bx)4-^_ajrn+_.
This equality is of the same form as (1.6), with the same conditions on the
a-(x, t, u, p) =» a - (x, t, u)pj + (frS" and a(x, t, u, p) as in (1.6), only in contrast
to (1.6) the function rj {x, l) here must vanish not on the whole surface but
only on Sj. \S'j,.
We put r) = •«(** (*, t) f 2 (*, t) in (7.8), where the value of k is arbitrary,
while ^ is an arbitrary smooth function with values between 0 and 1, that is
equal to zero on the lateral surface of an arbitrarily chosen cylinder Qip, r) =
x (tg, Jq + r) with 0 < < l0 + r < T, not intersecting Sj \ S'j. even for
* 6 Kp. We carry out an estimate in (7.8) with the use of inequality (1.2) of
Chapter II and conditions (7.4)—(7.6), as was done in §1 in deriving (1.11). As a
result we obtain the inequality
t
£
2
t
J l«W(*. t)l(x. +v J J (u<*f&dxdt<C
478
V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
J +S2 + + uW'\%\\dxdt.
where is the set of those * from = Kp fl ft fot which u(x, t) > k.
Hence fot k satisfying the condition
max u(x, () — *<6==2??uf ’ (?‘9)
QU>,V m
we have the inequality
B «•*>(*. +• Hi. Qp -f v || [j= Q(p tl
J J uW^ + ^Wdxdt
*% "
(,+ t
-f-j" J t,dxdt
■ ' 0(ti
which coincides with inequality (7-5) of Chapter II for r = 2 (it + 2)/n, k = 2/n. Hie
same inequality is also satisfied by the function - u(x, t). As was noted in Re¬
mark 8.1 of Chapter II, this is sufficient to determine if the function uix, t) be¬
longs to the class U r\ M, y, (2n + 4)/a, fi, 2/n), where F’ is the union
of S'T and the lower base of the cylinder Q-p.
Hence, on the basis of Theorem 8.2 and Lemma 8.1 of Chapter II, one can
draw the following conclusion.
Theorem 7.1. For any solution u (x, t) from C2,1(Qj.) of problem (7. l)-(7.3)
and an arbitrary cylinder Qfi- K p x Uq, Iq + p~) not intersecting Sj\ s’p the
estimates
osc (u, Qp fl Qr) < cp». (7.10)
J B|dJc<ff<ciP»+2« (7.11)
Opfl'Jr
are valid; here the exponent a > 0 is determined only by the quantity
= max^ |u | and the constants v and p from inequalities (7.4)—(7.6), c is
determined by M, v, ft, |u(*, 0)|*„* and the distance from the center of Kfi to
S\S', while c j depends on the same quantities as c anion max,, \ux (x, 0)|.
§7. OTHER BOUNDARY PROBLEMS 479
We now need a priori estimates of | and M*g!p5*- This will be done
in the following way. We first establish an estimate of the integrals \ut\^ dx
with any q > 1. Then we make use of it for q = n + 1. Namely, we consider equa¬
tion (7.1) for each t from [0, T\ as a quasi-linear elliptic equation depending on
t as on a parameter:
at](x, t, t, tt, 11,)=0,
for which the functions of-. and a(x, t, a, p) = - b(x, t, u, p) - «4(*, *) satisfy
conditions (7.4), (7.5) and
\a(x, t. a, p)|<p.ps + <p(x, t); max ||<p|| , „<li- (7.12)
' (f|0. ri
For solutions of such equations satisfying the boundary condition (7.2) the
estimates
maxla^Kc, osc \az, Q } <cpP, (7.13)
a
are valid if the boundary S of the domain Q belongs to the class 02, while the
function <ji from (7.2) is subject to condition (7.5). The constants c and a > 0
in (7.13) are determined only by the quantities v * and (i from (7.4), (7.5) and
(7.12), M = max„ |u] and the boundary S. Estimates (7.13) were derived in Chap¬
ter X of the book [<>5q] under somewhat mote rigid restrictions, viz., the differ¬
entiability of a with respect to x, u, p and the boundedness of </•>(*, t), but the
proof given there can be carried over without essential changes to the case of
conditions (7.12).
Thus, having estimates of the integrals /0 | u,|" + * dx, by virtue of (7.13) we
will have estimates of ma.XQ^\ux\ and the Holder constants (ux(x,
t 6 [0, 71, with respect to the variables x. Using next the estimate for
fa lu(l* dx with q > 2n + 2, as well as Theorem 2.1 and Lemma 3-1 of Chapter II,
we arrive at the estimate
| a < c. (7.14)
In deducing the subsequent estimates we require the following additional re*
strictions on the {unctions a^, b and tfr for (*, 2) € Qj* and |zt | < M-
!♦„(*. *. «). 1w 4>a(. atit- |
IM<> +-I/,l) + l*a| + l*<K^(i •+ p2). J (7-i5i>
480
V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
and
I &tjau' &i/at* alfux> (7.152)
Before estimating the integrals /|u{|^ dx, we will prove a lemma.
Lemma 7.1. For any ball Kp with center in Q which, together with the con¬
centric ball K-2p, does not intersect with S\S*t the estimates
T
J (7-16)
» %
max f u*(x, t)dx -<cp"~2+ai (7.17)
<f|n.nQJ
p
are valid', here the constants c and a > 0 depend only on M * ma|u|> v and
p from (7.4)—(7.6) and (7.15), max8 !»*(*, 0)| “nd the distance from the center of
Kp to S\S'.
In (7.8) we take t0 = 0 and tf m ut(x, t)£^{x, t), where £(x, t) is a smooth
noonegative function that is equal to zero on the lateral surface of the cylinder
&2pX (0, T) and outside it.
The fact that i»,£2 6 V^'0 (Qf) (more precisely, the existence of the deriva¬
tives utx. from Lj(Qf)) will he proved below in Lemma 7.2. Noting that
I d . ... 1
aijUX/tt,Xli —"2 IS ) 2 dt
and
=it i^2)- “*«nr®?)-
we obtain from (7.8)
t
J ju^dxdt + l | (a,jUt'Uxp + 2<|>«*n£2)dx
0 °Jp Q*p
t
— J C [(“I/#"/?2 4- «<,2K, + ~ aX(aZj — at]uXju,21£xt
0 %
— aa,£2 + uXa (t«a<£2 +- <f/£2 t 'i>2££,> — ^a,7^.x^ dx dt.
§7. OTHER BOUNDARY PROBLEMS 481
From this, by using assumptions (7.4)—(7.6) and (7.1$) and inequality (1.2) of
Chapter II, one easily deduces the inequality
J | «*(*. t)V(x. t)dx
» % °»p
<e ]>£(*. 0)+l)S*(*. 0)rfx+«,J J[|^|4S2
a*p " %
+ (1 + «“)(| | + £ -r £2)J dx dt. (7.18)
Let us show that for the iategral J* Jc dxdt we have the estimate
t t
J J|«x|t&dxdt^c^ f J {ulJZ+uVQdxdt. (7.19)
0 °*p " %
Indeed, by an integration by parts we find
f I ttxI4Pdx = - | (a-«0)(A«*p-f 2«, « p
flap °8p
+ «,«»2{£ )<** + J »*(«-«„)« gds. (7.20)
*,PrF
Here uq = u0(t) is the value of u(x, t) at the center of the ball virtue
of estimate (7.10) we have max„ ^|b(*, t) - UqU) | < cp“, where c depends on
the distance from the center of Kfl to S\S', Therefore the boundary integral in
(7.20) does not exceed
Wa f I H2 dS=C&« f Sr(\uje TO dx
*»pns' % "
Jd^zl^ + l^pti^Drf*
G2p
< J (wjp + i U*J? + ^ u%l) dx.
u*f '
The first integral in the right side of (7.20) is estimated analogously. Taking e
sufficiently small, we atrive at the estimate (7.19).
Let us now estimate the integral /„ .,^“xx£2 dxdt. To this end we put
482 V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
in the identity (7.8) and cany out an integration by parts. This gives
i »
£ J J [<*+«> (a*s>P+2u*J&*s)
0 %
— {aijUjcsij + OiJuuXsUjt) atjlaXi ~~ 6*^
X + «,/«*,)] dxdt = 0.
From this, by using conditions (7.4)—(7-6) and inequality (1.2) of Chapter II,
it is not difficult to deduce die inequality
J J £ '^iU\txpdxdt
o a,p s-i i-1
t
< c6 J J [«^-h|«x|<£2 +.(«£ + 1)(£ ±t?)]dxdt. (7.21)
0 %>
On the other hand, by solving equation (7.1) with respect to the derivative u ’■
*n*n
/ n-1 n \
2 “““v*4 * J ■
'• #-i /-i /
and taking into account assumptions (7.4) and (7.6), we see that
1 l'’v,+i*.r+')- <7-K)
From (7.22) and (7.21) it follows that
i i
j \u\j?dxdt^cti jf^+Kre*
-t(u2x+ l)(£2~K2)]d* dt. (7.23)
Consider the inequalities (7.18), (7.19) and (7.23). For p not exceeding a
sufficiently small number p0 (the quantity pg is determined by the constants c(
and a from (7.18), (7.19) and (7.23) these inequalities imply the estimate
§7. OTHER BOUNDARY PROBLEMS
483
J «;(*, 0P(*. t)dx
Cjp (
+ / J(Kl4+sL+“?)e!<**‘#<cs jK(^o)+i]
0% # %
XZHx, 0)dx + ctj J(«*-j-l)(|S,|S+£+£*)<**,«.
0 %
From it and inequalities (7.11) follow, as is easily seen, both of tbe assertions of
Lemma 7.1.
We will prove (he following proposition.
Lemma 7.2. For any q > 1 it is possible to determine a constant ciq) such
that
max f I u, I* dx < c iq), (7.24)
<flo,naJ
where Q' is an arbitrary subdomain of Cl that is separated from S\S' by a posi¬
tive distance d. The constant c(g) depends only on q, U, the constants u and
p from conditions (7.4)—(7.6) and(7.15), then aaxtt\uxix, 0)|, /n |u,(*, 0)|9dx and d.
We take the divided difference with respect to t of both sides of equalities
(7.1) and (7.2):
Q-
A {aiiUxtx,) kb
to
= 0. (7.25)
["ST (alja*j) cos ("• *<) + 4f] ~ 0-26)
then we multiply (7.25) by |Au/A«|'"”2 (Ab/A«)£ 2 (ac), r> 2, integrate over Cl and
over tbe interval [0, t], and then carry oat an integration by parts, taking into
account condition (7.26). This gives
ii\w*‘‘i+ii[2£Lk(\zrw
Q 0 Q
I & I dan \ A6 \ | Au I'-2 A« 1
+ ('3sr“jr/)‘*’_5r)l'S’l ~&r^\dxdt
+ J = <7‘27>
0 s
For r = 2 and £ s 1 it is not difficult to deduce from this equality, taking into
account conditions (7.15) and (7.4)—(7.6) and the membership of uix, t) in
484 V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
C2' ^ (<3y), die boundedness of the integrals
J / l](-5r) dxdt' o</<r-ft.
0 Q i-i
which is uniform with respect to |A*| < h, where h is any positive number. This,
by virtue of Lemma 4-11 of Chapter II, implies the existence of generalized deriv¬
atives of the form u%.t that are square-summable over tbe cylinder Q-p-
We will now assume that the function £(x) is a cutting function for some ball
Kp not intersecting with S\S'. In (7.27) we transform the latter integral into a
volume integral, as was done in (7.7), and then pass to the limit fot At ~~*0. As a
result we arrive at the equality
I
7 J I “t (*■ 0 W(x)dx\‘(>^ J J | (atjUz^t (\u, (-* uff)X)
+4r (-Sf *'«+*) i'“<r2“^
We expand the expressions for the derivatives with respect to t and x-:
t
7 j | a, (x. t)\r?(x)dx |'+ J J [(a,tuXjl + al)uU'UXj
% . 0 °p
+ 'Mi*,+ ♦#)(<»■ - >)|a,r2«<^2
+1 “f |r”'***^2CEjrf)+(bt + built -f- buxutxl +
-f"-f- tyutUX" + >W*„ + ‘^“ljltuxll«xj + OljuuU^U^U(
+ aijutUxjUxj + at jxyUzjUi -f a-ijx^Uxj) I f 2 «<£J] dx dt = 0.
From here, using conditions (7.4M7.5) and (7.15), it is not difficult to deduce
the inequality
t
fKrt?**u + J
s 4 °o(
<e,M J J |«,f(£ + +t?)dxdt, (7.28)
§7. OTHER BOUNDARY PROBLEMS 485
in which Cj (r), in general, tends to ■» fot t —► oo. Let us show that the integral
Sofa lIiJ|r+dxdt is also estimated in terms of the right side of this inequality and
fbC i.,r2-Lc2**
Indeed, putting 77 » |u{|r~1 a(f2 in (7.8), we obtain
t
J J [I */|r+1£a + atittXjr | utf~l u,xp
0 °p
■+■ aijaxj|»ifr 1 -+• a | ut f 1«(?? + ’l,r I “'I u,x^*
+t|> | ut f-dx dt=0,
from which a zero estimate for |a,jr*lC,2dxdt follows by virtue of (7.4)-(7.6)
and (7.28). Thus, in addition to (7.28), we also have the inequality
/
J\u<\r?dx\‘+ J J(K f - +1 u, r‘)£2 dx dt
% 0 °p
t
< c, (r) J J | a, f (£2 + UI& + &) dx dt. (7.29)
0 %
Consider the integral Jgfaf)U2 | ut\r(2dxdt standing in the right side. For
an estimate of it we take advantage of inequalities (7.17) (proved in Lemma 7.1)
and Lemma 5-3 of Chapter II. According to this lemma
J u\ I«, I' dx < cp* J (| a, dx.
% %
and therefore
t
J
0 fi
p
I
<c2(r)p2a j J[|.,|"2 u]g^\u^^\dxdt. (7.30)
The constant c2(r) here is determined by the quantities v and (i from (7.4) and
(7.6), M, r, the distance from the center of Kfi to S\S* and max,, M*. 0)1-
486
V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
From (7.30) aad (7.29) it follows that for p < p0(r), where pg(r) is chosen from
the condition
c2W(t,('-)c1(/’) = y. (7.31)
the inequality
/
j Ji«, r1 pax m
% 0 ° %
f
<c3(r) { J|«,|'(tf+tydxdt (7.32)
0 aP
is falfilled.
Suppose that we wish to estimate f\ut\1 dx for some q> 2. We apply in¬
equalities (7.32) to a sequence of contracting concentric balls Kp , pr =
Pq (q) (1 - r/2q), r= 2, • • • -, q, taking as ((x) for p = pr a cutting function for
the ball Kfl . Then, taking into account the estimate of fg uj dx dt obtained
in Lemma 7.1, we obtain the desired estimate:
J Iut(x‘ 0le^<c(?).
ap»w
2
Here t is an arbitrary value from [0, 7"], while pg(?) is a sufficiently small but
fixed (by condition (7.31)) number. Coveting Si’ by a finite number of the balls
KpQ(q)/2’ we “ftive at estimate (7.24).
Lemma 7.2 is proved.
As noted above, estimates (7.13) and (7.14) follow from estimates (7.24) with
q > 2n + 2. The constants c, a and 8 in them are determined by |ux(a;,0){ ^,
and also by the quantities determining c(q) in Lemma 7.2, in particular, the
quantity ||u((ac, 0)||? fl. The latter can be estimated, using equation (7.1), in terms
of Iu{x, 0)\(p. We formulate this result in the form of a theorem.
Theorem 7.2. Suppose the functions a -(*, t, it), b(x, t, u, p) and ifi(x, t, u)
for (x, t) 6 Qj., |b| < M and arbitrary p are subject to conditions (7.4)-(7.6) and
(7.15), and S€ 02. Then for any solution u(x, (j from C2,1(Qj.) of equation
(7.1) satisfying condition (7.2) and having max^ |it{ < M one has the estimates
§7. OTHER BOUNDARY PROBLEMS
487
max | |a|!’+0)<c.
(7.33)
where the constants Mj, c and S > 0 depend only on M, v and p. from (7.4)—(7.6)
and (7.15), the norm | u (x, 0)j(^ and the boundary S.
Remark. It can be seen from the conclusion of the theorem that if one assumes
in its conditions that not all of the surface S but only a part S' of it belongs to
02 and that condition (7.2) is fulfilled only on S'j- = S' x [0, T], then estimates
(7.33) will hold for any domain Q' C Qp that is separated from Sj-\S'-p by a posi¬
tive distance d, where the constants and c will in this case also depend on d.
We have assumed above in the derivation of all of the estimates that an esti¬
mate of \u(x, t)| is known. This latter fact is established by the same
technique that was used in the linear case in §2 of Chapter I. Namely, the follow¬
ing proposition, generalizing Theorems 2-2 and 2.3 of Chapter I, bolds.
Theorem 7.3- Suppose the functions a,y(*, t, a), b(x, t, u, p) and ifiix, t, u)
are subject for arbitrary u to the conditions
0 <.au(x. i. aH^Cix,!2 for (x, 06QX(0. 71.
— ub(x, t. u, +
for (jc, /) £ Qt \ rr, (7.34)
V|S2<«i/(*. — «>!>(-*. t, u)<c3«2 -f- c,
for (x, t)£Sr,
where jtj, = const > 0 and c( » const >0, i = 0, • • •, 4. Then the estimate
max|a(jc, *)K max max|a(jc, 0)|| (7.35)
is valid for any solution u(x, t) from C2'1 (Qj-\Tj.) that is continuous in Qj
and has continuous (right ttp to Sj.) derivatives ux, where the constants A. and
A[ are determined only by the quantities i/j, ji, cQ, Cj and Cj from (7.34) and
the boundary S of the domain 0, which is assumed to belong to the class 02.
If SB 0^ and in place of (7.34) the conditions
dijix.t, h)|,|/>0 for (x. f)6SX(0. T),
— ab (x, t. u, 0) < £,0* +- es for (x, t)£QT\ Tr,
— “$(.x, t. «)<0 for \u\ > 0, (x,t)£ST,
ct, c2 — const
(7.36)
488 V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
ore fulfilled, then
max
0
ax| u(x, *)l< min eKT maxi l/'j-—r; max) u(x, 0)|1. (7.37)
>T K>c, A. c> a J
(7.38)
In order to prove die first part of the theorem we consider the function
w(x, (I = e ^l<fi(x)u(x, t), where and K are chosen from the conditions
minqK*)^ 1; A(x. t)
o
*• “>cos(“- *i)
X
B(x, /)==minjx,-+• atj(x, t, a)—
Vxfix, Wl •,
- 2«,y (x, t, a) —^ ^-] > c„ x£Q
(the existence of such <f> and \ under our conditions on 5 and die is obvious).
Taking into consideration the relations
fx
Wf =—X«+e_w98(; wX[ = w--~--\-e-y‘^ttx;
«V, = —® + (V'y + V'^ +
we conclude that w(x, t) satisfies the equation
*Pjr x
w, 4- X® - a;/(*. *. «)«^ +
<fx. QxVx,
+ 2aliT-wx-2alJ-—r-w + e->''?b(x, t, u, «,)«<)
and the conditions
T <Pr/
[a,y(je, /. u)t»^cos(n. xt) — wat/ -^-cos(n, j:,)
4 e-'-'(p(jc)i|)(j;, /, «)] =0.
H=o —<P<*)“o(*)-
Since the form *s positive definite on Sy the vector I with
§7. OTHER BOUNDARY PROBLEMS
489
components a- cos (a, = 1, • • •, n, forms an acute angle with tbe outward
normal n to die surface 5. Suppose the maximum value of to2 (x, t) > 0 in Qj- is
achieved at some point (xg, tQ) 6 Qj. If xQ belongs to the surface 5 and tg > 0,
then at this point
—-=2®^^ cos (it, *()>0.
and hence from tbe boundary condition for w and (7.34) we deduce
w2{— otj ~ cos (n, xt) — e,) — c4e-W<p2 I <0.
' f ' J K«* w
It follows by the choice of <fi{x) that
2.
I«(JC0, *0) I < <-A ~ C3> 2<K*o)
If (xg, tg) £ D x (0, r], then, by taking into account the relations
f®. I > 0, ww I = 0.
' '<*» '»> < l(jr„ (,)
-<0, 9« e
1 J l(x«. it) 1 W '•) f kx,, i„)
we obtain from the equation for to and conditions (7.34)
Ix
r/ a./V/ vs \ ,
— e-5)V (c0«2 -f c,«2 -I- c2)J
f5 \‘
— 2a,
*i*t
(•*«* *nf
W2 —
lx8, /*)
^ •
*/
and this by virtue of (7.58) implies
1
1 «'(*!> f0)!< (B — c,)""2 tp(jf0)e-w».
Finally, if tfl = 0, then |«i(x0, «Q)| < max0 |u(x, 0)<£(x)|. From the three
cases considered we obtain estimate (7.35) with constant
490
V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
In order to prove die second part of the theorem it is sufficient to put <f>{x) - 1
in the preceding arguments. It is easy to see that in the case of conditions (7:36)
the function w2(x, t) ™ e {x, t) cannot achieve a maximum on the surface
Sj, while if maxQyu>* is achieved at a point tg) 6 Q x (0, T], then
J(A. — c,) ®s — c& ~n< | ^ w < 0.
so that l1"^, <0)| < yc j/(\ - Cj). By considering the last possibility for
maxQ^ = w2(xq, 0), we arrive at estimate (7.37).
Let us make use of the a priori estimates established in this section to prove
the unique solvability of problem (7.1)—(7.3) in the class
will assume that the initial condition (7.3) is homogeneous:
«|t*0 - 0 <7-350
(the general case of condition (7.3) reduces to this upon replacing u(x, t) by
v(x, t) = u(x, l) - ^fr(x)) and S € H2 + ^-
In the same way as in §6, we include the problem (7.1), (7.2), (7.39) under
investigation in a family of problems depending on a parameter r € [0, l], and we
will apply the Leray-Schauder Theorem. For simplicity we restrict ourselves to
die specific case
where
i’I«sti’s4-(l-t)^, xgJO, 1].
Jgtpu z=t^'s>u 4(1 — t) Jg^’a = 0,
«Uo = 0.
-S'oassa( —(lAa. + aj ,
(7.40)
with n being a constant from inequalities (7.4)-(7.5).
Suppose that the compatibility condition fot the initial and boundary data,
i.e. *(s, 0, 0)j * 0 is fulfilled for r = I. Then, clearly, it will be fulfilled
for all r € [0, lj. Ve assume in addition that the operators £ and satisfy
the restrictions (7.34) or (7.36) of Theorem 7-3. Then by tbe same token the re¬
strictions with the same constants ci and with t/(r) « min j»/j, jil, fi(r) =
maxlfip fil will obviously also be satisfied by the operators £r and for all
t from [0, I]. Therefore it is possible to assert on the basis of Theorem 7.3 that
the uniform (with respect to r from [0, 1]) estimate
§7. OTHER BOUNDARY PROBLEMS
491
max ) u (*, t, r)| < M
Qt
(7.41)
is valid fot all possible solutions u(x, t, r) from C2’1
(Qf) of problems (7.40).
Further, we will assume that inequalities (7.4)—(7.6) and (7.15) are fulfilled
for (a, l) 6 Qj., \u\ < U and arbitrary p. It is easy to see that the functions
<f.. = roy + (1 - bT - rb and i/ir = njt + (I - r)y.u satisfy inequalities of the
form (7.4)—(7.6) and (7.15) for all r 6 [0, l] with the same constants v and fi as
for t = 1. In view of this, Theorem 7.2 guarantees die estimates
for all solutions u(x, t, r) of problem (7.40), where the constants and c are
the same for all r from [ft 1]. We now formulate a theorem.
Theorem 7.4. Suppose the following conditions are fulfilled.
a) The functions a-(x, t, u), b(x, t, it, p) and <fi(x, t, it) satisfy inequalities
(7.34) or (7.36).
b) For (*, t) 6 Qj., |it| <U, where M is a constant from estimate (7.41),
and for arbitrary p the functions a-{x, t, u), b{x, t, u, p) and i/r(x, t, u) are con¬
tinuous in their arguments, possess the derivatives entering into conditions
(7.4)—(7.6) and (7.15), and satisfy these conditions.
c) For (*, t) 6 Qj., |it| < M and |p| < Jf p where Wj is a constant from in¬
equality (7.42), the functions a^x-(x, t, it) are Holder continuous in the variables
x with exponent fi, t, u) is Holder continuous in x and t with exponents
fi and f}/2 respectively, and b(x, t, u, p) is Holder continuous in x with
exponent fi-
max|ux(x, t, t)|<!Mt, |ac
Or
(7.42)
d) seH2**3-, <j>{x, o, oij^gj = o.
Then for each r from [0,, ll problem (7.40) has a unique solution u(.x, t, r)
in ike class H2*& * + ^2{Qj,).
Proof. We consider the problems
vt ~ [xai/(*■'. «0 + (1 — t) 6/nJ v
t, w. wJ — O,
{[■*«//(•*. *.«>)+(> — T)6/n]t»^cos(n. Jtj)
+ (1 — T)nt>4--n|>(*, t. w)l lsr = 0.
(7.43)
492
V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
As the principal space Ba we take the subspace of +a>/2 (Q^) con.
sisting of those functions that are equal to zero for i = 0. We put a = min IS; jQ(,
where the index 8 is taken from (7.42).
For any function w(x, t) from Ba satisfying the conditions max y I I < W
and mai^y Ikj^I < Mj, problem (7.43) is ■ linear problem of the form (5.5) of Chap¬
ter IV, for which all of the conditions of Theorem 5-3 of Chapter IV are fulfilled.
Namely, by virtue of the yet to be proved Theorem 7.4 the functions a^. (*, t) =
ra(y(*, t, w(x, t)) + (1 - r) Sj- p and 0^r(*, i) = ti//(x, t, w(x, t)) belong to
+ a,(1 + a^2(@r) , ar(*, f) = rb(x, t, w(x, t), wx) € Ha,a^2(Qj<) and the com¬
patibility condition tor the initial and boundary data in problem (7.43) is fulfilled.
It therefore follows from Theorem 5-3 of Chapter IV that to each function u>(x, t)
possessing the properties just described there corresponds a unique solution
v(x, t, r Jfromi//2 + a. 1 + a/2 (Qj.) of problem (7.43), and this determines a completely
continuous transformation v = ?’(»', r) in the space Ba, the fixed points of
which are the solutions of problem (7.40). With the use of the estimates (7.33)t
(7.35), (7.37) established above it is possible to prove the existence of fixed
points of this transformation and the membership of them in tbe space
#2 + $ 1 *fi/2(Qj.) in exactly the same way as in the case of the first boundary
value problem in §6.
The uniqueness of the solution of problem (7.40) for each r from [0, l] is
easily established with the use of Theorems 2.2 and 2.3 of Chapter I.
The propositions proved in the present section admit certain generalizations
and sharpenings within the limits of the methods set forth here. For example, one
can assume the existence of singularities of the functions o-{x, t, u) and
b(x, t, u, p) with respect to the variables x and t, as was done in §§1—4. Fur¬
thermore, all of the proofs given above survive almost without change for equa¬
tions (0.1) of more general form with o£(x, (, u, p) = Fp Ax, t, u, p) and tbe bound¬
ary conditions
a,(x, t, u, ux)cos (n. xt) -+- <p(at. t, u) = 0.
§8. THE CAUCHY PROBLEM
Let us consider the Cauchy problem for equation (0.1) in the layer
Rj- « I* 6 £„, l 6 [0, T]| with the initial condition
“l«=0 “
§8. THE CAUCHY PROBLEM
493
We will seek a solution in tbe class of bounded functions belonging to
jj2*0,l for any finite subdomain Qy of the layer Rj-.
With respect to the functions a-(x, t, u, p) and a(x, t, u, p), we assume that
they satisfy in any finite domain Qj. C Rf conditions a)—c) of Theorem 6.1,
where the constants in condition a) do not depend on tbe dimensions of Qj..
Furthermore, we suppose that ftg(x) 6 #Z+^(Q) in any bounded domain Q C En
and that max|^q(-*)| < “>•
The solution of the Cauchy problem can be obtained as the limit of a se¬
quence of soiutions-of tbe first boundary value problem for equation (0.1) in cylin¬
ders Qj. - Q x (0, T) under an unlimited dilation of tbe domain 0. Namely, we
suppose that iQ^l is a sequence of expanding domains with smooth boundaries
SN that tend in the limit to all of En. In each of the cylinders ~ OF x (0, T)
there exists by virtue of the theorems of §6 a unique solution u^ix, t) from
H2+% '*%(Qr\ S?) fl //“' * (Qr)
of equation (0.1) that satisfies the conditions
«"\sp = %(x)\s?- (8-2)
From the results of the preceding sections it follows that the solutions are
uniformly bounded:
max|(8.3)
0?
and, furthermore, for any Qj. and with N > m we have the estimates
I <«(«). (8.4)
■vr
in which the constants c (m) depend on m but not on N. By employing the usual
diagonal process we can extract from \u^\ a subsequence li/^l that converges
together with the derivatives at each point of Rj. to some func¬
tion u and its corresponding derivatives. It is clear that u(x, t) does not exceed
M, belongs to H21 *^2(QT) in each finite cylinder Qj. C Rp and satisfies
equation (0.1) and the initial condition (8.1); in other words, it is a solution of
the Cauchy problem for (ttl) in RT.
494
V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
If one assumes that the constants in conditions b)-c) of Theorem 6.1 do not
depend on the dimensions of Qf (i.e. conditions b) and c) are fulfilled in the
whole strip R j.) and W 6 H2* ^(En), then the constant c{m) in (8.3) will not
depend on m, and therefore tbe limit functions uix, t) for I1 will belong to
In order to prove the uniqueness of such solutions we assume in addition that
the functions
M*.p)^*a‘(X'u-p)>
1 (8.5)
A(x, t, u. p)s=a{x, t, u, P) — ^LPl — ^L
ate differentiable with respect to it and p, where the derivatives da^/dp,
da-^/du, dA/dp are bounded in modulus, while dA/du is bounded from below in
any finite domain of variation of (a, p) and (x, t) € Rf.
Suppose, moreover, that
ai; (*, t, u, p)(i€j >0 (8.6)
for t £ (0, T] and arbitrary *, u, p. Under these conditions it is possible to
apply estimate (2.23) in Theorem 2.5 of Chapter I to the difference u(x, t) =
ti'{x, f) — u"{x, l) of two possible solutions u and it" of problem (0.1), (8.1) that
are bounded, together with their derivatives of first and second orders.
Indeed, uix, t) is a bounded solution of the linear Cauchy problem
ut — alf(x, t) u ^ -{- a. (x. 1)11^ + a(x, t)u = 0,
H-o = °-
where
au(x, () = alJ(x, t, u'J;
f da, ,\x.t, o'(x,i). ul(x,t)\ C
0—<,,, j r^—+ J —L-ldr;
D xj U xi
O(x,0=-«" - Vi -*■ + 7rr~“
*l*) J da1 J da1 1
ax(x. t) = xu' (x. 0 + 0 —x) «" (x, t).
§8. THE CAUCHY PROBLEM
495
with die functions aj -, a(-, a being subject to the conditions of Theorem 2.5 of Chapter
I, and therefore u(x, t) s 0 in Rj. by virtue of estimate (2.23) of Chapter I. We
formulate the assertion just proved in the form of a theorem.
Theorem 8.1. Suppose chat the following conditions hold.
a) tjr0(x) £ H2*@CSX) in any Q C En and maxg^ I^q W| <
b) For t 6 (0, r] and arbitrary x, u, p inequality (8.6) and the inequality
A(x, t, a, 0)u>> — 6,«3 — b2. bt, bt — const >0,
are fulfilled for A(x, t, u, p) defined in (8.5). The latter inequality can be re¬
placed by the more general inequality
CO
A(x. t, u, 0)«> — 4>(|a|)|a| — b^.where J ^L._oo;
c) For any bounded cylinder Qj> of the layer Ry> and |u| < U, where M is
a constant from (8.3)t the functions a>(x, t, ut p) and a(x9 t> u, p) satisfy condi¬
tions b) and c) of Theorem 6.1 with the constants depending in general on Qj*.
Then there exists at least one solution u(x, t) of the Cauchy problem (0.1),
(8.1) in the strip Ry. that does not exceed M in modulus and belongs to
H2*!3,l+0/2(QT) f0T
any bounded cylinder Qy C Rp. It will be an element of
+ ) if in addition it is assumed that the constants in conditions
b)—c) of Theorem 6.1 do not depend on Qp.
If the functions a^-{xf t, u, p) and A(x, t, u, p) defined by equalities (8.5)
are differentiable with respect to u and p and obey conditions (8.6) and
j dan (x, t, «, p) dan (at, /, u, p) dA (x, t, «, p) I
35 ‘ ’ dp I
dA (*, t, u, p) ^ rfcrs
mm V ' ^ > — n2 (AO.
<jr. mnT,
IB, p) < N
for an arbitrary N and some constants ji j, [t2 depending possibly on N, then
problem (0.1), (8.1) in Rj has no more than one classical solution u(x, t) that
is bounded in R j together with its derivatives of first and second orders.
Remark 8.1. If condition a) is replaced by the weaker requirement of contin¬
uity and boundedness of <ft0(x) in En, then under the same assumptions b}-c)
496
V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
concerning the a( and a there will exist a bounded solution u(x, t) of problem
(0.1), (8.1) that belongs to p| C{Rj) lot any finite cylinder
Qj C Rj. In order to prove this assertion it is necessary to approximate (x)
by smooth bounded functions (x) converging to il/g (x) uniformly in £R,
and to take advantage of Theorem 8.1. The fact that the sequence of solutions
u(mHx, i) ot the auxiliary problems will converge to a continuous solution in Rj
of problem (0.1), (8.1) can be proved by means of the method of barriers.
Remark 8.2. it is not hard to see that the existence and uniqueness theorems
for the Cauchy problem (0.1), (8.1) are preserved under tbe assumption that the
known and unknown functions in problem (0.1), (8.1) have a certain growth when
|x| —* oo (see in connection with this the papers [123c, 63,62d] etc.).
§9. ON THE STEFAN PROBLEM
Let us dwell a little on the Stefan problem (more precisely, on one of the
Stefan problems). We will show that it can be reduced to the determination of
generalized solutions of boundary value problems for certain quasi-linear equa¬
tions of simplest form but with functions that are discontinuous in u. This is
done in primarily the same way as in §13 of Chapter III for a diffraction problem.
In either case we "hide” the first of the conditions on discontinuities in a require¬
ment of continuity (in one sense or another) of the desired solution, and the
second, in an integral identity in such a way that the surfaces of discontinuity
do not appear in the identity. This circumstance in Stefan problems has signifi¬
cance since tbe surfaces of discontinuity in them are unknown.
Let us consider, for example, the problem of determining the temperature
u(x, t) in a domain 0 of variation ol x - (xj, — , xn) for f € (0, T} when there
are changes of phase at the temperatures Uj, ■ ■ •, um- Let Uj < < • • ■ < um.
In those parts of Qj = ft x (0, T) where u(x, l) does not have the values
ul’ " ’' ’ um must satisfy the equation
a(«) at — t/i(u)ux \ =0, (9*1)
v ‘ xi
in which a(u) and k(u) are known positive functions that are smooth on each of
the intervals [uk, + and have discontinuities of the first kind at the points
u = u^, k - 1, • • •, m. On an interface of two phases we must satisfy the
two conditions
[«]|s(i) = 0 (9.2)
§9. THE STEFAN PROBLEM
497
and
cos (n, f)+ [*(«)«,,] cos (n. •K,)liSw =0* C9-3)
where the are given positive numbers, a is the normal to tbe interface 5^ =
|U, t): uix, t) = u^t in the direction of increasing it (along the gradient of a),
and the saltus [«J| is the difference between the limiting value of t; on
when approached from the domain )(x, t): u < it*} and the limiting value of v on
S(4) when approached from the domain i(*, t): u > u*l.
Finally, uix, t) must satisfy the initial condition
«|t = 0 = ^0^ (9‘4)
and some boundary condition, for example,
»|sr-°- (9-5)
By the same (classical) statement of the problem it is required that the sets
be piecewise smooth surfaces in the ix, e)-space. As a preliminary we simplify
the problem by introducing in place of it(*, t) a new unknown function
n (jr. /I
V{X, t)~ J k{\)di.
t)
For this function we have, from (9.1),
/3(v)vt - Aw = 0 (9.6)
with /3{v) possessing the same properties as a(tt) (i.e. f3(v) Is a positive piece-
wise smooth function having discontinuities of the first kind at the points Vj, ■ ■ ■
— > vm); the conditions on the surfaces ((*, t): vix, t) = v^} (these are the same
surfaces Sf-^) take the form
^ (9.7)
*» COS (n, t)+cos(n. x,) | 4) = 0. (9.8)
while the initial and boundary conditions retain their original form
v i,.o = 't’l (•*) aod v lsr ~ (9-9)
In the same way as in the problems of §13 of Chapter III we Impart to equation
498
V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
(9.6) a divergence form
^~i _At> = 0, (9.10)
where b(v) is a monotonically increasing piecewise smooth function, the deriva¬
tive b'(v) of which is equal to /3(f) on each of the intervals (vj, t>£ + 1). In order
to determine tbe behavior of b(v) at a point of discontinuity v = vk we pass from
equation (9.10) to an integral identity. To this end we multiply equation (9.10) by
an r){x, t) from ff'J’* (Qj) that is equal to zero on the boundary of Qy, and in
each term we carry out a single integration by parts, transferring the derivatives
d/dt and <?/<?*,- to r/ and bearing in mind that the function b(v) can have discon¬
tinuities oo the surfaces = |(*, t): v(x, t) = v^l, 4=1, , m (which we as¬
sume to be piecewise smooth). This gives us
Jl-
• b(v) ti + v ti } dxdt
*1 i*
Or
+ 2 J {l* («)] cos(n, t) — cos (n, *,)] T\ds — 0. (9.11)
*-1 sw
In this connection we have taken into account the choice of the direction of n
and the rule of composition for tbe saltus [ ]. If the function b(v) is defined so
that [£>(»)] = - 6j, then all of the integrals with respect to the surfaces 5^
vanish by virtue of conditions (9.8), and the identity (9.11) takes the form
Jj—b (v) r)( dx — 0. (9.12)
dr ‘ ‘
Thus we have shown that if v{x, i) is a classical solution of problem (9.6)-(9.9),
and consequently has critical values only on the piecewise smooth surfaces 5^,
then it satisfies the identity (9.12) with any rjix, t) from V^’1 (Qy) that is equal
to zero on the boundary of Qy, and with any monotonically increasing function
b(v) having at the points v^, k = 1, • • •, m, the saltuses bp and on the intervals
(«£, t>£ + j) a derivative b'(v) equal to f3(f). The function b(v) is determined
by these requirements to within its values at die points vk, k = 1, • • •, m, and to
within an arbitrary summand, which we fix by assigning a value to b (v) at some
point t>0 that is different from t>j, ■ ■ ■, vm. We denote by B(x, t, t>) an arbitrary
measurable function that is equal to b(v) for v £ v^, k = I, ■ ■ ■, m, and (a, e) €
Qy, and that for v - and (x, t) £ Qy has values from the interval
f lim b(v), lim b(v)'
[vh^-ci v~*vb*o
§9. THE STEFAN PROBLEM 499
(they can be different at different points (x, i)). The classical solution v(x, t) of
problem (9.6)—(9.9) satisfies die identity
f f— B (x, t, v) -f- ^jc^j dxdt — 0 (9.13)
«r ‘ ‘
with jj from V^’1 (Qj) and equal to zero on the boundary of Qj., and with any
such function B(x, t, v). Since the measure of the sets where v(x, t) has
the critical values «jj, —, vm is equal to zero, the infonnation which we obtain
from (9.13) is the same for any of the functions B(x, t, v).
By a generalized solution of problem (9.6)—(9-9) we will mean a bounded
function v ftom that is equal to t/r^ix) for t = 0 and satisfies the
identity (9.13) for any one of the functions of type B(x, t, v) and the above-indi-
cated jj. We will show that problem (9.6)-(9.9) can have no more than one gen¬
eralized solution. Moreover, we will do this for a wider class of generalized solu¬
tions, concerning which it is known a priori only that they are bounded, and not
that they ate elements of These solutions, in accordance with our con¬
ception of the definition of generalized solutions, are bounded functions satisfying
the identity
J tt v) ii, + v Ail] dx dt
Or
+ jB(xt t, ih(*))n(*. 0)dx*=0 (9.14)
0
° _
for any function ij from IT y iQf) that is equal to zero for t ~ T. Suppose there
exist two such solutions v and u", and suppose the initial state tfr^ (x) is such
that the critical values v^, k = 1, • ■ ■, m, ate taken by xjty (x) on a set of
n-dimensional measure zero in the x-space. We subtract the identity (9.14) for v
from the identity J9.14) for v , denoting the corresponding functions B(x, t, v)
by B" and B', and we write the result in the form
f [B' (x. t. ©') — t, o")}
Qj. ^
X { + fl, t, V") — ft" (X, t. v") An } dx dt
or, more coocisely,
J ^ (*. 0 K + « (*. t) Ar|) dx dt — 0. (9.15)
Or
500
V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
By virtue of the properties of the functions B‘ and B“ the function a(x, t) is
oonnegative and vrai t) s aQ <~. We take as r)(x, t) the solution of
the problem
0+e)A»|=/J(je. t). tlL=r = 0. *1 ljr = 0, (9.16)
in which f is a small positive number and Fix, t) is an arbitrary smooth function
that is finite in Qj,. By virtue of what was said on pages 179 and 180 such a
solution rfix, t) exists and estimate (6.25) of Chapter III is valid for it, i.e.
J (W + (° +-*>(At)8)2) dx dt (T) J F\ dx dt (9.17)
Qt Or
with c{T) depending only on T and aQ. *)
Substituting if ia place of tj in (9.15), we obtain
J B{P~-ii&if)dxdtz= 0. (9.18)
Qr
If we let e tend to zero in this equality, we arrive at the equality
J
BF dx dt = 0. (9.19)
Qt
Be At)8 dx dt
Qt
vrai max | B |
Qt
J a + £
Qt . Qt
dxdt (a-t- z)(^)'idxdt
for ( —>0. But from (9.19), since F is sufficiently arbitrary, it follows that
B ix, t) m 0, and hence that v and v" coincide for almost all (x, t) from Qq..
The uniqueness theorem is proved.
The existence of a generalized solution for any bounded function from
^(O) is obtained on the basis of some of the theorems in §6. To this end we
approximate the function b(v) by averagings bpiv) of it, obtained with the use
of an infinitely differentiable nonnegative kernel "*<l i>|) of radius p, and tbe
I) We note that if 5 € 0^ then V( will be an element of (Qj<); otherwise it
belongs to (Qf) and the integral (&r}*)^dx dt is finite for it. Such an r/e(x, l),
as is easily seen, can be taken as r\ in (9.15).
§9. THE STEFAN PROBLEM
501
function ^r^(x) by some uniformly bounded and infinitely differentiable finite (in
Q) functions that converge to it in the norm of f ^(Q). If S G fl**a then
we also approximate S by surfaces Sp of the class J?2 + a. The auxiliaiy problems
— At» = 0. v 1^ = 0. c|,_o = 'i’f (9‘20)
have solutions vfl ftom #2 + a,l + a/2 (Qf>) by virtue of Theorem 6.1. Tbe uniform
estimates
max | v" | < max | <|>f | < e,
<%■ xi.sP
and
(9-21)
are valid for them, where i/j is a lower bound of the functions b'p (v) on the in¬
terval [- c, c]. The first of them is a special case of inequality (2.31) of Chap¬
ter I, while the second is a special case of inequality (6.6) of Chapter III. The
number i/j is positive by virtue of die strictly monotonic increase of the function
b(v). The inequalities (9.21) permit us to choose a subsequence |/H k = 1,
2, , which converges almost every where in Qj. and weakly in * (Qj) to
some bounded function v from W2’l(Qf)- Furthermore, the p^ are chosen so
that the sequence of functions bp^(vP^ (x, t)) converges weakly in L2(Qj) to
some function b {x, t). This is possible since the functions bp (,vfi(x, t)) are uni¬
formly bounded. The identity (9.12) is valid for each of the v^k. By passing to
the limit in it as k —► », we arrive at the identity
J {— b (x, t) ri( vxi\x \dx dt = 0. (9.22)
Or ‘ 1
»v.
It is easily verified that b (*, t) is a function of type B{x, t, v(x, t)) and conse¬
quently v(x, t) is the desired generalized solution of problem (9.6)-(9-9).
We will show that outside of the sets where V{x, t) * the smooth¬
ness of die function v(x, t) (and hence also of u(x, t)) is determined only by
the smoothness of the function b(v). Suppose the values of the function v(x, e)
on a set Q' = O' x [tj, tj] belong to the interval [t»ft + e, »t + 1 -*], e > 0, and
b(v) is a smooth function on it. On Q' the function B(x, t, v(v, ()) is equal to
b(v(x, t)) and possesses a derivative with respec.t to t from L2(Q'). Conse¬
quently, the identity (9.13) for r)(x, t) equal to zero outside of Q' can be
502
V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
transformed to the form
J [6' (•») vti) + «* 11, ] dx dt — 0.
This identity indicates that, for almost all t from [(j, t^], the function v is a
generalized solution from {£3*) of the equation Av = b'(v)vt with free term from
Z-2 (£&*)• By virtue of Theorem 10.1 of Chapter III in [65q], for such t the solution
v has derivatives with respect to * of second order, and, for any interior subdo¬
main 0" of the domain fi',
|| v (x. t) ||<%, < c {d") || b' (v) v, ||2i B.. (9.23)
where c{d“) depends only on the distance from 0" to the boundary of O'. It fol¬
lows that
IIv li^V ^ C ^ II V W V‘ H2. O' < °°> (9-24)
where Qn » flwx [t^f Thus v is an element of IFor almost all
points (x, t) from $Mit satisfies the equation
b,(v)vt~~ Av =» 0. (9-25)
The function w(x, t) = b(v(x, 0), as is easily seen, satisfies by virtue of (9.25)
the equation
W( i)^ = 0. (9.26)
Since tbe function l>' (vix, t)) satisfies the inequalities 0 < i/j < b' (v(x, ifi <y.^
on Q" and w 6 ^'2 [Qn), by virtue of Theorem 10.1 of Chapter III the function
u>(x, t) is an element of Ha>a/"2 (Q“) with some a > 0. But then v{x, t) also be¬
longs to Ha’a/<2 (<?")• If we assume that the function b'(v) is Hdlder continuous
on the interval [v^ + e, vj^+i ~ t]» £ > 0, then b'(v(.x, <)) will be an element of
fja,a/2^q«) wjt[, some a > 0, and hence v(x, t), as a solution from
of the linear equation v( - (l/b' (iAx, «)))Au » 0 will be an element of
^y2 + a, 1 + a/2(lj") (see Theorem 12.1 of Chapter 111). The solution v(x, t) will
be a smoother function fot smoother b(v); this is guaranteed by Theorems 12.1
of Chapter III and 5-2 of Chapter IV on the solutions of linear equations.
Thus we see that the existence and uniqueness of generalized solutions of
the Stefan problem (the case considered admits certain generalizations) is estab¬
lished in primarily the same way as the unique solvability of the diffraction prob¬
lems in the generalized statement in (see §13 of Chapter III). This was
§10. ESTIMATING THE HOLDER CONSTANT
503
done in [55] (see also [9M]). But the question whether ot not these solutions are
classical and, in particular, whether or not the sets S^ of points at which
v(x, t) is equal to one of the singular values form piecewise smooth n-dimen-
sional surfaces in the (*, e)-space remains open and its solution is of unquestioned
interest.
Id conclusion let us mention the works [102, 108, 31, 58]etc., in which one-
dimensional Stefan problems in their classical statement are studied. The methods
of these works are different from the methods presented here. In them the prob¬
lems are reduced to an investigation of nonlinear integral equations. In [31] and
[102b] there are solutions of some of these problems in the large.
§10. ANOTHER METHOD OF ESTIMATING
THE HOLDER CONSTANT FOR SOLUTIONS
In the present section we will estimate Hdlder constants for generalized so¬
lutions of equation (0.1) by a method that is different from the one presented in
§1. This will be done without making use of the theorems of §§7—8 of Chapter
II on functions of the classes 3 2' The method to be used is an extension to the
general case of the method described by tbe example of an elementary linear equa¬
tion of parabolic type given at the end of §10 in Chapter III.
We will give here another proof of Theorem 1.1 (more precisely, of its first
part) concerning interior estimates. Estimates near the surface Sj. are estab¬
lished analogously, but with the enlistment of the lemmas of Chapter II relating
to the behavior of functions in cylinders intersecting with the surface Sj.
Thus, suppose the conditions of Theorem 1.1, viz., the inequalities (1.1)—
(l.j), are fulfilled, and suppose u(x, t) is a bounded generalized solution of the
class Qj) of equation (0.1), so that uOt, t) satisfies the integral identity
(1.6) for any bounded function tj(x, t) from W^’^iQj,). From (1.6) and conditions
(1.1)—(1.5) there follows the validity of inequality (1.7) for a{x, ():
J -f J J [— if), + at (x, t. a, ux)r\x]dxdt
a '■ i, a
t>
< f J + t)\\n\dxdt,
<*, a
0< *,<*;>< T.. (10.1)
504 V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
01 1
where tj (x, t) is a n arbitrary bounded function from IT (Qf)> while the func¬
tions o; and cf>2 satisfy restrictions (1.1)-(1.2), (1.4)—(1.5). As in §1, this in¬
equality is the starting point for all of the subsequent arguments. In §1 we de¬
duced from it that mat u{x, t) is subject to inequalities (7.5) of Chapter II:
llW<‘V <» 4 T)C(X, *0 + T)H2. Kp 4" vIf /:2t <J (p, t)
< || wm (x. t0)£(x, t0) fl’ ^
+-Yi( f (■&'*'?{£ + l\l,\)dxdt
U<p,t)
/ ’
1-0+*)
t
'?T / <■ \?
+
J I J l£ld-*| dt
*-'• \p« /
for any values of k > vrai max^^ r^«>ix, «) - S. Here
x = — x,. q = -%v-(l +k), r=_iL_(l -f-x),
ft " * /■ — I
8 is a positive number determined by if = vrai max^|u|, », y, fi, /ij, ft2, r and
?> (Hp. r) ” Kp x (<q, tp + r) is an arbitrary cylinder contained in Qj., A^^ p(t)
is the set of those points x from Kp at which mix, t) > k, and £{x, t) is an arbi¬
trary smooth function that is equal to zero on the lateral surface of Qip, t). This
guaranteed the membership of the solution uix, t) in the class %2iQj, V’
from which the desired conclusion was drawn concerning an estimate of the
Hdlder constant for uix, t).
As already noted above, we will not employ here the theorems on functions
of the classes Sj. ^ all of the propositions proved in §§7—8 of Chapter II we
will make use of only the simplest of them, Lemma 7.1, which follows from in¬
equalities (10.2).
The estimation of the Holder constant will be carried out in the following
way. We consider an arbitrary cylinder Q2g = K2R x - *o^> w**ete *^*e
constant 6> 0 will be fixed later, and the coaxial cylinder 9ft 55 x (*o ~ 9R2/ 4, *0)
and we prove the validity of the inequality
osc {“• Qffl <(1 -6,))OSC {a, Qw) + /?*■, (10.3)
with die constant Sg > 0 being determined by M - max^lul and the known
quantities from conditions (1.1)—(1.5). This by virtue of Lemma 5-8 of Chapter II
§10. ESTIMATING TOE HOLDER CONSTANT 505
gives the desired estimate for \u)(q), Q'CQj*.
If osc to, QgI < S*1, then (10-3) is valid. We assume that oscla, Qg\ > J!*1,
and we introduce the notation o> - osc (a, Obviously o> > if*1. Without loss
of generality we can assume that in the function uix, t) varies within the
range 0 < uix, l) < 6). At least one of the two following inequalities is.always true:
mes | x £ Ka : u (x, t0 — 0R2) ■< j } >• -j mes K# (10,4j)
or
•mes | x € Kr : o — a (jc, t0 — 0/?2) mes Kk. (10.42)
If die first of them holds, then we deduce from (10.4 j) and inequalities (10.1)
with 17 > 0 that die difference in between uix, t) and <u is not less than a
certain fraction of <u. If on the other hand (10.4Z) holds, then instead of n(x, t)
we take the function uix, t) = a> - uix, t). Besides (10.42) inequalities of type
(10.1) with t) > 0 are valid for it, as follows from (10.1) for uix, t) and the fact
that if(x, t) can have any sign in (10.1). Therefore, as with uix, t) in the first
case, the difference in Q^ between it and its maximum in Qjr (*-e- &>) is not
less than a certain fraction of to- Thus in both cases we arrive at (10.3). Suppose,
for example, (10.4j) is true. Then a fortiori
mes |x£Kk : u(x, t0 — 0/?J)<(l —m,)(o| >---mes
where ft0 = mini 1/2, S/2M\ < min{l/2, 8/<ul. According to Lemma 7.1 of Chapter
II there exist for any ( G il/y/Y, l) positive numbers 0(f) and b (<f) such that
for all I € [ij - OR2, tQ]
mes [x g K„ ■■ a {x, t) < (1 —Ho)® 4- |W) > b (|) mes Kg, W0.5)
as long as
H= max uix, t)—-(1 — m)®
*C*I>
/„!
does not exceed § and H > R* The first of these requirements on H is ful¬
filled since H < a> - (l - while < S/m. If the second requirement on H
is not fulfilled, i.e. H < i?K1, then u(x, t) < il - jiQ)ca + R*1 and hence
(10.3) is fulfilled with SQ = fig. If on the other hand ff > RK1, then we use the
506
V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
assertion of Lemma 7.1 of Chapter II. As £ we take, for example, 7/8 and corre¬
sponding to it 6 = 6(£) and b = 6(f). Since (l - n0)a> + 7H/8 < co - nQa>/8, it
follows from (10.5) that
mes j jc £ : u (*, /)<;<» — to j > i mes KR
t£{t0~QR\ /y.
Consider in QjR t^le function vix, t) = t/>(u(x, t)), where
(10.6)
$(«) =
In!
It is bounded from below by the number -ln(l6/ft0) and is obviously nonpositive
at those points where uix, t) < co — ft^oi/8, and this set of points constitutes ac¬
cording to (10.6) a definite fraction of for any t 6 - 0R2, iQ]- If we prove
that maxq^ v(x, t) < Mj, then this will guarantee the estimate mauix, t) <
o)(l - /i0e Ml/8) + R*1, and thereby also (10.3) with Sg «■ ^l/8.
Thus, in order to obtain estimate (10.3) it is sufficient to show that v(x, t)
is bounded in Q^ from above by a quantity determined by M, v, n, 9> r
only and not depending on R. The estimate of vix, l) from above will be based
on the use of Theorem 6.2 of Chapter II. Namely, we will establish the estimate
*3
rt — n—2
J J v1 dx dt c
KiR
1 2
and prove that the function vix, t) satisfies the inequalities
'o
max J (o(")J)2 dx 4- J J | ^S,[2S2 dx dt
<„U ,o_t Kf>
*0
<Y J j\^>p(tl+^li,\)dxdt
0
r
J (mes A„'P(t))<i dt
(10.7)
<l+x)
(10.8)
for k exceeding a certain value kg-, here fiit) = 1*6 Kp: vix, t) > Al, £(*, ()
is an arbitrary smooth nonnegative function with £{x, t) < I that is equal to zero
§10. ESTIMATING THE HOLDER CONSTANT 507
on the lateral surface and lower base of tbe cylinder Q(p, r) = Kp x (lQ - r, £q),
<M?2/4 < r < 36R2/4, R<p< 3R/2, and q, f and k are determined, as in (10.2),
from the relations k = 2«j/b, q = (2q/(q - l)) (l + k), r= (2r/(r - l)) (1 +■ n)- We
can easily see, by using assumptions (1.5), that the parameters r and q are sub¬
ject to the restrictions (3-3) of Chapter II.
The desired estimate for vrai max^ v(x, t), and thereby inequality (10.3),
will follow from (10.8) and (10.7) on the basis of Theorem 6.2 of Chapter II.
We begin with a derivation of inequality (10.7). To this end we first estab¬
lish the following lemma.
Lemma 10.1. If u(x, t) 6 Vj'® (K2/j x ^)) ■ = M
and for all^ tJ € (0, 7") inequalities (10.1) hold with an arbitrary bounded nonnega¬
tive rj G IPl"! (Qf) while the at- and <j>2 satisfy conditions (1.1)—(1.5), then for
any < tj from [0, D the estimate
*2
J j ulS>dxdt
'| *2/?
< cRn + c(t2—ts)R" [max & + (it - /,)^ (10.9)
holds, where the quantity c depends only on n, M, v, fi, ft|, ^ r, while
£ = £(*) is a smooth function that is equal to zero on the boundary of the ball
^2ft’ with 0 < £ < 1.
To prove this we put ij ~ («^U*£2)£" in (10.1) and transform the term contain¬
ing the derivative r\t in the following manner:
'a
u (e^X2)*! dx
<1 k2R
l2
= f J h*,/X2 dx- f \a(x,t+ k) ex“i>i\ dx |'J
h Ki/t k?r >l
“X / — J l“(*« 1 -t A) e>J'h^X dx l'5 . (10.10)
Now in (10.1) as transformed by means of (10.10) we can pass to the limit for
h —> 0. As a result of the passage to the limit and of estimates made with the
use of conditions (1.1), (1.2), (1.4) and (1.5), we obtain
508 v. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
X" J eKu^ dx f* + vX f / tip dxdt
'I K2R
f J* (M-1aijrI~MPi)£iSjc
'i *2* 1
<| J [(H+*P? + %)P+£i+ j-\^+^p]dxdt.
‘1 *2#
(10.11)
Choosing A = 4t'-Vi ‘n (10.11) and making use of die condition
vrai max |u(*, t)| < M
Qt
and (1.4), we obtain
/j
» » 4)xtM
Jt,e v J J updxdt^~e v y.nTRn
h *m
n s i 1 , *
+ e v [lh 2j (t2—tt) ~'(x,2’R*) ~
+C1 + lr)max ^ v* ~ '
KU) J
from which follows (10.9). Lemma 10.1 is proved.
Let the ball K2n have its center at xQ.
Consider the inequality (10.1) with «t > efl - OR2 and t2 = cg. We put
t) m (t/t’ (uh ) 3l2(x)y2 (<»£- in it, where
1. 0<| X — XoKJ-/?.
St(*) = 5R(|*|)= 2^-i--^^1), jR<\x-x0)<£2R,
0. 2/? -< | x — xa |;
[ o, *<«„—e/?2,
x(0=i vpV-to + w*)- h— QR2<t<t0 — jM2-
1. t0-~QR*<t<t0.
We carry out an integration by parts with respect to t in tbe second summand
of (10.1) and we reduce its first term to the form
§ 10. ESTIMATING THE HOLDER CONSTANT
509
J “fti'f1' (“») Si* (*) X2 (0 ^ dt — J (a**i|> (a„) SR2*^ d* f*
<?2# AT2r
— J J («+VK)J?Y)A-^f‘
*2«
— J 2®*9tJxx'rfxrf/,
Qz#
where u+* = u(x, t + h) and v* =
We then pass to die limit in (10.1) as h —* 0. The legitimacy of this passage
is easily established by means of the lemmas in §4 of Chapter II. As a result we
obtain the relation
J J a, («,y9iy + 2VmXlx*)dxdt
Km <hR
< J [MI+ %(*. t)\VW%*dxdt
Out
+ J2®9l2xx'<**<«• (10.12)
QtR
We carry out estimates in (10.12) by means of inequalities (1.1)—(1.2) and
(1.4)—(1-5), taking into account tbe definition of v(x, *):
J t>9rl2x2 **•* | -t~v J »J9t2X2 <<■*
*2i? Qir
< f 2vW%x' dxdt
Qvt
+ J [ <Po>t>"^sX2 4- |5ir|X2GM«rl + <Pi)
<?2#
4-(H,«2 + <P2)>t>W]^^.
We estimate the first summand of tbe right side of this inequality by means of in¬
equality (1.2) of Chapter II:
J 2v’R2ix'dxdf-C™ J vWx*dxdt 4-~ J W*dxdt
<hn Out
and we take advantage of inequality (5-4) of Chapter II. By virtue of assumption
510 V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
(10.6) and the fact that v > - In (16/fxg), this gives us
J 2v^l2xx'dx dt ^ ep3 J v2^ dx dt + ^ R“.
QiR 02R
Ps = P2 (*>£„)-Vn+3.
The remaining terms are estimated as follows:
J wTW dx dt < R~'2Xi J %'Jiy dx dt < c2R" || % ||^^
C2« Oft?
2 J ili'SR 15y x2l “* I rf* <# < e J ^X2
$2R QiR
J Wxt2dx dt<e J dx dt 4- ~ R".
QlR ®2 R
f </?"*' max |9tx| f «pt djc dl?
Qvt KiR
c^Rn II Vl It}. 2r. Q21),
J u^’Wx1 dx dt — J\vx\\ux\^h2dxdt
Qm 02*
<4 J v>xydxdt+ ~ J a^dxdt.
Qir 02«
By Lemma 10.1 it follows that
J «J*|>'9i2X2 dx dt < J vixJl’1%7 dx dt 4- R";
Q2r Qm
f W dxdt^R" f dxdt^CsR'"' *' ||«, II,,«,
L <4
Upon reducing similar terms after these estimates, we obtain
| 4 v j" v7JSl’2'l3 dx dt
k“ir QiR
< e ^3 -f- H +- j (i,) J v-Jkh^dxdt
QiR
4_ (_|l 4- a || <j>01|?> r, +- n -j- 4- 2c4 11 <p, ||2?i ^ <J?
■+• l1! -y- + II <p2 II,, ,, Qj.) fi"-
(10.13)
§10. ESTIMATING THE HOLDER CONSTANT 511
Let us choose e = v/2(/9j + fij + fi/2) here. Then from (10.1}) we obtain
J vimYdxdt^CjR",
QtR
which together with the inequality
mes (x£Kr] v(x, f)<:0} >Ames K#, t£\t0 — QR1. /„],
implied by (10.6), the inequality (5.4) of Chapter II and the estimate v >- In(l6//tQ)
gives us
i8
*0
+ &R2 J J v]m2dxdt^cR',+2
Thus estimate (10.7) is established. It remains to show the validity of inequalities
(10.8). To this end we put in (10.1) =■ - r, t2 6 U0 -.r, tg] and
n(*. 0= It'(«*)maxl^f(«*)—*. 0j&2Jj = Hj!/(aA)^'S^,
where £ is a cutting function for the cylinder Qip, r) = Kpx itQ - r, t0), and we
transform the term with 17 in (10.1) to the form
J f - J
= J f [-0* <*>£]2 rfjc |/l — J f IvWyiZ, dx dt
'-**p
- j «>V <*>£%-f’.
*p
we then pass to the limit as A —» 0 and carry out estimates in the resultant equal¬
ity by means of conditions (1.1)—(1.2) and (1.4)—(1.5). As a result we will have
an inequality of type (1.10), namely
512 V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
4- max f
KP
+j J [jvFF**V + ltppt?]*xdt
Q (t>, t) ^
< J J f folP(*e*>*'+
0(R fl <.-r Ati p
-4-2$V*>Ct^|(|i|«J| • f <Pi) + Vv(k)£(H,a| + <P2)]dx dt, (10.14)
where Ak^pU) = t*€ Kptv(x, t) > ii-
We estimate tbe summands of the right side of (10.14) in the following manner:
J 1|>" («) dx dt
Q<p, t)
I
r. Q(p. t, q,p, p- *»?
< f «<* + ,
X(|i(ft. p. T))r
2 (i + it)
(here and below
|*(&. p. t)~ J me$|i4ft(P(0)V4f/
/,-t
with q and r” defined above and k = 2k j/n).
r r t. J-0+*)
J ] <wrf?dxdt<R--2x'||%S||,.r, Q(p>t)nf (*.p. T).
2 J mj>'ti<*>£|£.r||a.r|</jE*rt
<e J <** + -£ J (*«>)*
C<P. t)
<?((>, t) <?<p. t)
§10- ESTIMATING THE H5LDER CONSTANT 513
2 J </*<«< j (v^YQdxdt
QU-ti Q(p,x)
U
4-/?-*“■ J J <f$?dxdt^ J (v^fQdxdt
'•~T «(p. t)
+ P.T).
Qto x)
J Qti^dx dt
“ft J {vx'V£‘v(k)W'ytdxdt
<?(P. T)
"la* f |vPfgdxdt.
x) «(ft v
Since f»0<S/<u and max^ ^ r) < e-1-*,
J dxdt<C ■g-mte-1-* j* tyx'Y&dxdt;
QIP,T1 <?(ft t)
J *V*> C*ft <**dt < #r * II*<*> til
C(R fl
i (l+2x|
X ll^ll,, r, Q,p. »>>*' <*■ P- t><-2 11^ EH?, r. Q<p, x)
+ f M.r.Qte.flf*' <*• P.*)-
We substitute these estimates into (10.14), replace Jv^^H “ pQ^/^r) ^ l^e
larger quantity fi |t>^ ^Iq^ r) (see “equality (3-4) of Chapter II), and take e
equal to (1/2(1 + fi2))min (1/4; v/2\. After the reduction of similar terms, this
gives us the inequality
min \ i; jll itux f lttk'Q*dx f
+ f W!Xfdxdt\ <Y, [ J (f(*’),(£|£,| +%®)dxdt
Q(P, X) J *-Q{£, X)
+ “T“(*. p. X) +4 PL,#-*-* I yptfdxdt (10.15)
Q(R fl
514
V. QUASI-LINEAR EQUATIONS IN DIVERGENCE FORM
which is valid for any A, with y j determined only by JI/, v, fi, /Zj, q and r.
We now impose on k the requirement, k > Ag, where kQ = In [(8/8)max (8;
Under this requirement inequalities (10.8) follow from (10.15)- Thus the first part
of Theorem 1.1 can be considered proved.
Remark 10-1. In the derivation of (10.3) presented above, instead of the as¬
sumption that u(x, i) is a generalized solution of equation (0.1), it is only neces¬
sary that uix, t) satisfy inequality (10.1) for any bounded 7)(x, t) > 0 from
V I’1 (Qj-) and ai and <£2 subject to conditions (1.1)—(1.2), (1.4)—(1-5) (see in
connection with this Remark 1.1 of the present chapter), and chat condition (10.4 j)
be fulfilled for die function u(x, t). The latter assumption could be replaced by
the condition
mes {jc £ Kp : u (x, t0 — 9R2) •< max a
QiR
&2osc [Wt | ^ (10.16)
with certain positive constants S2 and Sy Hie proof of estimate (10.3) from this
goes through with only a few obvious changes.
By taking into account what has just been said we can assert the validity of
the following lemma, which will be used in Chapter VI.
Lemma 10.2. If a function uix, t) from (Qt) has vrai I ■
< oof and satisfies inequality (IQ.I) for any bounded nonnegative rj(x, t) from
17 2**^7*) witk ai and $2 subject to restrictions (1.1)—(1.2), (1.4)—(1-5)*
and if condition (10.16) with S2* ^3 > ® *s fulfilled for u(x, *), then inequality
(10.3) is valid for u(xr *) with the constant > 0 determined only by the quan¬
tities M, 1/, ftq, r from conditions (i.l)—(1.2), (1.4)—(1.5)* ^ and By
CHAPTER VI
QUASI-LINEAR EQUATIONS OF GENERAL FORM
In the present chapter we investigate the quasi-linear equations of general
form
*»-«!/(*• *’ “• “^==0‘ (0J)
For their solutions u(x, t) we establish a priori estimaces of the norms
which together with the results of Chapter IV on liaear equations with smooth co¬
efficients and the Leray-Schauder Theorem lead to the classical solvability in the
large of the first boundaty value problem for the equations (0.1). These estimates,
as was shown by the examples in §3 of Chapter II, do not hold for (he whole class
of uniformly parabolic equations (0.1). To ensure their validity it is necessary
that we impose certain restrictions on the behavior of the functions oit{x, t, u, p)
and a(x, t, u, p) for large a and p. This is what we do in deriving estimates of
max |u,|. (Estimates of max Q y\}A were given in §2 of Chapter!. For them it is
also necessary that certain conditions be fulfilled.)
Estimates of the Hdlder constant in terms of M - juj and
Mi -■ maxp^,|uz| and certain numerical characteristics of the known functions
oi;(*, t, u, p) and a(x, t, u, p) are given for the whole class of nonsingular para¬
bolic equations. We begin the present chapter with a proof of this important fact
(§§1 and 2). In §1 we present interior estimates of for all of die solu¬
tions u(x, t) of equations (0.1) that are continuous in QT and have it, from
Ij2(Qt) and bounded derivatives ux from Vj’0 (Qr)- (The class of such gener¬
alized solutions u will be denoted by 58.) We are able to obtain these estimates
as consequences of the results of §9 of Chapter II on functions of the classes
since, as is shown in §1, the vector function ux = (uxi, ■ • •, uxJ belongs to
die class ®2n> ***e parameters of which can be calculated. In §2 we estimate
515
516 VI. GENERAL QUASI-LINEAR EQUATIONS
\u%/ w^°^e cylinder Q f lor it satisfying die first boundary condition.
The following section (§3) is devoted to the derivation of local and total
estimates of max |uz|. For solutions u of equations (0.1), as opposed to equations
of tbe preceding chapter, we have not been able to give an estimate of the Holder
constant (it)’^ (or at least of the modulus of continuity of a with respect to x)
in terms of mai^ |u| and known quantities, and hence in proving tbe solvability
of a boundary value problem it is necessary that we have an estimate of max |u
in terms of max |b| (and not in tenns of |u|^, as was done in Chapter V). In
order to obtain such an estimate we have had to impose an additional condition on
the functions °j/(*, t, u, p) and a(x, t, u, p). It is removed in §5 for the case of
a single space variable. Finally, in §4 we prove some existence theorems.
§ 1. A PROOF OF THE SMOOTHNESS OF GENERALIZED
SOLUTIONS OF CLASS ® AND AN ESTIMATE OF (bx)q?
Suppose u(x, t) is an arbitrary generalized solution of equation (0.1) of class
S, i.e. it is continuous, has generalized derivatives it,, uxx from L2(Qt>i its
derivatives ux are bounded in modulus and continuously depend on t in the norm
of Lj (O), and it satisfies equation (0;1) almost everywhere. We will show that its
smoothness with respect to (x, t) is actually determined only by the smoothness
of tbe a£j (x, t, it, p) and a(x, t, it, p) as functions of x, t, it and p, as long as
equation (0.1) is parabolic with respect to the solution in question. The whole
problem here really comes down to a proof of the Holder continuity of the deriva¬
tives ux, since if this is known, then the rest can be established on the basis of
the results of Chapters III and IV on linear equations with smooth coefficients. In
fact, it is possible to consider equation (0.1) as a linear equation with smooth
coefficients Of/(*, t)=aij(x, I, »(*, 0. «*(x, *)) and with free term o(*) =
a(x, t, u(x, t), ux(x, I)) and to reason in the following way.
If the 0,7 (*, t, u, p) and a(x, t, u, p) are Holder continuous functions of their
arguments, then the «(,(*, I) and a(x, t) will belong to with some
a> 0, and hence (on the basis of Theorem 12.1 of Chapter 111) each solution of the
linear equation will actually be an element of 0,1+a/2 (@ j-). If the functions
Oij(x, t, it, p) and a (x, t, it, p) possess a greater smoothness, then the membership
of u(x, t) in H2+a,1*a/2lQr) guarantees a greater smoothness of the functions
oi;(x, i) and 3(x, {), and this, by the same Theorem 12.1, means a greater smooth¬
ness of any solution of the linear equation, i.e. of the function u(x, t) in question.
§1. SMOOTHNESS OF GENERALIZED SOLUTIONS
517
In the present section we prove the smoothness of ux inside Q7 and we
obtain an estimate of for any cylinder 'Q'C Qj not intersecting Sj-;
namely, we prove
Theorem 1.1. Suppose u(x, t) is a generalized solution from the class SI of
equation (0.1), for which the functions a;^.(x, t, u, p) and a(x, t, u, p) are defined,
while the a t, u, p) are also differentiable with respect to x, u and p in the
domain
|(x. *)€<?»•• |a|<max|B(jc, <)|, | P | <max| ux(x, f) | = M,)
Qr Qr
and satisfy the condition
vl“ < aij (.x. 0. ux (x, t) )1,|;< n!2. v, (i == const >0. (1.1)
If for u = u (*, t) and p = ux (x, t)
I day (x,t,u,p) I , , , .
max 5- -<(1), I, j. R=1
Qt I apt I
I^ST’ B>
(1.2)
where
<*•»)
and j anti r satisfy relations (7.2) of Chapter III, then the derivatives , —
— ,ux are Holder continuous in Qj. and for any Q' C Qj, the quantity v*x/^i
does not exceed a constant c depending only on M j, n, v, p., (ij, q and r from
conditions (I.I)—(1.3) and on the distance from Q ' to Tj.. The quantity a>0 is
determined by Mj, n, v, p., p.q and r.
If the norm jux(x, 0) , fi > 0, is bounded, then the estimates < c,
are also valid for cylinders Q’=(l' x [0, 71, where (l‘ is an arbitrary interior
subdomain of 0. In this case the constant c depends on U j, n, v, p, fij, q and r
from (l.I)-{1.3), on the distance from fi' to S, and on \ux(x, 0)|^ and f$,
where the exponent a< fi.
Without loss in generality we will take the functions uxi = 1, 2, • • •, n, to
be normalized in the sense that their values do not fall outside of the interval
[0, l]: 0 < ux. < I, i = 1, 2, * • •, n. Otherwise in equation (0.1) we could replace
the functions u(x, t) by
5X8
VI.GENERAL QUASI-LINEAR EQUATIONS
u(x, /)-|-A4, 2 ■*/
*<*./)- m
aad conduct all of our arguments with tbe function v. It is obvious that v satis¬
fies an equation ot the same form as (0. 1) with the same properties of the a..
and a.
Let us show that the vector function U(*, 0 with components
A * 1
belongs to the class Sj (Qf, M j, Sj, S2> ^3> ®> «)» the parameters M
y, 8j, i52> Sj, r, 5 and k of which are detetmined only by the quantities M j,
v, /i, fij, y and r. By virtue of Lemma 9.3 of Chapter II it is sufficient for this
purpose to prove that the functions
A
•w' (x, ()= I0nar (x, (•*• 0.
+ Xl iml *i
to'_ (x. /) = 10/1 p — (x, /)] + u'^ (x, t). I —I
° 1 1
satisfy for any bounded nonnegative function tj(x, t) from ((?y) equal-
J w(x, t)r\(x. t)dx\' + J J { - Wt], 4 [o,;(x, t)wXj
-f 2c* (x, 0] ^() dxdt^C ^ | [y®J-*-%(*. i)]f[dxdt.
(1.4)
(» Q
in which ai-(x, t) = a--{x, t, u (x, l), ux(x, t)), while the c\(x, t) and t)
a re subject to the conditions
ii(ci)2' <PS||7,,,<?r<^ l- '=> «• (1-5)
Here the meaning of } and r is taken ftom (1.3). Inequalities (1.4) actually
have the same form as inequalities (1.7) of Chapter V, with the same restrictions
on the functions a{{x, t, u, p) = <fy(*> OP; + 2c\(x, t) and $2(x, t) as in §1 of
Chapter V. Ftom them, as was shown in §1 of Chapter V (see Remark 1.3), fol¬
low inequalities (7.1) and (7.2) of Chapter II fot tv -v}l±, / = 1,•••,«, which
guarantee the membership of the function U (*, t) in die class S|n.
Thus, we ptoceed to a derivation of (1.4). To this end we consider the inte-
§1. SMOOTHNESS OF GENERALIZED SOLUTIONS 519
gral identity
J J (^8)1^ dx dt = 0. s=l
(1.6)
where t) is an arbitrary function from !^ we assume that tbe solu¬
tion u(x, 0 Is thrice continuously differentiable, we can transform (1.6) by an
integration by parts into the form
t
°= J J [v-vW “
Vl
«xxl — al ,\dxdt
du dxs I -V; ■cjJ
#
J J k/ +VVAr,
Q L
+*«% i
\ ^Ujcm Xm*t X* 1 J *^f
- (iS^ “+-5 %+is;) V/6"
We introduce the Dotation
afjerf/. (1.7)
da‘j da'» i2£ „ + Jffii _ fl/
’ “i /• “r. T — r
du, da,. </’ da -«i d.*;
*»>
>$
djtr, du Ujcs
&*//
dxs du
Then the identity (1.7) will appear as follows:
t
JI [«,/+au\*S+(a”uVluVi+
i, a
+ bUuVj)^ -«6rJ dxdt = 0. s=\ n. (1.8)
Here we put
K*, t)=uJl (X, t)n(x, t),
II
where i]{x, i) € (Qf){ and we sum the resultant equalities over s from 1 to
n. Then, letdag
520
Vt.GENERAL QUASI-UNEAR EQUATIONS
/I'
f?l XS
we will have
t j
+f(«rA/vl ^aS + 2^A,V>
— au t) — au t\ ]dxdt=Q. (1.9)
Vi * V
We combine this identity with tbe identity (1.8), in which s is taken equal to I,
while £ - 5 »!}(*, (), and we write tbe result as tbe following relation with respect
to the function ui -w^ = 10nuxl + u:
t
J J [iwtx\ ■+• + J
+ J a?JWxla*a*P + \ “Vr/l + ^V/l + C'\] <*■* dt = °* <U0)
where
Cij — b’i ]Uxs — <*&/ + 5nb[j,
— c\ — aaX; 4- 5nab}.
Id the first term of equality (1.10) we carry out an integration by parts; this
gives
t
f w(x. t)rt(x. t)dx\' 4- J J [-«n,+2<,i7Bv/V«n
a *• /. a
+ (“„«v-+-2ci) \]dx dt
t
— J J 1“^, “V( + a‘Wr. + 2C'(A,*,] r\dxdt- (1.11)
i. a
The terns ia tbe tight side are estimated by inequality (1.2) of Chapter II:
I °‘"/WW/n I< h ! fam*,+ KI *T, f <) •
| a'wxp| (< (j wll 4- -I | a} p) | tj |.
|c5/V;nl<(e0Vy+ I C'‘/!2)lT>|-
§1. SMOOTHNESS OF GENERALIZED SOLUTIONS 521
For f = u/n and rj > 0 inequalities (1.4) for the functions ui - I = 1, • • -
• •*, n, will follow from these estimates and (1.11) by virtue of assumptions (LI)—
(1.3)* In this connection in (1.4)
£ Wtf+T*
T I, h nt
n
q)j(jr, 0=y£ + 17 S (CI/)2' a ^x-
j-1 1, )-i
• O 1 *
and Tf{x, t) is an arbitrary bounded nonnegative function from *2* (Qf)*
In deriving tbe identity (1*11) aad inequalities (1.4) following from it we have
made use of die existence of the derivatives u , D^u of u (x, t). These deriva-
XV X
tives do not appear in the identity (1.11) itself nor in the original equality (1.6);
the integrals there make sense for arbitraiy functions ( and bounded 77 >0 from
$2* ^iQf) if **(*» 0 possesses only those properties which are prescribed by the
conditions of Theorem 1.1.
We will show that (1.11), and hence (1.4), is also valid for any such solution.
To this end we approximate u(x, t) by thrice continuously differentiable functions
up{x, t), p * 1, 2, * * •, in such a way that 0 < up <1, the functions up converse
X( —
to u uniformly in Qy., the derivatives converge to ux.t i -- I, • • •, n, almost
everywhere in Qj., and the derivatives and ^xtxj converge to and
i, / = 1, • • •, n, respectively in From the i»P we compose die expressions
S’, (“"> = ai/(*•<• «. “,) + a (Jt. t. «. “,)•
It is clear that j£j)(itp) - ^12 qt —• ® for p —» «o. In accordance with (1.6) we
consider the identity
J /[“?-"</(*• l< “* *,Hir,+ aiX- *- «. “X)]l,dxdt~ j \StW\xdxdi
tg Q 1 fi
and we transform its left side into a form similar to (L8):
Jfh
£ f- wiffa
6aU ,
+ laj- '/VP1
- «6,J dxdt,** J J JST'Ml'dxdt.
522 VI.GENERAL QUASI-LINEAR EQUATIONS
Ftom here we deduce relations similar to (1.9), (1.10) and (1.11). The latter of
these relations will appear as follows:
t
f w(x. t)i\(x. t)dx|Jt + J J (- W’n, 4 20,
Q *« u
2C^%\dXdt
t
If we pass to the limit in (1.12) as p —■* <*, we obtain equality (1.11). The pos*
sibility of such a passage can be easily proved if we take into account that vrai
maxp^,|u£| < 1, die it?, converge almost everywhere in Q^ to and
_ “*,■*’!I2 Qf —y ® ^or P °°* ^*et us *>y way of illustration, one
of the terms in (1.12):
y'“ jj
and show that
, . r r
jp~>J^\ J uXiXuXst\dxdt
t9 u «
for p —►». Indeed, by virtue of (1.2)
U-J,\<clil2,Qj- 2,I It, Qr •
VPe take an arbitrary f and, corresponding to it by Egorov’s theorem, a set
<rT with mes (fy < e such that on Qj-\(?y the functions u£; converge uniformly
§1. SMOOTHNESS OF GENERALIZED SOLUTIONS
523
to a , £ = 1, • • •, ft. It is cleat that the nonn
does not exceed die sum of the norms
2. «r\ Qr
where in the right side the first two summands tend to zero for p —> while the
last summand can be made arbitrarily small by die choice of t. From what has
The {act that one can pass to the limit in the remaining terms of (1.12) as
p —> oo is verified analogously.
Thus it is proved that equalities ( L11), and hence inequalities (1.4) with
r)> 0, hold for the solutions u(x, t) of equation (0.1) considered in Theorem 1.1
and the functions
constructed with respect to them.
In completely the same way we can establish that inequalities (1.4) with
Tj > 0 are satisfied by the functions
As was noted above, the existence of inequalities (1.4) with rj(x, t) > 0 for
w = wl±r I = I, • • ■, b, petmits us to conclude on the basis of Lemma 9.3 of Chap¬
ter II that the vector ftinction U(*, t) ■= (u*j {x, !),•••, ux (*, £)) belongs to the
class $la(Qr Wj, Sj, §2- y, Ti k),where Afj « 11 n, Sj = l6n, 5^ = 1/4,
8j =- 1/2 and the parameters y, r, S and K are determined by the quantities y,
?, r and fij from inequalities (1.4) and (1.5). It follows by virtue of Theorems 9.1
and 9.2 of Chapter II that the assertions of Theorem 1.1 are valid.
Instead of reverdng to results concerning functions of the classes '8^1 we
can take advantage of Lemma 10.2 of Chapter V in order to conclude from inequali-
been said it follows that |/ - /| —> 0 for p —> «•.
a
I— 1, .... it.
524
VI.general quasi-linear equations
ties (1.4) and (1.5) that the assertions of Theorem 1.1 concerning an estimate of
(q) ate valid. In fact, inequalities (1.4) are a special case of inequalities
(10.1) of Chapter V (or, equivalently, inequalities (1.7) of Chapter V). For them
fil = y and the functions a{(x, t, u, p) = o.-(x, t, u, p) p. + 2cj and <j>2(x, *)
satisfy by virtue of (1.5) and (1.1) the requirements of Lemma 10.2 of Chapter V.
In Qj. we take an arbitrary cylinder Q2j{ = &2R x t£o ~ an^ the
coaxial cylinder Q^ = x U0 - 6R2/A, JQ], where 6 is a number detetmined by
n> M p v, ft j, r and <j in accordance with Lemma 7.1 of Chapter II. Among the
functions ux.(x, t) we select that one which has the greatest oscillation in Q2R’,
let it be a,f, so that &/ = max^.j ... noi‘ = maijOsc )u%.; According to
Lemma 9.3 of Chapter II,
05C
and the inequality
mes {*£/(£ : wr (x, t0 — 0/?J) max®'—62 osc {w\ Q2A,j} fl3 mes KR
Qw
is fulfilled for at least one of the functions u>r = wr±.
Therefore this function satisfies all of the requirements of Lemma 10.2 of
Chapter V, and hence also the inequality
osc {w', <?*) <(1 — 60)osc {wr, QygJ (1-13)
with Sq>0 and *j>0 depending only on S2, 8^ and the quantities in inequality
(1.4). The desired estimate for {«,.) qi1, Q' C Qj, follows from (1.13) and Lem¬
ma 5.9 of Chapter II.
§2. AN ESTIMATE OF {ux)fr
In the preceding section estimates of (a^! were obtained for any interior
cylinder Q' = O' x (f, T), ( > Q, S' C fl. In order to prove a theorem on the solv¬
ability of a boundary value problem we must have estimates of in the
whole domain Qj. This will be done according to the same plan that was used in
tbe case of equations with principal part in divergence form in §5 of Chapter V.
Namely, we will first establish the inequality max^|u(| < c, and then consider
u (x, l) as a solution of the elliptic equation
at} (x, t. u, ux)uVj + f(x, f) = 0 (2.1)
with the free term f(x, t) = -0(1, I, u(x, t), ux{x, ()) - u((x, t) depending on t as
on a parameter. The following assertion holds for the solutions of such equations
§2. AN ESTIMATE OF (“jr>(^ 525
(see Theorem 1.1 of Chapter VI in
Theorem 2.1. Suppose the function u(x) belongs to V2(Q), has vrai maxft|u%|
< M1 and satisfies almost everywhere in 0 equation (2*1) {in which t is
fixed: t = £q). If the conditions
vl2<aij(x. t0, u(x). «,(«))$£/<p|*. v > 0.
M"!
max
Q
datj (x. a (x), ux (x)) datj day
dux ' da ' dxi, '
b
are fulfilled and if u(x) |j = ^r(*)|j, where </j(x) € Wq.iQ) with q > n, and S C O2,
then u{x) 6 #1+^(0), /3 > 0, and the quantity \u% is estimated from above
by a constant determined only by v, (t, fij, U j, q, || i/j ||^2g and the boundary S.
Theorem 2.1 gives an estimate of (itj.) ^ in terms of known quantities and
n»ax^ j, |“,|i which together with (he inequality max^ juf [ < c permits one to
estimat e ^ (see Lemma 3-1 of Chapter II). We will first obtain an a priori
estimate of max^luj. We assume that u(x, t) is a classical solution of equa¬
tion (0.1) and that J/j = max^^,|ux| is already known to us. We further suppose
that maxr |u I is known.
T '
We will assume that the functions (x, t, u, p) and a(x, t, u, p) are con¬
tinuous and continuously differentiable with respect t» all of their arguments in a
neighborhood of the manifold Kx, i) 6 u = u(x, t). P - ux (*> t)In order to
obtain an estimate of max^^, |u{| it is sufficient to estimate the maximum in Qj
of the expressions v = u( + A2" = 1uXi. and w - -ut + A2"s1b^, where A> 1 is
a certain constant. Let us consider, for example, w(*, t).
If the function u(x, t) bas not only continuous derivatives of the form u(
and «* . but also the derivatives it(lD^u, it is then possible to proceed as
follows. We differentiate equation (0.1) with respect to t and x ^ I - 1, • • •, n,
and form the sum
n
(&a)t + X 2 2u = 0..
;•»! *1 I
The result is writteo to the form of the following relation for v:
vt — atJ(x, t, u. ux)VjClXj + 2kaijaX[XiuXltj
526
VI. GENERAL QUASI-LINEAR EQUATIONS
+ 2 E “I, + 2X tt*i+ * ~
a
+ a-kYal = 0.
M '
We multiply this equality by v - k and integrate the result first with respect
to x 6 A^(t), where <4fc(0 is the set of points * £ fl at which v(x, t) > k, and
then with respect m I from 0 to T. The number k is assumed to be positive and
greater than max v.
The term
T
y, = J J _ (V _ k) dt dt
o Akm
is transformed by means of an integration by parts into the form
T
— J J + [-gjp + -g~- 4--ftf) Vx, (« — *) dx dt. (2.2)
0 AtW L \ *1
Then we obtain
1 * f r
2 J I®r> —*)*<<*+• J J (a(/w, v, ~f- 2Xa!Ju..uXAAv — k)
r
+ •<•-*)]*«« — J J U^-a,
“ «> V \ *I
+ ~dT a*i + Tsf) V*1 +[~^7t a*<xi+ v*‘
- sr(a"-“ ~°)■+ ~sr+~~gu~2*S
/-I
d&u 1 do da
+ 2lUxi\ *Vy + W~ “)+ IT
/I
+ "S7 “■r<~ + °
“V*
§2. AN ESTIMATE OF<Ux>® 527
We estimate the right side in (2.3) from above, taking into account that M j =
mxQT Kl is known and that we viU take A and k efficiently large in the
sequel. From (2.3) we have
7
J I lal/\v*l + 2kavttVia*r'liv~k)+Hv~k)\dxdl
0 V*>
7
<*f
0 As (II
-+ X|uxx|-+ U(v ~k)dxdt
<f f J [^'i +-^ |u„- kf -f- tv\ + -~-(v~k)1
u A„m
*+ "”*) +'f*,2C»—ft) -|~ J-(t» — ft)J dx dt.
The constant c is determined fay the quantities Si j and
max max
l, i. *-i n qt
OO.; j
atj(x, t. u(x, <). #,(*• t)). a. ,
•*»
da datj datj datj da da da
dt fa ’
We take e and A so that
dt ’ du ’ dt ’ dx„ ’ dux^ ’ du ' ax^
2 c = Xv, 2ce = -J.
2 1
Then, transferring the terms with v2 and “**(•> - k) to the left, we obtain
r
J J |jL -|- Xv| uxx |s(» — k) ^- v(v — *)] dx dt
o At (0
* r
u m
-j- (v —- ft) (1? — /c -j— -f 4cX)j dx d£»
The constants c, e and A are now fixed in their dependence on M ^ and
The parameter k on die other hand is subject to only the one requirement
528
VI. GENERAL QUASI-LINEAR EQUATIONS
Let kmax be the maximum of v in Qj. and let it be greater than MQ. Io (2.5)
we take h = k - 8, where S = 4 cvX/c. For such k inequality (2.5) implies
r t
J j v(v - k)dxdt ^.c j J(v — A)-f- jX2 + dx rf/,
<> Ak (II 0 Ak Id
or, equivalently,
7
J j {v — k) |v — c -+• ^ + *■)] dx dt ^
u A^\t)
k S£Z /Emju ■— 6.
We only decrease the left side if we replace the v in brackets by &max - &
T
[*»*-6-*(^ + x+X)] J' J *)dx#<0.
U A fill)
i.e. we obtain
*ma* < * + e + — + ^] •
Finally, we can write
maxT»-< ttiax [Mq, c,J, (2.6)
Qr
where
„ v , „ . c> . 2cl
c,==274-2c + -^r + —.
This is the desired estimate of v and, consequently, of ut from above. The posi¬
tive maximum of the function w = - u, + A u^, which gives an estimate of u( from
below, is estimated analogously.
But all of our arguments are legitimate only for u possessing the derivatives
utx, D^u and the derivatives subordinate to them, even though the identity (2.3),
from which we obtained estimate (2.6), contains only the derivatives D*u, ux(, u%.
Let us show that (2.3) is also valid for solutions of equation (0.1), when it is
known only that they belong to C*'l(Qj.). Such solutions have the generalized
derivatives u(x, and these derivatives belong to L2(Qy). Indeed, we form a
divided difference with respect to t from both sides of equation (0.1), then we
multiply both sides of this equality by the divided difference Au/Af from »(*, ()
and integrate the result over Qt^, tj <T - At. We obtain, by taking into consider-
§1 AN ESTIMATE OF (u*)^ 529
radon formulas (4.7)—(4.9) of Chapter II,
■ii
f* f A(JPb) AaJJ4 f ( ( d tSu _ *“*(*/
0— J J 5? &Tdxdt— J J to al) to
o & o «
—Uxf^x. / -f- A/) —
1 a /A«\* „ *“'< *“*! , *“'/ A“ ^
T * \ A< / v to to *t-~S SF dxt
[(da,. A», da<| A» 0“//\
— «*,*, (*. * + sf-H &r dT)
from which, by making use of the boundedness of \ux | and j u%x j, we deduce
t J (ir)’dJC [ + / Sa‘t (Ir), (4r), dx dt
tt 0 fl *i *}
<‘/;H(^Lr+a+')i-&r+(i+-)]‘"'-
where e is any number from the interval (0, 1). For e = v/2e we obtain
2 J (^)2rfjf|0' + v J dxdt^c' J J (41-f <txdt + c\
a 0 fi 0 u
From this, on the basis of Lemma 5-5 of Chapter II, we deduce the uniform bound*
edness of the integral
/ J\(ZU dxdt^. const
0 2
for all A t and I t<T - £tf. Inequality (2.7) guarantees the existence of the
generalized derivatives ux.t of u and their square summability on Qj-
In order to derive a relation of type (2.3) we take the function
530
V!. GENERAL QUASI-LINEAR EQUATIONS
i-i
and proceed with it in the same way as above with the function v. Namely, we
consider the relation
■£■(*«)+ X%Mgp-2toL+j?u = 0 (2.8)
fml
in tbe domain Qj 4 = 0A x [0, T - A], where 04 still belongs to fi after a
translation by A along any of the axes. This relation can be transformed into a
form similar to (2.3). Indeed, by virtue of (4.7)—(4.9) of Chapter II the equality
(2.8) can be written as follows:
A«, Au Aa,. As
+ 2Xa0 -JJJ- — (X, t -f Ao —jj- + sr
As#* Att Aa Att
(*+**!• a5t+A7r2i^r+-2’“=0- (2-9>
Let
Aft = max j v |.
r r
By virtue of the continuity of t>(x, l) the set of points (x, l) from at which
v(x, t) > Uq + 1 is contained in a cylinder whose distance from Vp is equal
to h > 0. For sufficiently small A (A < h) (he functions t>4 are defined in
and the maximum of v& on the lower base and lateral surface of does not
exceed Mq + 2.
We multiply (2.9) by - k, where k >M0 + 2, and integrate the result first
over the set A%(t) of points x from fiA at which vA (x, t) > k, and then with
respect to t from 0 to ly < T - A. The second term is replaced in accordance
with equality (2.2). This leads to a relation similar to (2.3):
~ { t,)-kpdx
4('i)
■+1 J h' (**),, +2Xa‘j k) +-2’“ (v* ~ A)ld
At<‘>
§2. AN ESTIMATE OF
531
j"f[
0 a£(»
Sf ~ u*‘xi{x' *+'A° +isr
-*V,(* + A*r 0^2^ + l£ 2*^K ~ k)dXdt- <2-10)
We will pass to the limit in this equality fay letting Ax^ and At tend to zero.
The divided differences Au/A*p Au/At, lSux./!Sxl tend uniformly in to
du/dxp du/dt, d2tt/dXjdX[ respectively, while the Aa*./At tend to d2u/dx-dt in
L20?<*>) (see Lemma 4.10 of Chapter U). Consequently, the wa converge uni¬
formly in to v, while the (v&)x, converge to vx. in I<2(0*^ )• By virtue
of inequality ( 4.1) and Lemma 4.3 of Chapter II we conclude that the continuous
functions
*2* (■*• 0 — max («a (Jr. t) — k; Oj
tend uniformly in to the continuous function
©<*>(.*. t) = max {»(■*• 0—k\ 0)
aid
All of the integrals in (2.10) are extended from onto the whole domain
fi by taking in place of (v4 - tc) and ( v&)x, in them the functions and
(v&’hx. respectively, so that the domain of integration will now be independent
of A. From what has been said it follows that we can pass to the limit in all of
the integrals in (2.10) as At = A*j = A —> 0. After such a passage to the limit we
obtain
Tf J l«(*. t,)— kfdx
jl
+ J J [aijv^x. + 2XatJuxlxltt*fSV~ k). + J3^u{V — k)]dxdf
0 Ak <<)
= ~“J f [ldfv*/~uVj ~3T + W- — ^u*i~^~~dX[ 2Xa';]^“ k)<ixdt.
oA„m
532 VI. GENERAL QUASI-LINEAR EQUATIONS
Here we take t ^ •= T and develop tbe total derivatives of the a., and a, while
writing £ u in the form
«
S>u=-V A. 2 aimar, +®-
l-l 1 l'“ xl*m
This gives
~ | b(jc, T) — k\2dx
A„m
T
+1 I K/V',+2Xa‘V/ (*-- *>+ « (’- *>J dx dt
o AfiiO
I (da‘> - .t!L\v
xixi &u xi &xt J *f
/ dan , dan datj \ da
~ V,(^«v+■sru- + sr)+ss^“y
do da / day <*“(/ daj/\
+-5T •<+tt~2X*V/*.j (s^- V, + *r «*, + -s^rj
_2J.« (-J^— a + -I1 < 4~ (v — k) dx dt.
*t \ duXn vi ^ du *, ^ dxi Jj
We combine die underlined terms, noting that
u , 4-2A,u, , =v ,
and we replace the derivative u(, using equation (0.1), by a[muXlX - a. As a
result we obtain an equality which exactly coincides with (2.3). It is deduced by
us for all values of k satisfying the condition
ft>M0 + 2.
In accordance with (2.6) we can assert the validity of the estimate
maxt/-<m4x (jM0-f-2; £,}.
Qt
where the constant c j is the same as in (2.6).
Thus the estimate of maXq^, |u(| is established. The result obtained can be
formulated as a theorem.
Theorem 2.2. Let u[x, t) be a solution of equation (0.1) belonging to C2,1(<?r)-
§ 3. THE ESTIMATION OF mai |Ujc| 533
Suppose that equation (0.1) is parabolic on u = u(x, f) (i.e. inequality (1.1)
is fulfilled) and the functions {%, t, u, p) and a(x, £, u, p) are continuous
and continuously differentiable with respect to all of their arguments in the domain
!(*• OCQr- 01- IP l< = ®?x Itt* (•*•
\ Qt Qt 1
Then the quantity maxQ^ |u( | is estimated solely in terms of the constants v,fi
and M2 from conditions (1.1), (2.4), M j and maxrr I“(l-
From this theorem, Theorem 2.1 and Lemma 3-1 of Chapter II follows
Theorem 2-3- Suppose the conditions of Theorem 2.2 are fulfilled and S 6 O2.
Then for some cl> 0 the quantity (u^/ q]j, is estimated by a constant depending
only on = max^lii^*, »)|, the quantities v, ft and M2 from conditions (1.1)
and (2.4), the norm |u(^ and the norms in O2 of the functions defining the
boundary S. These same quantities also determine the exponent a.
The derivation of this theorem from Theorems 2.1 and 2.2 aid from Lemma
3.1 of Chapter II is literally the same as for the equations with principal part in
divergence form in §5 of Chapter V.
Remade 2.1. The assertions of the theorems remain in force fot tbe solutions
described in Remark 5.2 of Chapter V.
§3. THE ESTIMATION OF max\uj
In §§3—4 of Chapter V we estimated max io terms of maxjuj for the
solutions of equations with principal part in divergence form. Here we will do the
same for the solutions of die general quasi-linear equations
_S?u==«( — au(x, t, u, ax) -f a(*, t, u, «,) = 0 (3.1)
under the assumption that (hey are uniformly parabolic:
v<l*l)(l+|/>|)ffl~2!2<ai,(.>r. t. a, p)|IS/<H„(|«|)(l +-|p|)'”-2|2. (3.2)
As the examples in §3 of Chapter I show, for this purpose it is necessary
that the order of growth of a(x, t, u, p) with respect to p exceed the order of
growth of the t, u, p)^-^ with respect to p by no more than two, i.e. that
|a(x,t, u. />)|<M(I«I)(1 -H/»l)ra- (3-3)
la addition, it is necessary that we impose certain restrictions on the growth
with respect to p of the partial derivatives of the functions t, u, p) and
534
VI. GENERAL QUASI-LINEAR EQUATIONS
a (x, t, u, p) with respect to the variables *, u and p. These restrictions have
usually been formulated as follows: under a differentiation of the a., and a with
respect to x and u the orders of their growth with respect to * are not increased,
while under a differentiation with respect to p they are decreased by at least one
(it is clear that, in the case when the a.y and a are polynomials in x and u for
large |p|, these assumptions concerning their partial derivatives ate a conse¬
quence of conditions (3.2) and (3.3)).
We will assume that these restrictions are fulfilled for the partial derivatives
with respect to the pk'.
I J?ffi
I *P»
(3.4)
We weaken them for the partial derivatives with respect to the xassuming
<H(|«|>+P<| t\. 1.1)1(1(3.5)
where c (r) is a continuous monotonically increasing nonnegative function of r,
while P(p, r) is a continuous function on the set ip > 0, t > 01 that is monotoni¬
cally increasing with respect to r and tends to zero for p —* oo uniformly with
respect to r from [0, ’j], where is any number. As for the partial derivatives
with respect to u we will assume that
- da«ls(a)4-P(|p|. |«|)ld+|p|)m.
(3.6)
au
where £ (r) and P(p, r) possess the same properties as in ( 3-5)-
Under tbe fulfilment of conditions (3.2)—(3-6) there will be obtained an esti¬
mate of max|ux| in terms of maxQj\u\ = U and the known quantities
v = v(M). (i^i^/W), H-i = Hi('W).
e = e(Al). P(\p\) — P(\p\, M),
in C3-2)—(3.6), if ((M) is sufficiently small. If the constant e(M) is arbitrary
(not small) in inequalities (3.5) and (3-6), then it is possible to estimate max|ux|
in terms of |n| any a> 0 (or in terms of max^|uj and the modulus of
continuity of «(*, i)) and the same quantities v, fi, fi0, y. j, e and jP(Ip|) •
In the case of equations with principal part in divergence form (see Chapter
V) the norm with some a> 0 was estimated in terms of M = max^?|i
§3. THE ESTIMATION OF max 1^1 535
known constants without any assumptions on the smallness of <(M), and conse¬
quently max|uxj of theit solutions was estimated for any f(M) from (3*5) and
(3-6)* It is not cleat whether this can be done for all equations of form (3-D under
the fulfilment of just the. conditions (3*2)-(3-6) with arbitrary € (AO.
Exactly the same situation exists In die case of equations of elliptic type
(see [65q], Chapter VI, §2).
In this section we will follow die account of the analogous estimates in
Some difference will be caused, first, by the appearance of the term in the
equation and, secondly, by the fact that in comparison with [^ql we weaken the
a priori assumptions concerning the smoothness of the solution u(x, t). Namely,
in
[65*1
it is assumed in obtaining estimates of max |u | for the solutions of the
elliptic equations
“i }{*■ “■ **) “ V/+ a (*• *• “■*) “ 0
that uix) € ir|(Q) and vrai maxy ju^l < ■*>, From the results given below it fol¬
lows that we can replace the requirement that vrai max a |ux| be finite by the
condition
jMt’(.\“xxf+l)dx<oo, (3.7)
a
where £ is a sufficiently large number depending on the Oy and a. Ia particular,
all of the assertions from §2 of Chapter IV of [^5ql concerning interior estimates
of max are valid for solutions u(x) from W%(Q) with n>2; for them, by
virtue of the embedding theorems, the integrals (3-7) are finite for any s.
We will first estimate |uz| on the surface Sj. In this connection we will
assume that the function u(x, t) e and at each point of Qj-\Vj has a
second differential with respect to x, a derivative with respect to t and satisfies
equation (3.1). We let
max |« I = M. v(.M) = v, =
«7
We will assume that
«lr7. = 'K*. 0 lrr> while if (•*, 0) £ 0l (5) and «|> g 0% 1 (Sr). (3.8)
Lemma 3* 1- Suppose the function u is a solution of equation (3-1), with u
and tfr possessing the properties just described. Then max^ |»x| is estimated
from above by a constant depending only on M and p (M)/v(M) from (3-2) and
( 3.3), on the boundary S of the domain 0, which is assumed to belong to the
536 VI. GENERAL QUASI-LINEAR EQUATIONS
class 02, and on mazx€u\>l>x{x, 0)| and
Let us take a not large piece S * of the boundary of the domain Q. Introducing
new coordinates y j. • • •, yB, we transform fi so that S * lies in the plane y = 0
and the whole domain fl is situated on the side of positive yn. Equation (3-1)
will have the same form and the same properties (3-2)—(3*3) in the variables
y j, • • •, yn, as long as the functions y(. = yt(x), i = 1, •• •, n. Have bounded deri¬
vatives up to the second order inclusively in Q. Our assumptions concerning S
imply that it is possible to divide 5 into a finite number of pieces and to introduce
for each of them nonsingular coordinates jr j, • • •, yn possessing the properties
just described. In view of this we can assume without loss of generality that
already in die coordinates * j, • • •, xn the piece Sl of the boundary S lies in
the plane %n = 0 and the whole domain Q is contained in the half space *n > 0.
In addition, we will consider the case of tbe homogeneous boundary condition
Blsr = °-
The general case of condition (3.8) reduces to this case upon replacing it(x, t) by
!»(*, i) *= u(*, l) - <p(x, t), where tji(x, t) is a function ftom that is
equal to tfi on Sj..
Let us construct with respect to u(x, t) a function v(x, t) which will be
equal to zero on Sj. and for which the expression
(i + K ij-'-W +KI)*“" [*«-•</(*• *• “JV/1
will be less than some known constant. We introduce v(x, t) by means of the
equality
u = ip (v).
Then
and equation (3.1) gives
jsr(w, v)~svt-au(x, t. a, i
“ + “• ux) = °- P-9>
If <£' >0 and <£" <0, then
vt-aiJ{x, t. u.
+ + ^-^(1 +1^1/]
< [(-*r v + 2n<p'j v\ + |£-] (1 +1 ux |)m"a. (3.10)
§3. THE ESTIMATION OF max|uj 537
Ve select <f> in such a way chat
<P(0) = 0.
For example, we take
q>(t>)= In(1 + O),
so that v = -1 + e^/and v|= 0. Then from (3.10) it follows that
(1 +|«,|)*-" [«,-«„(*. <. «,
» . , 2|Uf , * 2ji>M
—-^-(1 +«) — ■—e v <-—« v sbc.
For i = 0 we have ^ 0) im
2pt „ v AI
~~~ max * v Mq < ——
Vo *
max| vx(x. 0) max e v Alo<——2e v ssc,.
a *0 v
Now consider die function v(*, t) + Ae~*B. Fot A > c jerf, where d is the dia¬
meter of Q, its maximal value 00 Fj. is taken for x € S1, t 6 [0, 71:
max (w -f- Xe~*") ■< (w + Xe-jr») | , „ = X. (3.11)
r
Indeed,
tnax(© + A.e_jr") = max Xe'*" = Xe~*n |x,s, = A,
Sf Sf
and, furthermore,
0) + ke Jr',]|Jcjo'^<;I Xe -^0.
from which it follows that
max [w(jc, 0) -f- Xe~x*] < max [t>(x, 0) + < X-
We put
in which case
.*(«.+- atl (x. t, u. ux) {v + <0- (3.12)
since 1
‘Ot~au(x- “• “x)^-*-4-|“^l)ra_2c
- \anae~*n <(j +1 Ux| )»>-* (c _ vw"-) < 0.
From (3.11) and (3.12) we conclude that die value of the function v + A* *" for
* € S1, t € [0, 71 is its maximal value in 9p> ®>d hence
538
VI. GENERAL QUASI-LINEAR EQUATIONS
vrai™,X.*‘+ ^ max{”T“°• ^r}- (3.13)
'£10.71
This gives an estimate from above for du/dxn on S1 x [0, 71, since
du I , . . dv I v do I ^ vX
dxn Ijrf i-i ^ dxn ji 2|i dx„ $i '''■ 2[i
In order to estimate du {x, t)/dxn from below for * 6 S* it is sufficient to apply
the preceding estimate to the solution -it(x, t) of the equation
(—a), — atj(x, t, u, ux){—it) —a(x, t, a, ux) = 0.
irj
From mis we get
— vrai min 4^- — vrai max —C .
x£S* x(.S> da 2(1
<?io.rj /ei«,n
Thus we obtain a two-sided estimate of du/dxR on S1 x [0, 71.
Estimates of this type of max|ux| on S j have their origin in the works of
S. N. Bernstein [s^l. In these works elliptic equations with two independent vari¬
ables are considered, but the transition to any n and to equations of parabolic
type does not present any difficulties for estimates of max \ux\ on a boundary.
A different situation exists with interior estimates. In Bernstein’s works such
estimates were made with the use of the fact that at a maximum point of some deri¬
vative i»x, it is possible to exclude from consideration all the derivatives of
second order ux^x^, anc' “*2*2 ^ exPrcss'n8 d,e,n terms of the deri¬
vatives with respect to * of first order, using three relations: the two relatioas
obtained by setting the derivatives uxix^ and a*.*2 equal to zero and the equa¬
tion itself. Such an exclusion is impossible for n > 2 in the elliptic case and for
n > 1 in the parabolic case (it is clear that such an exclusion is not required for
parabolic equations with one space variable
The first interior estimates of max |ux| for quasi-linear parabolic and elliptic
equations with arbitrary n were given in [6 5h1. The method proposed there for
estimating max |bx| is related to Bernstein’s method for making estimates by the
idea of introducing in place of u a new function t> m for which it is impos¬
sible to estimate max|vx| by using only the equation for u. In regard to the solu¬
tions investigated in [65h] it was assumed that they are continuous in Qy together
with their derivatives of the foim ux., a(, utx., Dxtt, Dxu. Let us modify the pre¬
vious arguments in such a way that they become suitable for solutions u(x, t)
§3- THE ESTIMATION OF maxluj 539
having only generalized derivatives entering into the equation (the case when
these derivatives are continuous was handled in 1).
Thus, we will assume that the solution »(x, t) of equation (3.1) is bounded
in Qj and has generalized derivatives of the form ux , ux x , u , in Qyr with
\nxYs*2 e V\'°(Qt) and ‘ *'
J Id +1«J )4,+'“ I *« f+(>+1 «* I )4I+J K I2
Ot
■+• I uxl41+m+A]dx dt < 00, (3.14)
where s is a sufficiently large number, the magnitude of which will be further
specified below as a function of the constants u(M\ jl j(M) and e(M) from condi¬
tions (3-2M3.6).
We will give two kinds of estimates of |i» |: a global estimate for the whole
cylinder at once and local estimates for subdomains Q‘ C Q-p- We begin with the
first, assuming that maxrj. |ux| = Mq is already known. We again inttoduce a
function v(x, t) by means of the equality
a = q)(o), (3-15)
assuming <f> (») > 0. Relation ( 3.9) is valid for v and, consequently, so are the
identities ,
J ^(B; v)t]^ dxdt =0, /=1, ... n. <3.16)
where rj(x, t) is an arbitrary function that is continuously differentiable in Qj,.
If nL =0, while the solution u(x, I) has continuous derivatives u , u B,.,
- IT X XX Ai
D^u, then, by carrying out a double integration by parts in (3.16), we can trans¬
form it into the form
nf dau dat,
+ [isr "v, - <V/
Ot,
—4z($atw*i-Y)b)dxdtmm0' i==i *- 3-17)
Let w = 2“=;1 v* and 2 = »**’. We put
ri(x, t)~2vxwsz''ltK (3.18)
where
z**> = max (z (x, t) — ft; 0}
and
k >- vrai max z, (3.19)
r r
540 VI.GENERAL QUASI-LINEAR EQUATIONS
and in (3.17) we carry out a summation over I from I tq n. Then, noting that
=•'/* 2vV/vty=-J-,. zXj,
we deduce from (3.17) the equality
•j—y J OP dx + 2 J 12
O ft,
+ 4- «(>*?/*?]
f </a/< dan a / 'P”
L dx] Vxt*l <<•*/ rf-t/ \'y aiJv*jVXj
— ~r'j2vXlwszil,']]dxdt = 0, /,g[0, T|. (3.20)
In the intermediate transformations: we hare made use of die existence of the
derivatives ux(, D^u of u(x, t), although in equality (3.20) itself, as also the
initial equality (3-16), the integrals have a meaning for any solution satisfying
conditions (3.14) only. Let us show that (3-20) also holds for any such solution
if Ihe values of k ate subject id inequality (3.19).
We introduce averagings of the function v(x, t) oi the form
vp(x- *)= J %(lx— yl)v(y. t)dy.,
l-r-yl
t+h
vh(x> t)wkJ v(xt x)d%
t
and in accordance with (3.16) we consider the identities
J jv« — fli/(x, t, u, ttx)vpxixj t, u. ux)vxvXj
""j— / a {x, t. u. u j.t|j[^ dx dt
= J — v' + ai j(x' *• u’ u*> (Vjci*j — ®pVr/)l ***/dx
Qt.
1= 1 n, tx £ |0. 7* — A|, (3.21)
witb a smooth function i){x, t) that is equal to zero in the vicinity of Sj*. We
carry out an integration by parts in the left side of (3-21) in the same way as was
done in die derivation of (3-17) and then pass to the limit as p —* 0. As a result
we obtain the identities
§3. THE ESTIMATION OF maxlaj
541
vrai max z ■< vrai max
sT-h sT-h
j+i
vrai max | vx |2t+2 = vrai max z.
St. st
» r r dan <* Iff"
J +[”5^7 **!*) —IF, li7 a‘^iVxi
—v) - -S Adxdt^l{v,u ~v,) TU< dx dL
1=1 »: (3-22)
Here we can take as ?/(*, () any function from V2(Qj) that is equal to zero on
Sj, and for which ail of tbe integrals in (3*22) have a meaning. In particular, we
can put
T|(j;, t) — 2vhxt (®*)* Zh {t>, (3.23)
where »
= 2^v «"=(®y+,1
and the number k is subject to condition (3-19)- For such k die function
vanishes on Sjsince
'+*
I J \vx(x, x)fdx
t
We substitute the indicated 17 (x, i) into (3.22), cany out a summation over I and
write the first term in the form
4= J vAjtl,2vhXl (whf zh <*> dx dt = 2(jq;yj J [z* <*>]2 rfx |J.
Q
We let h tend to zero. Ia this connection we have
II 11^-* 0.
by virtue of Lemma 4.7 of Chapter II, and since k is chosen from condition (3-19)
it follows that
lira || **<*>(.*. 0)||2>a = 0..
H-+-Q
Consequently,
{^dx fot h-+o-
a
The remaining terms in the left side of equality (3-22) with tj from (3.23) are
transformed in the limit into analogous terms with rj from (3-18), while the rijfii
side of ( 3*22) tends to zero for h —> 0. In this way, on passing to the limit, we
arrive at relation (3.20).
Let us put ty = T in (3-20) and rewrite it as follows:
542
VI. GENERAL QUASI-LINEAR EQUATIONS
1
14-T f + 2 J J { Zai/Vgrftf/af (z — k)
AfiiT) 0 Ak{t)
+ sal(wxjB!xTi>!-x{z — k) 4- -jqrr allzJ,lzXl
\ datj dan lV’Yn
+h—Er^Pv-w) 2wa‘iv*iv*,
©' dan y*
7 1vxv,vx 7 2 a,jw,vx ^ 2 wa
dxt * • / q>' / 1 y'*
4. J_ J^L.2oA(]ots(z — k)}dxdt = 0. (3.24)
Here A^(t) is the set of points * 6 0 at which z{x, t) > ft. We will choose a
function $(v) in such a way that <j>' > 0 and /</>')' - (<f>">0. In the
left side of (3.24) we leave only the positive terms
T
7i-T \{z — kfdx + 2 J f [2auvXlXlvX{X)ws (z — k)
S Atp) « -AftV)
1
-f- sa; /®s ~ %’Wil'WXj (2 — *) -f aUz*lz*i
—2 (;jr) 1 'i* -
while the remaining terms are transferred to tbe right side and estimated from
above with the use of inequalities (1.!)-(1.2) of Chapter II and conditions (3-3)—
(3*6) as follows:
ff 4lt itVx. Wr, (©”
V1"'8'/' < - «■ »' Mf) a,i’Vx^)w-
The total derivatives d/dx^ in (3.24) are represented in the form
da,j da,, dat, dot,
dxt ~~ duXm Ujcmxi du xi dx,
+ ^mVxt) + -3T <P'tS + 7^
rW 1
and analogously for da/dxFurthermore, we note that the derivatives da^/dx^
and da/dx^ in (3-24) are multiplied by 2vx^ and summed over I from 1 to n.
These sums are represented in the form
§3- THE ESTIMATION OF max\uj
543
WVX )-
171/
+ 2-^<p'*+2-g®X(.
2 2 is; V'I •* i^(<p^ra +
As a result we deduce the following inequality from (3*24) with tbe use of
conditions (3-2)—(3.6):
r
V j j + + —($)***
0 Ak it)
__ 2 j *0* (2 — ft) £JC
r
f J.*’+ l*-,y“*{'-'llcT+^FSSr‘*rf'—l
“4“ 1 yf* Iw) -f" (® “f" p) *?' V® *4“ (® 4“ 0 4-<p' V**®)]
+ (l»«l + 1^-w)[ (<P' I •, 1+ I q>" I <**)
4-2(e + P)<p'w 4-2(e 4- P)(1 + <p' V») Vw]
w
4"jr L*i(l 4- «P' Vr»)(q>/|«’»l4- 21 <p" | w5)
-f- 2(e-f- /*)(! 4- q»/
4-2 (e 4- P)(l 4- <p' /5)3 V®!) w’tz — ft) dx dt. (3.25)
Considet the terms in the braces. We estimate those of them which contain |vxx|
by the inequality 2ab < 8a2 + S 162, S > 0, assigning a small £ to the factor
1
da,)
dx.
+ 2<f
544
VI. GENERAL QUASI-LINEAR EQUATIONS
**• wy <i|Wx_rP+4^-:
^ 26 (14-<p'y»)3 2 xxl 23 <»
1 v**1 [ (r+iV^T ^1 ®'1++ 2 (e + P) <f'w
II 2
<36K,P+|I-^-
2 / »\J . 2
+ (Jr) W* f j(8+ P)2(p'V + -g(£ +P)2(l 4 <p'!w) «.
The terms wilh |u>z| ate estimated as follows:
1
|®xl(e+ P)(\ 4 -h(e-f P)s(l +
• 2 ^
Mi 0 + <p' y®) I l< 4-«(l 4~ <P*®) ■W.
The remaining terms do not exceed the expressions
(*i (--r)2 + 4 (e 4-I <P" I wz + 2 (e 4-P) a» ^
4- 4|i| (ff |w24~ 4jt -S|-®4 2(i, | <p" |ot24- 2(i, i2lL wv
4-4(e4-P)(<p/J™24-w)4“ I6(e4-P) ^~r- 4- <p'2®2J.
We choose 5 equal to v/Tn^, while s is fixed so that
-(4^+'+$<-?• («>
We substitute the above estimates into die tight side of inequality (3-25) and
then transfer the terms with ItJ^)2 and ui2 into the left side of the inequality
and combine similar terms. In (he right side the principal terms will be die teitns
containing tos+^, with the remaining terms containing smaller powers of w,
namely ws +^2 and lower. We will take k > 1; then the function z, and hence
also w, will be not less than 1 on the set A k (t), i.e. all of the smaller powers
§3. THE ESTIMATION OF max|uj 545
of w can be estimated in terms of u>* By virtue of all that has been said
we will have from (3.25)
r
o A,(l) *
— •a^ws(z — k) dx dt
<f f (i + l«,l)ra-2{^[(f)’ + l<flO-+«+^
M Ak it)
4-(eJ+Pajr 2e-+-P)<p'!] w! -J- c2w*\ws(z~k) dx dt. (3.27)
Here
P = max P(|p|).
IPl>0
The constant e j is determined only by n and die values of viM) = v, p(M) =
(tj(Jtf)= lii from (3-3)—(3-4), while tbe constant c2 also depends on <(ilf) = (, P
and tbe function tjy[v), which we will choose in a special way. Namely, we will
choose <fA,v) so that
-{&)'> c' [(£)’ +1T"I(!+e + P)+(2s2+3e)<p'2], <3-28)
<p'M>0. (329>
In addition, it is necessary that die inverse function for <fh>) be defined on the
interval {-if, Ml. The construction of such a function <£( v) will be taken up below,
but now, assuming that such a function has already been found ,we continue with
the estimation. Consider inequality (3.27) for k> ftj, where 4j > 1 and satisfies
the condition
p(h~%+? min (p'(i/)')-f P2f&*STr min <p'0))<e for *>V (3.30)
Such a fcj exists, since P(p) —> 0 fot p —> ~ and
min <p' (*>) > const > 0.
It follows ftom (3.27) by virtue of (3.30) and ( 3.28) that
T
v* I J (* -tr\ttx\)m~2— ^
o Akm
546
VI. GENERAL QUASI-LINEAR EQUATIONS
T 3
<c2J j (\+\ull\)m~iws*'5(z — k)dxdt, (3.31)
0 A„lt)
where
v,= n,i„ (-£T>0.
From (3.31) we obviously have
1 T ,
— c2)J | (1 +l«xl)”’_2«'S+1(*— k)dxdt^0.
o **(<>
If the value of k exceeds
*2 = max(*,; (■£) '
then this inequality is possible only in tbe case when the integral is equal to
zero. In other words, the measure of the set of points (*, t) from Qy at which
z > k2 is equal to zero, i.e.
vrai max z (x, <)< max max*}. (3.32)
H>is yields an estimate of
vrai maxTO,+1(jc. /) — vrai max \Vj, j2j+2.
Qt Or
and hence also of vrai max^ }ux|.
For (he completion of (he proof it remains to find the function cf>. As $(i>)
we take the function
V
, q>(v)= — 2Af 4- 6Aie J ed\.
o
We will verify that it satisfies all of the requirements for sufficiendy large
q. Let us determine the range of variation [vj, t/jl 0 when a = varies
from —U to M. The integral
I
e~l* dl.
will obviously vary in the interval [l/6e, l/2e], so that
jWrf 6-^..
0 0
-
For any q > 1 the values of die function e 5 are included between e 1 and 1
on the interval 0 < £ < 1 and between 0 and 1 for all £ > 0; thus for all q > 1
§3- THE ESTIMATION OF max^l
547
?' ®* 2
6e < J <*6 — ®, and -g- = J g-S® = J e-^ d%
u oo
-f J #-»*d| > -1-+ j
T T
i.e. Uj < 1/2.
Thus, fot all g > 1
1 ^ - 1
•gj <®,<w,<T.
Let as now verify celadons (3-28) and (3.29):
<p' = 6 Mee-*". q>"= —6 Megv^~ie~'^,
^=. — 9^-1, =_?(9_- l)©«-s.
Inequality (3-29) is obviously fulfilled. To verify (3-28) we calculate its
tight side:
c,{?V«-s-1 (1 + e + P)6Meqv<-te-v''
+ (3e -+- 2e2) 36^2-2^) < c, V22-«
-f-(l +e-f- P) 6ATe?K*- ‘e-'W4 +(3s -f-2e2) 36yW^2-2<6^>_<').
From a comparison of this expression with it is clear that inequality
(3.28) will be fulfilled if e is sufficiendy small and q is taken sufficiently large.
It is here (and only hete!) that we encounter the need to assume that c is a suf¬
ficiently small number. Its maximal possible value can be calculated exactly, but
we will not do this here. Thus an estimate of vrai max^j, Wx(%, <)| has been ob¬
tained. We formulate it in the foim of a theorem.
Theorem 3.1- Suppose a function u(x, /) 6 ^^((Jj.) is subject to conditions
(3.14) for all s up to the order indicated in (3.26), and satisfies equation (3.1)
almost everywhere in Qj . and suppose M = vrai max^ |b| < oo and A/Q = vrai
maxjY \u-x\ < Then, if conditions (3-2)—(3-6) ore fulfilled and c (M) in (3.5) and
(3.6) does not exceed a certain number eQ determined only by M, v - u(M), y. -
ft W), fij = /i j (AO and
P=i max £>(|p|)= max Pdp], M)
\P\>0 !?l»0
548
VI. GENERAL QUASI-LINEAR EQUATIONS
from (3.2)—(3.6), it follows that vrai mai^ \ux\ does not exceed a certain con¬
stant depending only on M, v, fi, (iv e, />(|p|) and UQ.
From this theorem and Lemma 3.1 we get
Theorem 3.2. Suppose a function u(x, t) belongs to
O,0(Qr)n«1,l«?r>
and at each point of the cylinder Qj. it kas a second differential with respect to
x, a derivative with respect to t and satisfies equation (3.1). Suppose further
that conditions (3-2)—(3.6) are fulfilled with a sufficiently small t (U) determined
by M = maxQj.lu], v{M), fi(AO, H j(W) and
M = max|«|, v(/M), n(/W). (M)andP— max P{\p\, M).
Qt l/»l>o
Then, if u|r^ = </>(*, t)|rj., where <p(x, 0) e Ol{Si) and <//{x, t) € 02,1(SjO,
with S belonging to the class O2, it follows that vrai max^lu^l does tux exceed
a constant depending only on SI, v(M), it (M), fl^(M), e(Sf), />(|p|, M), the boundary
S of the domain Q, maxieiJ lifix(x, 0)| and the norm •
Tbe assertions of Theorems }.l and 3.2 remain in force if instead of the
smallness of e (M) one requires a sufficient smallness in the oscillation of it
in Qr
Indeed, suppose eiM) is arbitrary. We denote by u j and itj the least and
greatest values of u in and by <u = u^ - “ j the oscillation of u in Xhf- We
will verify that conditions (3.28) and (3.29) and the requirement that the inverse
function v = 0_1(«) be defined on the interval [u j, u2l are satisfied by tbe
function
V
y („) = 3a». + 3wr J e~ S’ dl (3.33)
o
fot sufficiently large r > 0, as long as cu is sufficiently small (the maximal pos¬
sible value of <d is deteosined by the constants c j, P and e {com (3.28)).
Let us determine the range variation of v when u ~ <f>(tO varies on
the interval [uj, u^]. From (3.33) we will have
t', v*
I e~l'dl=lF- J e~l,dl=^W'
Since 1
it follows that the numbers tij and »2 will be small for sufficiently large r:
§3. TOE ESTIMATION OF maxlaj 549
1 f6r < t> j <v2< e/2r. The derivative <f>'(v) - $tare'”2 is everywhere positive.
The difference between the left and jight sides of (3.28) for the function (3.33)
( for arbitrarily fixed constants c j, P and f)
2 — c, (4®® 4- (1 + e -f- P) 6me - **© 4- (3e 4- 2e2) 9wVs«-2,jJ
will be greater than zero if r is taken sufficiently large, for example so that v2
is less than 1/2yrCj, and to is assumed to be sufficiently small.
Thus we have proved the following theorem.
Theorem 3.3. The assertions of Theorems 3-1 and 3.2 remain valid if instead
of the smallness of e(Jlf) in (3.5) and (3-6) one requires a sufficient smallness in
the oscillation o> of the function u(x, t) in Qj,.
An admissible value of a> is determined by the constants n, U = maxQ |«t|,
v(M), ji {M), p l(M), e(M) mid
P = max P( | p |, M)
lp\>o
from (3*2)—(3.6) and it can easily be calculated explicitly, starting bom the method
just given for estimating |»x|.
We now proceed to obtain local estimates of |» |. We will show, for any in¬
terior cylinder Q' = Cl' x («, T), ( > 0, that vrai max^> can be estimated in
terms of it, the distance from Q to r'p M = max^^,|u| and the constants m, v =
v(M), pt0 = n0 Wt fi = p.(M), c - f (M) and P(jp|, AO from conditions
(3-2)-(3.6), if m > 1. Ttiese estimates do not depend on the properties of S and
the values of u(x, t) on Vj.. If on the other hand the boundedness of
vrai max a !“*(*» 0)| = MQ is presupposed, then vrai max^, ju^l with Q' - x
(0, 7”) can be estimated for any m in terms of Mq, M, v, fi0, ft, p.j, e and P(jpj).
Without loss of generality we take as Q' an arbitrary cylinder Qp= Kpx
(r — p2, r), where KpCU and 0 < r - p2 < r <T. Let £(*, t) be a cutting function
for Q that is equal to one in (he coaxial cylinder x P^/4, r) and satis¬
fies the condition
lt,l<«p-*. I5«l. I^Kcp"2.
We will estimate vrai taaxg^z, where
Z(x, *) = (£Jw)J+1=(s2St£
Let us deduce an equality for z(*, l) that is analogous to (3.24). To this end,
in the identity (3*22) with t j = r we take the function tj{x, t) equal to
550
VI. GENERAL QUASI-LINEAR EQUATIONS
2 vkxt?s+\wh)s zhm
where k is an arbitrary positive number, and carry oat a summation over I from 1
to n. Then the first term in (3-22) will have the form
“smjV'*’<*• <x**- J(•rv,vi*tA
a eT
and therefore, passing to the limit as A —> 0, we obtain die following equality,
similar to (3-24):
t
-k- J {z—kfdx +-2 J' J {-®”ie+IS,(^-ft)
Ak, p(tl t-P‘^ p«l
+(2» - *> + «,/•,
+1... 1 w't2s+2 (* — *)) dx dt = 0. C3.34)
Here the contents of the brackets [ —1 are the same as in (3.24), while p(t)
is the set of points * from Kp at which z(x, i) > k.
We carry out estimates in (3-34) with the use of conditions (3.2)-(3.6), im¬
posing the restrictions (3.26) on s, as was done above in deriving ( 3.27). This
leads us to the inequality
J J (1 +l«*l>m~2[^^ + ?i*~|2-2('$r) **]
T-p* Aki p(/)
t
Xtfs+2ws(z-k)dxdt^\ j J
+ (2s + 2) MdCl +• | «, I )m-! I «x iS2J+I I tr I «* U - *)
t
-auwxw%2"Wdxdt + j J (1 -HM)""2
X { c, [(J)' + |<P"| 11 4 e 4 P\+(^+f» 4- 2e4-P)<f'!]
+ c&k } fs+V (z - ft) dx dt; (3.35)
§3. THE ESTIMATION OF ma*|uj 551
here it is assumed that 4 > 1. Tbe {unction <£(t>) is determined from the condi¬
tions formulated above (see (3-28), (3.29) and following). For it
- 2 (-£)' - c, [(£)*■+1 <) (1+ e+ h
4- (2e 4 2e2 + P (I «* I) + 2/» (|a, |)) (p'*] > es > 0.
as long as tbe level k in (3.35) is taken sufficiently large, aamely k > ftj, where
ftj is chosen from the condition
P<|«Jt|) + 2#*(|«,|Xi. <3’36)
Since P ip) by definition tends to zero for p —> «>, while
l
i _2(«+l> >
\ux | = q>' (w) | vx | = <p' (ti) •& 2 = q> (t») —j > const 22<i+1>,
i.e. fot an unbounded increase in z we have an unbounded increase in |ux|, it
follows that inequality (3-36) can be satisfied by taking the levels k larger than
a certain number kj > 1. For such levels k we will have from (3-35)
/ J (i+KIr~2 [ir^-+t I I2+«*•*]
t-K>
t
x?*+v<*-*>*««< i J J *)
+ (2s + 2) nt (1+1 ux | )~21 | £2i+1£X (* - k)
— a. twr w%2s+2zr | dx dt
ij Xj xt\
t £
4-c, f f ■ {\+\ux\)m-'lw*^2(z — k)dxdt.
T-P«4*.p(«>
We concern ourselves with a transformation of the integral
t
/,== — J J atiw w%Ss*2zXtdxdt.
bearing in mind that
2 = (<J/+I, zXi = (s -f- 1) \ti>swx ^■s+j -f- +I2g2'+'£ 1;
t 1 “
/l== La$ S~^ha‘Jz^,+a‘/wS+l2^^a)**"-
& Ak, pW 7
552 VI. GENERAL QUASI-LINEAR EQUATIONS
t
"I J M+ U —
**,pw
xMa^+l2e%)}dxdt=- J J {iTTW'i
+ 2 t*% + «„ (• + »
-f (2s 4-1) a,fi%tZXj 4~ «,/** ’tvJ ®+‘ C*— A) | rf* dt.
We estimate this expression ftom above, using assumptions (3.2)—(3*6)i
t
/,<2j J (i+!«jr-!{[TT!^(<P'i^i+«P^)
t-P»A,p(0
+ (e + P)«p' +(s + P)(14 <P' )] £5,+* IC*I
+{#+iu -+ ty?**’ ie«i}®1+1 (* - *>•***•
From this and ftom (3.37) we get
/ f d+l«x!),"s[^!4+Tv«+C3H
tip* ^,pw
X ®i£2l+s(* — *)rfJS dt < J J ■a’,+I£2,+ l^il (z~ *)
T-(* **,„<»
4- f J 0+l«,l)"‘*[c«
t-o1 A*,p«>
4-Kxl®,+^,+1 l5»|4-«,+^5,*+,l5*l 4-C’I+' l?„l»+T)
4-ejtw* 5£55 +2](■* — *)dxdt.
We replace the tenns
^IWxl?2145!^!®’ X®S+T^,+ I 1^1
§3. THE ESTIMATION OF max |«J 553
in the right side by the larger terms
v-' +2T,!T)^ xe a«a j|t^ve+2+cy+l$21
respectively. After this we transfer die terms containing I®.,,!2 and mi2 r> die
left side.and reduce similar terms.
As a result of all that has been said we obtain tbe inequality
J J 0+|«*l
t-p> Jt,,0(0
< / J (l+I^D-^^^+^U^V
t-p» At'Pm 2
+ «,S+V+1) + 4®+,£2!+1™^]<*-*> dx dt.
(3.38)
The constants cp and c'p depend on £2> and and tend to infinity like
p”2 as p—• 0, bat this is not important to us since, although p can be taken to
be small, it is fixed. Noting that ws+1^2s +2 ■= z, we can rewrite (3.38) as follows:
J
f (1 -{-\iix\)m~iwz(z~ k)dxdt
x-»‘ Ak.0^>
r 1 .j. 1
t ^3 ( _s_ l42\
< J J (l+\ux\)m-2w[czs+l +cp\z*+l +z*+1 J
t-1*
+ ci,z
(z — k)dxdt.
Since k > 1 it follows that z > I and tv > 1 on tbe sets A^ p(t), and there¬
fore »<I-m>/2 < 1 for m >, J. Thus inequality (3*39) is consistent with inequality
(3-31) for m > 1. It yields an estimate of vrai m&tQZ in the same way as (3-31)
yields an estimate of vrai max^w. The quantity vrai maxq z is determined by
the constants c, cfi and c 'p in (3-39) and by the height of the level > 1, be¬
ginning with that at which (3*39) holds. Knowing vrai nwa^ we wjjj aiso know
estimates of vrai max ui and vrai max \ux\ in the cylinder Qp/2~
Thus, by introducing the cutting function £(x, t), we have estimated |« | in
any cylinder Qp « Kp x (r - p2, r), using the value of max |u| in a somewhat
larger cylinder Kp% x (r- p2, r), p1 > p. In this connection we must assume that f
554
VI. GENERAL QUASI-LINEAR EQUATIONS
is sufficiently small in inequalities (3.5)-0-6). This condition of smallness can
be dropped if it is assumed that we know the modulus of continuity of the function
u(x, () in Qy, i.e. die function co(p) ■ sup osc I u; Qpf| Qy), where the supremum
is taken over all of Qp and to (p) —► 0 for p —> 0. As was shown in the proof of
Theorem 3-3, it is possible to pick out the necessary function (see (3.28),
(3-29) and following) for any f and c j if it is known that the oscillation of u(x,t)
in the domain under consideration (in this case the cylinder Qp/%) does not ex¬
ceed a known quantity <Oq. Therefore, assuming p so small that coip) < tug, we
can estimate max|ux| in the cylinder Qp/2 without any assumptions on the small¬
ness of e. If it is assumed that we know max |u| and the modulus of continuity of
«(*, t) for the whole cylinder Qy, then it follows from what has just been said
that we can estimate max|t>x| for any cylinder of the form Q' *= fi' x (8, T), where
O’ C H and 8 > 0. To do so it is necessary to cover Q' by a finite number of
cylinders Qp/2 sufficiently small p (so that co (p) < <u0) and to make use of
the estimate of max |u,J for each of these cylinders Qp/2 ■
If max£ |“x(*> 0)| = W0 *s Steady known, then we can also give estimates
of max|»I| for the cylinders Q' = Kp x (0, f), t <T, and in this case we can as¬
sume that m is arbitrary. Indeed, in this case it is necessary in equality (3-34)
and below to subject k to the restriction k > Mg and to take a function £ that is
independent of t. The term containing c'p in (3-39) will then be equal to zero ,
and this was die only term whose estimate above required the restriction m > I.
We can also give local estimates of max |»x| for cylinders intersecting Sy.
If it is known that max r, lu*l 5 % then it is necessary to assume in (3-34) and
below that k > max0 nr? z. We note that it is also possible to make an estimate
of |ux| on Sy not for the whole surface Sy at once (as was done above in Lem¬
ma 3.1) but locally, on a separately taken piece Sfi = Sy f) Q of it, without the
use of any information on the behavior of u(x, t) in the vicinity of the remaining
part of Sy.
We formulate what has been proved on local estimates in the form of two
theorems.
Theorem 3*4. Suppose a function u(x, t) is bounded in the cylinder Qy, has
generalized derivatives of the form u%, «xx> ut in Qy and satisfies equation
(3.1) almost everywhere in Qy\Vy. Suppose further that for (x, t) E Qy the coef¬
ficients of the equation are subject to inequalities (3-2)—(3.6) with m > 1 and
that for some s satisfying inequality (3*26) conditions (3-14) are fulfilled with
§3- THE ESTIMATION OF max |Ujtt 555
respect to u{x, t). Then for any cylinder Q‘ = Q1 x (5, T), O' C 0, S > 0, the
quantity vrai max^> \ux(x, t)| does not exceed a certain constant depending only
on the distance front Q' to Ty, n, M - v - v(M), fig * fi0(AO, ft = ft (JO,
ft j = ft jUfl, f = e (A/), P (|p |) = P(|p|> M) and m from inequalities (3-2) —(3-6),
where e(M) must be sufficiently small. If u(x, t) is continuous in Qy, then
vrai max^> |<*x(x, *)| tan be estimated in terms of these same quantities and the
modulus of continuity <o (p) of u(x, t) in Qy for an arbitrary (not necessarily
small) number t{M).
If vrai masu 0)| < Mq , then, regardless of the value of m in conditions
(3.2)—(3.6), vrai max |ux| in the cylinder Q' = fi' x(0, T), where Q' is an arbi¬
trary interior subdomain of fl, is estimated in terms of the distance from 0* to S,
n, Mq, M, v, ft0, ft, ftj, e and P(|p |) if e - ( (M) is small, and also in terms of the
modulusyf continuity ojip) of u(x, t) in Qj. if e(M) is arbitrary but u is continu¬
ous in Qy.
Theorem 3-5. Suppose all of the assumptions of Theorem 3-2 are fulfilled
except the conditions on the boundary S and on the function 0(x, t), which are
replaced by the following conditions: a section S' of the boundary S belongs to
the class 02 and on a certain part S j C S' x [0, 71 the function u(x, OI5' =
*l>(s, t) 6 02’\Sy). Suppose further that the number m in conditions (3.2)—(3-6)
is not less than one. Then, for any part Q' of the cylinder Qy that is separated
from rj,\Sy by a positive distmce d, the quantity vrai max^i |itx| is estimated
by a constant depending only on. S', l^l^, d, n, M, m, u(M), n0(M), ft (A/), fi^M),
t(M) and P(|p|, M) from (3-2)—(3.6) if (\U) is sufficiently small, and also on
the modulus of continuity &>(p) of u(x, t) in Qy if f(,if) is arbitrary but u is
continuous in Qy•
Let Sj = S' x [0, r], r< T and vrai mxs lux(*> ®)| ^ and let denote
the union of the lower base of the cylinder Qy and the section Sj. of the surface
Sj~ Then, for any subdomain Q' of the cylinder Qy that is separated from S’r\Sr‘
by a positive distance d, vrai max^< |ttr| does not exceed a constant depending
only on S ', n, d, M0, M, m, v{M), p0{M), ft(A'), dM), and P(|p|, M)
if f(M) is small, and also on the modulus of continuity &»(/>) of u(xt i) in Qy. if
*{M) is arbitrary but u is continuous in Qj>.
Remark 3*1* For certain classes of equations (0.1) it is possible to obtain
an estimate of max^^lu^l in terms of maxr7|ux| or the known quantities in the
proUem by imposing no restrictions on the orders of growth of the functions a..
*/
556
VI. GENERAL QUASI-LINEAR EQUATIONS
and a with respect to p or by imposing weaker restrictions than the conditions of
Theorems 3* 1—3.5. As an example we take an equation of the form
= ut—atj(t, ux)uXlXj a(t, ux) = 0 (3.40)
and we will assume for tbe sake of simplicity that all functions and solutions are
sufficiently smooth.
Ftom the equality
I 1 d7u\
0 = «*»= 2-3T — 2ai) Txtdxj
1 da,, du? 1 da du*
~ IJ ""X
+auu*l',uxlxJ—y -+-2"*rr a*
**
*k
we get the inequality
vi — aijvXjXl -f- AkvXk < 0,
in which t> = u?, while
* . oa,/ aa
A*ms~ 1^7 “vv “duT’ ‘
* *
and it implies, as is known (see §2 of Chapter I), that v takes its greatest value
on
r j, i.e.
v(x, 0<max». (3.41)
This inequality reduces the problem of estimating maxq to that of estimating
the maximum of u2 on Vj.. But in order to estimate ma^ u* we must take into
accouat the boundary and initial conditions. (See in this connection §7 of Chap¬
ter V and Lemma 3-1 of this section.)
§4. EXISTENCE THEOREMS
In §6 of the preceding chapter we showed how the solvability of nonlinear
problems is established by means of (i) the Leray-Schauder principle, (ii) a priori
estimates of 1“*!^ fot all possihle solutions of these nonlinear problems, and
(iii) solvability theorems in Holder classes for linear problems. Here die question
as to whether or not the equation in question has its principal part in divergence
form is not important. The arguments presented in §6 of Chapter V for equations
with principal part in divergence form remain valid for the general quasi-linear
equations
JPusm, — ait (*, t, u, ax) uJjt. -+■ a (x, t, a, ux) = 0, (4.1)
“lr, =>Mrr. (4'2)
§4. EXISTENCE THEOREMS
557
The only difference lies in the conditions under which the a priori estimates of
the solutions of the various equations are establishedr and in the equation with
which the given equation is connected by a parameter. In view of this we will not
repeat here all of the arguments in §6 of Chapter V but formulate only the final
results (and only those that are fundamental, without their corollaries), using the
a priori estimates obtained io the present chapter for the solutioas of problem
(4*1), (42), and give whatever explanations are necessary.
In the same way as in §6 of Chapter V, we consider problem (4*1), (4.2) not
by itself but along witfi a one-parameter family of problems of the same type:
^’tusul—[tal/(x, t, a. ux) -+ (I — t)(1 4- uJ)T” 16/] *
+ ia(x, t. a, ux) — (l — —(l + <j£)T~' Ai|>] = 0.
“Irr = ♦ liv 0 < T < 1,assumingi|> £ 1+2 (Qr). (4,3)
The linear problem
V,— t, W, Wj-Hl— T)(l -f- W>x)1"~ 1 6/] V*i*j
xa <x, t, w, wx) — (1 — x) [)|>, — (l 4- 'Kr)2 At))] = 0,
®lrr = <l’lrr. (4.4)
determines a transformation r) = v of u> into v, to which the Leray-
Schauder principle is applied.
Concerning the functions t, it, p) and a(x, t, u, p) we assume ful¬
filled either conditions (2.29), (2-30) of Chapter I or, more generally, conditions
(2.29), (2.32), (2.33) of Chapter I. Either of these conditions, as was proved in
Theorem 2.9 off Chapter I, yield the estimate
max | bx | ■< M (4.5)
<?r
for all possible classical solutions of problems (4.3).
We suppose further that for (x, t) 6 Qj, ji»| < M and arbitrary p the functions
o;^.(x, t, u, p), a(x, t, u, p) are continuous in x, l, u, p, continuously differ¬
entiable with respect to x, u and p, and satisfy the inequalities
v(14-bl)m_!|2<a,/(*. t. u. />)Uj«n0 4 l/H)01-2!2.
v, |i = const > 0, (4.6)
Id&i j | j da j
+|pl)34-|o|4|-g^- I (1-f II X Hi 0 4 \P\)m. (4.7)
558
VI. GENERAL QUASI-LINEAR EQUATIONS
ISI'.+i'iH £1
<Ie + P(|p|)J(, ■4-|p|)m+I. « > 0, (4.8)
|-5|«!e + P(|/H)Kl + |pl>m~S. (4-9)
-^<le + p(l/'l)K1 + l/'l)m- «•»»)
where Pip) is a nonnegative continuous 'fraction (hat tends to zero for p —»»
and m is an arbitrary number. One can easily verify that the functions
m~ 2
afjix.t, u, p)=xa[J(x,t, «,/>)-+-(!—t)(H-p*) 2 6{
and
— (1 — (l-fil)J.)2-1Ai|)]+Ta (j:. t, a, p),
are subject to inequalities of the same type, where the constants v, /i, ji j, t and
the {unctions P()p|) in them can be taken to be the same for all r 6 [0, 1}. In
view of this, as was proved in Theorem 3-2, when S 6 02 the \uTx\ are uniformly
bounded for all solutions uT possessing the smoothness indicated in this theorem:
maxlujK/M., (4.1!)
Qt
as long as the number t in (4.8)-(4»10) is less than a certain number determined
by the numbers M, u, ft, jij and
P = maxP(p),
p>0
or if all of die solutions uT are known to be equicontinuous in Qj..
We now formulate the existence theorems.
Theorem 4.1. Suppose that the following conditions are fulfilled.
a) For (*, t) e Qj. and arbitrary u either conditions (2.29), (2.30) or, more
generally, (2.29), (2-32), (2.33) of Chapter I are fulfilled.
b) For (*, t) C Qf, |nj < M (where M is taken from estimate (4.5)) and arbi¬
trary p the functions a..(x, t, u, p) and a(x, t, u, p) are continuous and differ¬
entiable with respect to x, u and p and satisfy inequalities (4.6)—(4.10) with a
sufficiently small t determined by the numbers M, v, /t, jtj and
P = maxP(p).
#>o
c) For (*, t) e Qj., |u| <M and |p| <Mj (where Ml is taken from (4.11)) the
§4. EXISTENCE THEOREMS
559
functions a^ix, t, u, p) and a(x, t, u, p) are continuously differentiable with
respect to all of their arguments.
d) The boundary condition (4.2) is given by a function t) belonging to
f]2+P, 1+^/2^^) an£ satisfy^ on Sg = {(*, l): i € S, I = Ot equation (4.1), i.e.
i>/ —0. 0), <)>,(*. 0))^(r, 0)
+ aix- O' $ (x- 0)A>x0*. °))L€J = 0 (4.12)
(in oiAer wards, the compatibility conditions of zero and first orders are assumed
to be fulfilled).
e) S e H2*p.
Then there exists a unique solution of problem (41), (4.2) in the space '
ff2+0,1+0/2 (^,). fhis solution has derivatives 11( from L 2(Qf).
Using this theorem, die theorem on estimating nuot^ |vx| and theorems on
interior estimates of tbe solutions in stronger norms, we see that the following
theorem is valid.
Theorem 4.2. Suppose conditions a) and b) of Theorem 4.1 are fulfilled, and
that
c) For (*, t) € Qj,, |b| <M and |p| <Mt the functions a.,(x, t, u, p) and
a(x, t, u, p) are continuously differentiable with respect to x, u and p and
Holder continuous in t with exponent (3/2.
d) i|j |s £ O2,1 (ST)t max | v|)* (x, 0) | < oo, and
Q
t e c(Qt)
e) s e o2.
Then there exists a solution u{%, t) of problem (4.1), (4.2) which belongs to
C(Qt) n H2*fi’**@^2(Qj.) and kas a bounded supqt.Ux\
'llieoreni 4.1 is proved analogously to Theorem 6.1 of Chapter V. Under the
conditions of Theorems 4.1 and 4.2 die requirement that r be small in inequal¬
ities (4.8)-{4.10) can be replaced, following Theorem 3.3, by the condition that
all of the solutions uT be equicontinuous in Qj,. This change in Theorems 4.1
and 4.2 leads to the following existence theorem.
Theorem 4.3* The assertions of Theorems 4.1 and 4.2 remain valid if the
condition in them that t be small in inequalities (4.8)-{4.10) is replaced by the
560
VI. GENERAL QUASI-LINEAR EQUATIONS
requirement that all of the possible solutions of problems (4.3) be equic ontinuous
in Qr
Also we can weaken our assumptions on S and ^ at a cost in smoothness of
the solution in Qj,. For example, die following theorem holds:
Theorem 4.4.|Suppo«e conditions a)—c) of Theorem 4.1 are fulfilledwitk m > 1
d) t|>6C«r)n«2+M+^<Qr)>
e) Each point of S can be touched from without by a ball (or cone) of fixed
size in such a way that the ball (cone) does not have any points in common with (I,
Then problem (4.1), (4.2) has a solution from C(Qy) f]H2+^,t^^2(Qj.).
Theorems 4.2 and 4.4 are proved in the same way as Theorems 6.2—6.4 of
Chapter V, by means of an approximation of the equation, the
boundary and initial conditions and the domain itself by a smoother problem satis¬
fying the conditions of Theorem 4.1, and with the use of the theorems of the pre¬
ceding sections on a priori estimates in Q j. and in Q' 6 Qj. The specific char¬
acter of the conditions on S and <J> in them is such that we will now have
for closed domains Qm a single uniform estimate
max | um (x, t) | < M (4.13)
(«)
instead of < M as in previous cases. Estimate (4.13) does not yield the
conclusion that the limit function for tbe um is continuous in Qj.. This must be
proved separately by using the fact that the um are solutions of parabolic equa¬
tions. In this regard one employs the classical method of barriers, which was
developed in the first decades of our century. The method of barriers for parabolic
equations differs little from that for equations of elliptic type. In either case it
is based oo the "maximum principle,’’ which is inherent in equations of second
order of both types. We will not present the proof of Theorem 4.4, but refer to §3
of Chapter VI in our book where this is done in detail for quasi-linear el¬
liptic equations of general form.
§5. EQUATIONS WITH ONE SPACE VARIABLE
For tbe case of equations
J2ju = ut — a„ (x, t, a, ux) uxx -J- a (x. t, a. ux)~0 (5.1)
with one space variable the arguments and results mentioned above admit certain
simplifications and sharpenings. As above, we confine ourselves to tbe domains
Qj, = (-1, D x (0, T), although the methods presented here are applicable without
§5. EQUATIONS WITH ONE SPACE VARIABLE
561
any essential change to domains of the form ((*, t): ^jO) <* <^2(*)l with
+;v>* oo. Moreover, in this section, to save space we confine ourselves to clas¬
sical solutions and continuous functions Oj j(z, t, u, p), and o(x, t, u, p). From
what has been stated above it is clear as to what generalizations of the proposi¬
tions given below are possible in regard to the admission of various singularities
of the functions <*ij(*» *> », p) and a(x, t, u, p) and ot the solutions u(x, {).
For a smooth function t, it, p) equations (5.1) can be reduced to the
form of equations with principal part in divergence form, namely
u<~~lxai (X' *' *' “• u^=s0’ ^5'2^
ia which
P
av(x, t, a, p)— j an(x, t, a, x)dx,
0
ft x , dat ix% t. u, p) , dai (*, #, p)
b(x. t, a, p) = a(x. t. a, />)-1 .
In view of this the theorems proved in Chapters V and VI can be applied to equa¬
tions (5.1). Moreover, for their solutions u it is possible to estimate |<tx|^p with
respect to the whole domain Qj. without a preliminary estimate of max^jaj and
hence without any assumptions concerning the differentiability with respect to t
of the functions making up the equation. Let us demonstrate this fact.
Theorem 5.1. Suppose that u(x, t) is a solution of equation (5.1) that belongs
to C2,l(@j,) and suppose that for (*, t) £ Qf
V < fl„ (x, i, u (X. <). (■*. <))<!».
|a(x, t, u(x, t), ux(x, /)|<(i, v. n = const>0 '
Then it is possible to estimate the quantity <*>0, for any rectangle Q'=
(-/ + «, / — <) x (f j, n with (, (j > 0, from above by a constant depending only on
v, fi, Jfj =maz^ji|ux|> t and tj. For (j = 0 the quantity (ux)q) is esiimated
from above by a constant depending only on v, p, M j, e, («,.(*, 0))^ ^ and /3>0.
For the whole rectangle (,ux}q^. ** estimated from above by a constant depending
only on v, p, Mv(u£x, 0))((^(), fi and max(f |() ^ |«f<—/,f) j, |ut(/, l)|). The con¬
stant cl is determined by v, fi, U j and fi•
For a proof we consider the relation
t I
— J J S'u(umi?\jxdt*=0. (5.4)
i, -i
where a^ (*, t) - max \ux{x, *)- 4; 0 i, i is an arbitrary number, £(*, t) is a
562
VI. GENERAL QUASI-LINEAR EQUATIONS
smooth function with values from [0, l] that is equal to zero outside of Kp =
{x: |* - *q| < p|, while (*0, <„) € Qj. and p < /. If the segment Kp is contained
in (~l, I), then, by carrying out an integration by parts in (5.4), we obtain
/ ( i I
1 J t)t(x. tffdx\ + J J [-(^
2
+ («„«„ ~ «) +'2«<f>Kf)] dx at = 0. (5.5)
But if one of the ends of the interval [-2, l\, for example * = l, is contained in
ransfonn the first term
i i
J J ut{a^)xdxdt
Kp then we transform the first term of (5.4) as follows:
= f f [[- ut(x, t) + u,(i. 0](UW^-U<(/, 0K>£\)
<• -* 11
-J J ~“<K‘,£JL]dt
i ~l , 11
--\ J (•?I - J J + ut (/. OK‘^1 dt.
As a result we arrive at equality (5-5), in which a(x, t, u, p) is replaced by
a(x, t, u, p) = a(*, t, u, p) + ut(l, t). Hie fact that in deriving(5-5) it was as¬
sumed that the solution has derivatives uxt from i^Qy) is nonessential. As
was explained many times above, this can be avoided in various ways; for ex¬
ample, one takes instead of (5-4) the relation
J J[-“M + (anuxx — a)](*8F)xdxdt
h-t
= j
where
f + fl
mh(x, t) = j- J «(•*. t)dx.
transforms its first term in tbe same way as above, and then passes to the limit
as h —> 0.
From (5.5) and the assumptions of Theorem 5.1, as can easily be seen, we
§5. EQUATIONS WITH ONE SPACE VARIABLE
563
get the inequality
/
J (u<*Kf dx |' -f v J J (u%if dx dt
<Yi
mes v4yji(1 (t) dt
in which Op =* Kp x (-1, I), Ak p(t) is the set of those points x from Qp at which
ux(x, t) > k, while the constant yj depends only on v and fi for Kp C (-1, T), but
also on max ^ (| |u((±/, *)| if * = I or * = -I lies in Kp. Analogous inequal¬
ities are valid for the function - u% (x, ()• In view of this (see §§7 and 8 of Chap¬
ter II) the function /) belongs to the class y> 6, oo, 2) with y =
2yj/min{l; vl, and therefore the assertions of Theorem 5-1 hold for ux by virtue
of Theorems 7.1, 8.1 and 8.2 of Chapter II.
Let us revert to the estimates of max |ux| for solutions of equation (5-1). In
§3 we gave estimates of max |»x| directly for any n > 1. The assumptions under
which such estimates are possible can be weakened somewhat for n = 1. If n = 1,
m = 2 in conditions (3-2), then instead of conditions (3.3)-(3.6) we can take the
condition
I|(1 +1p I >3 +1
| 1 “■..El [ (1 +1 p [) +1 a (x. t, u. J»)|
<ii(!“l)(i+|p|)J. (5-6)
for under these assumptions equation (5.1), when written in the fonn (5-2), satis¬
fies all of die conditions in §§3 and 4 of Chapter V, and therefore tbe theorems
concerning the estimation of maxp/ |«xj and established in these
sections are valid for it.
For it m 1 and m ^ 2 we can eliminate from assumptions (3-2)-(3-6) of this
chapter the requirement that f(|u|) be small in condition (3-6) with respect to
dajj/da. This can easily be seen by tracing the estimates of the corresponding
terms in inequality (3- 25) and noting that for n = 1 it is possible to represent the
function \vxx\ in the form(l/2)M>~*/,*|w | and to cancel the terms corresponding
to it at the expense of a term existing in die left side of the inequality.
We will now formulate a theorem on the solvability of the first boundary value
564 VI. GENERAL QUASI-LINEAR EQUATIONS
problem foe equation (5.1), which follows from die just-described estimates of
maxerl“*l 30 ^ above, we take a family of boundary value problems
in Qj,-.
-2>ss«, — [«„(*, t, a, «,)4-(l -T)(I ']«„
+ ta(x. t. a, «x)-(l — (1 +^)^‘^J==0.
“ li»o = <i> (•*. 0), u ( /, = (— I, t), u (I, 0 = <|) (I. /).
0<i<I, (5.7)
where iftix, t) is a known function defined in l)j.. Tbe following proposition is
valid for than:
Theorem 5.2. Suppose the following conditions are fulfilled.
a) For (*, t) 6 Qf and arbitrary u
«„ {X.t.u, 0)>0 (5.8)
and
-a(x, t. u, 0)u<|«|®(|«|>- = (5.9)
{this yields the a priori estimate max^^,|t>f| <M for all possible (classical) solu¬
tions of problems (5.7)).
b) For (*, t) e Qj., |b| <U and arbitrary p, the functions Ojj(*, t, u, p) and
a(*, t, u, p) are continuous, differentiable with respect to x, u and p and satisfy
the inequalities
vd+lpD—^Ont*.*.*. PX^CI+IpD”'2. *>». (5.10)
| aa"(^--g}-[(» +|Pl)3 + |-^f-|(i + [P1)2
+ |^|(i + ipi) + w<M,a + wr. <5. H)
|^-|(l+|j»D, + |-S-|<l«4/,(|pl)10 + |P|)*tI (5.12)
_ < [E + P (| p |)] (i +1 p | f, (5.,3)
where m is an arbitrary number, P (p) is a nonnegative continuous function that
tends to zero for p —> oo, and e is a positive number that is less then a certain
§5- EQUATIONS WITH ONE SPACE VARIABLE
565
number determined by the quantities M, v, ft, ft j and
P = maxP(p)
p>0
(inequalities (5.10)—(5*13) guarantee the estimate maxQj^l <M j).
c) For (x, t) 6 Qr, |u| <M and |p| <Mj the functions au(x, I, u, p) and
a(x, t, u, p) are Hdlder continuous in the variable t with exponent jS/2, and in
the variables x, u and p with exponent /3.
d) The function </)(.%, t) 6 ff 2 + ^’onrf
[>|)((x, 0) —a„(x, 0. 0), 0))vtJ^(-c, 0)
+ a (x, 0.. (x, 0). «|>x (x, 0)] |jr_1, = 0. (5. J 4)
Then problems (5-7) are uniquely solvable in for all r from [0, lj.
Remark 5.1. Condition b) in Theorem 5.2 can be replaced by the following
condition.
b') For (x, t) 6 Qf, |u{ <M and arbitrary p, the functions ajj<*, t, u, p) and
a{x, t, u, p) are continuous, au(*» £> “> p) ** differentiable with respect to x, u
and p, and the inequalities (5-10) with m - 2 and (5.6) are satisfied.
Remark 5.2. Condition b) in Theorem 5.2 can also be replaced by the follow¬
ing condition.
b") For (x, t) € Qj, |u| <M and arbitrary p, the functions Oj, j(*, t, u, p)
and a(x, t, u, p) are thrice continuously differentiable with respect to x, t, u and
p, a j j(x, t, it, p) > v, v = const > 0, and the function q(x, t, u, p) =
a(x, t, u, p)/<Jjj(*, t, u, p) is subject to the inequalities
l<le+/>(lpl)1(1 +l^l>3- <5>16)
where € and ^(|p|) are the same as in(5.12).
Upon comparing assumption b) with b") it is seen that in the latter more
stringent conditions are imposed on the smoothness of the functions a^ t, u, p)
and a(xt t, u, p), while the restrictions on the growth of and a with respect
to p in b*) are somewhat weaker than in b) (namely, it is not required that e be
small in (5-13))> On the other hand assumption b') is che weakest of ail the con¬
ditions b), b') and b"), but it involves only the case m * 2.
566
VI. GENERAL QUASI-LINEAR EQUATIONS
In order to show that Remark 5.2 is valid we must prove that condition b”)
ensures an estimate of raaxgr|</|. Estimates for max^^ t)| follow from
Lemma 3.1 of the present chapter. And an estimate of |ux(x, t)| at interior points
of Qj. under conditions b ) has been established by A. F. Filippov in the note
l2^]. We will give a proof of it.
Lemma 5.1. If assumptions b") and (5.10) are fulfilled, then is
estimated from above by a constant depending only on v, p., t, P(p) and max _ |u |.
*7* *
We introduce an auxiliary function w(x, t) by means of the equality
w{x, t)*z=<p(x, t, u(x, t), ux(x, t)), (5.17)
where <f>(xP t, u, p) is a certain twice continuously differentiable function of its
arguments in die domain f(*, tf u, p); (x, t) € \u\ < M, - *> < p < 00} that is
strictly monotonically increasing with respect to p and tends to 4*00 for p —+00
and to — oo for p > - ». In the following we will impose several other restrictions
on <j>{x, l, ur p) and then construct it.
In order to simplify the calculations we assume that q(x, t, u, p) does not
depend on %. Then we can also take <j> to be independent of x: u, p).
From (5* 17) we have
w, = <PA + (P>“«. (5-18)
w, = <P, + +VpHxi- (5.19)
From (5. IS) and equation (5.1) it follows that
«, = ~r — <W>-
and from this and (5*19) we get
w, = <P/ + tWjc — (Vx + <M)1
<6-M>
We require that for large |p| die function u, p) satisfy the equation
«» />)?(*. «. P) — 0 (5.21)
in partial derivatives of first order. It is well known that such an equation is
satisfied by the first integrals of die ordinary differential equation
dp = g«. P) (5,22)
du p
in which t enters as a parameter. The right side of equation (5.22) has a singularity
§5. EQUATIONS WITH ONE SPACE VARIABLE
567
fot p = 0. We therefore take instead of (5.22) a somewhat different equation:
(5.23)
where F(£, u, p) coincides with q(t, u, p)/p for |p| > 1 and is a thrice continu¬
ously differentiable function of its arguments in the whole domain
[(<, u, p)\ t £ [0, 7"], |«| ^ M, — oo < p < oo}.
We will take as u, p) one of the first integrals of equation (5.23) that satis¬
fies for £ 6 [0, 71, |u| < M and arbitrary w die inequality
OO
l<Pi(*. «. Pit w))K®(|w|). j ~^=oo, (5.24)
where p=p(t,«,«i) is a solution of the equation
w — <f (t, a, p). (5.25)
We will construct such a function u, p) below, but let us now show how
it is used to estimate |“x(*> t)|« Ve consider relation (5-20) as an equation for u>:
w, — auwXJC — b(x, t w, w*) = 0, (5.26)
in which
b(x, t, ®. = + (5-27)
for large |p| and
6(X. t, ©, «Jt) = f< + (JgU.+^^.i.)«jr
— + <P,?)-^p-h [•^•(Vx + 'Pp?)] (5.28)
for all ocher p. Here in the right sides of (5-27) and (5.28) we must assume that
uxx has been replaced by (wx - 4>uux)/4>pi that p and ux have been replaced by
tbe function p(t, u, w) which is a solution of equation (5-25), and that u has been
replaced by the solution u (x, t) being investigated. For u>x = 0 and large w
(i.e. large p) the function b(x, t, w, 0) satisfies by virtue of (5.24) the condition
\b(x,t, w. 0)|<<»(|®!), (5.29)
i
and therefore max^|te| for a solution to of equation (5.26) can be estimated
from above by a constant.depending only on 4(r) and mur |u>|, i.e. on known
quantities (see Theorem 2.9 of Chapter 1). This in turn yields the desired esti¬
mate for max^ |ux|.
568
VI. GENERAL QUASI-LINEAR EQUATIONS
Thus it only remains id construct a function u, p) having (he properties
enumerated above. Let p =■ p(l, a, pQ) be a solution of equation (5.23) satisfying
the initial condition
P\u=-M~ P(1' ~M' Po) = Po- (5'3°)
Such a solution exists on the whole interval -M < a <M for any pQ, and the
curves p = p («, u, pq) fill without intersections the whole strip -M <u <M of
the (u, p)-plane. These assertions are valid on the basis of (i) a well-known
theorem on the unique solvability of the Cauchy problem for ordinary differential
equations, and (ii) the properties of the function F (t, u, p), namely its smoothness
and its not greater than linear growth with respect to p (see [117, 551). The lat¬
ter follows from assumption (5.15), more precisely from the fact that
for |pj > X, while for |p| <1 we have
max \f,
Owing to these inequalities we have, as is easily calculated, the following esti¬
mate for the solutions p(t, a, pQ):
\p(t,u, PoJK8®1*"(po+-^-) foc (5-31>
The function p(e, a, p0)is a strictly monotonically increasing function of p0
(for fired t and u); this follows directly from the way in which the integral
curves p <• p(t, a, Pq) are distributed in the strip —M < u <!H. Therefore the
equation p « p (/, a, pQ) can be uniquely solved with respect to pQ:
Po = Po(t> «. P)< (5.32)
where Pq —> ±oo when p —► ±oo respectively. We will show lhat tbe function
Pg(t, a, p) can be taken as the desired function <j>(t, u, p).
As is well known, dp (t, a, Pq)/<?Pq satisfies on die integral curve p =
p(t, u, p0) tbe linear equation
d dp dF dp (5.33)
du dp0 dp dp, '
and becomes equal to unity for a s —U, i.e.
4£-| =1. (5.34)
Op* lu= —iM
Therefore dp/dpn and hence also <9p0/5p = (dp/dp0)'1 are strictly positive.
§5. EQUATIONS WITH ONE SPACE VARIABLE
569
Moreover, we can obtain an estimate from below for dp/dp0, starting from
(5.33) and (5-34) and tbe assumptions (5.15). Namely, from (5.15) it follows that
and
said hence
dp (<, u, p„)
fol lPl>1
dF_\
dp I
w*—“P
Jf4
-M J
j (5.35)
-M
Further, on the basis of theorems from die theory of ordinary differential
equations on die nature of the dependence of the solutions of the Cauchy problem
(5.23), (5.30) on the argument u, the initial value Pg and the parameter t (see
[117, 55])
we can conclude that p(i, a, Pg) is a twice continuously differentiable
function of all its arguments, and from the positiveness of dp/dp0 it follows that
these same properties are possessed by die function Pg(t, «; p) inverse to it
Ve will show, finally, that property (5*24) is valid for u, p) a p0(t, a, p),
or, more precisely, that
| <Ml+IPol). (5-36)
p-p(tt U, Pt)
Indeed, upon differentiating with respect to t the equation
p = p(t, u, p0(t. u, p))
(considered as an identity in the independent arguments t, it and p), we obtain
n dp . dp dp,
dt ' dpQ dt *
ue.
— _i£./j£.'r‘ (5.37)
dt — dt\dpj •
Owing to (5.23) (he derivative dp/dt satisfies on the trajectory p =
p(t, u, Pg) the equation
JL ~ j?£_ 4£. (5.38)
du dt dp dt * dt
and the initial condition
*2.1 _o (5.39)
* L-M
and since by virtue of (5.15) and (5.31)
570
VI. GENERAL QUASI-LINEAR EQUATIONS
j Up I c’ j nr j ^ ^ I p 1 c Ci I c*'
where c2 — 4fX,es>tM, and c3 = c-\-ceSllia, it follows that
I i£-1 £d-£ai±£i .2 (^»+c) M
| dt I ^ 6|i c
This together with (5-37) and (5.35) yields the desired inequality (5.36). Thus
we have shown that Pg(J, u, p) possesses all of the properties which are required
of the function <f>(t, u, p), and this, as was shown above, implies that it is possible
to estimate max^^.|u>|, and hence also maxg^,|ux|, in terms of known quantities.
Lemma 5-1 is proved.
CHAPTER VII
SYSTEMS OF LINEAR AND QUASi-LINEAR EQUATIONS
We will consider in this chapter a particular class of systems of differential
equations of second order and of parabolic type, linear and quasi-linear, and we
will investigate the same questions for it that were investigated for one equation
in the preceding chapters. It is selected on the basis of the following criterion:
the systems are those for which a weakened form of the maximum principle is
valid. Namely, if one rejects the minor terms from all of the equations of the sys¬
tem and freezes the coefficients in the major terms, then, for the modulus |u(», «)|
of the classical solutions uix, t) of such a simplified system the maximum princi¬
ple holds in the same form as for the heat equation. A11 of the equations of such
systems have the same principle parts. It turns out that all of die basic assertions
established in the preceding chapters for one equation are also valid for these
systems (see [65°]). We will prove them, but only in detail at those points that
essentially distinguish systems from the case of one equation.
§§8-10 contain a survey of the basic results that have been obtained on gen¬
eral boundary value problems for linear systems of equations of parabolic type.
§ 1. GENERALIZED SOLUTIONS OF LINEAR SYSTEMS
We will consider in Qj systems of the form
u< - jrt ia‘1 (x- 0+ A‘{x- n a)+B' (x’ t]
+ Aix, f)n==n_f. (1.1)
where u(x, t), f;(*, t), f(x, t) are the /V-dimensionalvectors (ul(x, f), ..., uN(x, ?)),
if}, • , ff). (/*. • • •. f1) respectively, A^x, t), B^x, t), A(x, t) are N x N mat¬
rices with elements a^Kx, t), bfHx, t), Hx, t) respectively, and c) are
scalars.
571
572 VII. SYSTEMS OF EQUATIONS
We will use the following notation in this chapter:
The membership of a vector function in some space will mean lhat all of its com¬
ponents belong to this space.
We will assume that the system (1.1) is such that the inequalities
vis<ayl,|;<H|2. V> 0, (1.2)
are valid for any real f = (f1P - • •, <fn). Condition (1.2) means that the system
(1.1) is parabolic.
For (1.1) we will consider the first boundary value problem, i.e. the problem
of finding a function u(x, «) = (u*, ■ - - , u^) satisfying the system (1.1) and the
conditions
«lsr = 1>(s. 0. (1.3!)
H-o = *<>(•*>• (1.32)
By a generalized solution from ^ 2^C^rom ^2 or
P»-1/2«?r» of
problem (1.1), (1.3) we will mean a vector function u(*, t) € V^Qt) (I^|' °tC?-r)
or ^’l/2^r)), satisfying condition (1.3j) aod *uch that for a°y vector function
ij(*, t) 6 the integral identity
t
J u(x.t)i\(x,t)dx+ j J J—ui),-f-(a;/U -j- j4(u + f4)tj
.8 0 Q '
-f- (£,u,. i--4u+f) y\\dx dt-j 0)dX"ft
Q
is fulfilled for almost all (for all) £ from [0, T].
In this section we will discuss the unique solvability of problem (1.1), (1.3)
in the space j).
As was shown in § 2 of Chapter I, in order to retain the basic properties in¬
herent in the problem in its classical statement it is necessary to impose certain addi¬
§1. GENERALIZED SOLUTIONS
573
tional restrictions on the coefficients of the system (1.1), which we will formulate
as usual in terms of their membership in the spaces L^ r(Qj>). These restrictions
are the following:
S l^l3'. S|S,|2; Ml
where l/r + n/2q = 1, and
1. r, Qt
?€(§•■ oo];
rg[l, oo)
for
n> 2.
f o°l;
r en. si
for
n— 1,
*«6 M«r>
+ n/2?j - (n + 4)/4,
and
*€[*■ »]■
r.en. 2i
for
B> 3,
»«€(». 21.
r.en. 2)
for
n = 2,
?,€!». 21,
r,6 Ml
for
n— i.
(1.5)
(1.6)
In conditions (1.5) and in the second of conditions (1.6) the indices q and r can
be assumed to be different for each of the elements of the matrices and each of
the components of the vectors on which these conditions are imposed. Nose of the
assertions is essentially changed in this connection; their formulations and proofs
are merely lengthened.
In addition, we will assume for the sake of simplicity that t/r(s, t) - 0.
Just as for one equation, the following theorem is valid for systems of the
fonn (1.1).
Theorem 1.1. Under the fulfilment of conditions (1.2), (1.5), (1-6) and the
initial condition ui, = ^r„(:
Pl,lh{
If furthermore the |B.|2 and \A\ belong to £* (QT), r>q and f e L\ q.(QT), r > q
then u 6 V\-l/\Qrl
iJ=0 ■ <lrj.x) e L^tl) the system (1.1) has a unique generalized
solution u(x, t) from P^>l'2{QT). Any solution from V^Qj) belongs to V\’1/2(.QT).
The proof is carried out in exactly the same way as in the case of one equa¬
tion, and is based on the a priori energy estimate
574
VII. SYSTEMS OF EQUATIONS
which holds for an arbitrary generalized solution from V^(Qf) of problem (1.1),
(l.J) with ifris, t) s 0.
The constant c in this inequality is determined only by n, v, ft, (he norms
(i j of the coefficients and the indices q and r in conditions (1.5).
The proof of inequality (1.7) does not differ essentially from the proof of ine¬
quality (2.2) in Chapter III; it is only necessary to replace the usual multiplicatior
of functions by the scalar multiplication of vector functions.
Uniqueness in V^QT) follows at once from inequality (1.7) applied to the
difference of two possible solutions, while existence can be proved, for example,
by Galerkin’s method.
§2. ON THE BOUNDEDNESS OF max^M
In this section we will estimate the maximum of the modulus of a solution
u(*, l) of the system (1.1). According to the examples of Chapter I, for this to be
possible we must impose additional restrictions on the coefficients of (1.1) and
on die f. and f. The conditions under which the boundedness of generalized sol¬
utions from will be proved are formulated as follows:
si*»p.
gup.
i*i
9. r, Qt
► 1*1.
where l/r + n/2? « 1
*e[-
with
3TT^x,)1 °°]'
*€U. oo].
*1
^•4
0<x,< 1, for n 2.
■T^d-
0 < K, < -j. for n = 1.
(2.1)
(2.2)
The numbers q and r can assumed to be different for each of the elements of the
vectors f(. and f and for each of the elements of the matrices A-, Bi and A in
conditions (2.1). It is sufficient that they be subject to condition (2.2). For the
sake of simplicity we will assume that a\sj - 0. The basic idea for obtaining an
estimate of max^ |u| is the same as for one equation in §7 of Chapter III. But
its realization for a system of equations is complicated by the fact that our a
priori assumptions on the solution (that u 6 {^^((Jj,)) do not permit a develop-
§2. BOUNDEDNESS OF 575
meat of it in the same way as for one equation. Let us explain. For one equa¬
tion the inequalities underlying the definition of the class — ) and there¬
by permitting a determination of the boundedness of u were obtained from a basic
integral identity satisfied by the function u, in which the arbitrary function jjU, t)
is taken equal to u^\x, t)= mar |u(*, () - Oj. For a system of equations quite
similar inequalities are obtained from the identity (1.4) if the tector function
rj(x, t) in it is taken equal to u(*, t) max\v{x, t) -k; Ol, where v(x, l) = a2{x, t).
But we are not justified in doing this if we do not know in advance, for example,
that |u| is bounded or at least that u2 6 K*>°(()j.). In view of this the function
tj must be taken in a more complicated form, by introducing a "slicing off" of
v(x, t) at large values of |u(x, t)|. In addition, as an intermediate step, it is
necessary here as elsewhere to carry out a smoothing with respect to t. Let us
show how this is done.
First we note that, in the same way as for one equation, tbe identity (1.4)
implies tbe identity
J [U*<H + (aVuxj + -ditt + h)h
Qt,
+ (S/Ox(+/4u + f)(>Tj]rfxrf< = 0 (2.3)
with any vector function ij from P*>°(@j,) and tj < T - h (see §2 of Chapter
III). The notation ( )A means an averaging over £ (see (2.10) of Chapter HI). In
(2.3) we put
ij(jr, /) = 2uA (jc, t) max [^(x, t)—k; OJ == 2uhvl'Ml'‘\
where uA(*; i) is ao averaging of u over t, v(x, t) = u2(*, i), = (u^)2, the func¬
tion Vy(x, t) is equal to Vh(x, l) for vh(x, t) < M, and to M for vh(x, t) > M,
while k is an arbitrary nonnegative number. We transform the first teim obtained
in this connection in accordance with the equality
2“«“*<(4)= - jw W (2.4)
which can be verified directly from the definition of the functions v\ and
After substituting the right side of (2.4) in place of uA(1j in (2.3) and car¬
rying out an integration with respect to t in these terms, we then pass to the
limit in all of the integrals of (2.3) as A —* 0. This, as can be easily verified
(see Chapter III), is admissible and leads to the relation
576
VII. SYSTEMS OF EQUATIONS
J dx I'’ — i J (v%fdx £
+2 J [(«,,« ++
Q»,<*r 1
+ (Bu^-(-i4u + f) ut(<g>]dxcf/ = 0, (2.5)
where Qtl(k) denotes tbe set of points (*, t) of the cylinder Qt at which
Vjj(x, t) > k. In order to estimate the different terms of (2.5) we make use of the
following inequalities:
< +1S m* f ^+«w+12 Mi i2 ^
< «*ff + 7 2 |*. l^+'WBJ* + 7
1*1 1-1
n
< eu>!3’ +8 W5i)*•+-7 21 f< 12(w* + o.
n
<e^«S*+l2l
2 j A\iuv{$
2|fut-W|<2|f||n(^'<2|f|(TOyl,)7<2|f|(W;H-4- 1).
Here we have taken into account that v"'vjJcJ. = for any m and < v.
By virtue of these estimates (2.5) implies the inequality
T f «WwS5» dx f'+v | J2b*«5») + OT dx dt <
a Qt,<*)
§2. BOUNDEDNESS OF 577
<yJ («<*)),^f“0+ J [3en^+2e(t.wy
(n C<1< n
2Sm«p+Sib'I8+2£I/4i
i-i f~i
+ Slf/P + 2e|f|j«w* *-
i-1
from which fot c = v/4 we in turn obtain
dxdt, (2.6)
±J f [2ny$+W\J\dxdt
S! Qt, m
<y f (®l*,)J«?Jc| + J ^(mtM+l)dxdt, (2.7)
n * rt.'t*\
where
2Si^i2+Si5>i2+-Ji>li+Sif<p+j|fi
' /-i /-i z«i '
and from (2.7) the inequality
Vl{o<<<J dx+QJj2u^+W]dx"j
<||*W(jt. 0)|g q+2 J ® (wjH-1) dx <if <|| ti<*> (X, 0) II* a
+ 2||®llc.r,Q,i(i!)(ll/wil, || - 11||- Q, ,*)). (2.8)
in which ul = min {%, v/2\, while q = 2?/(? - 1), ~ = 2r/(r - l). From this, by
virtue of the inequality
» *
£ [(V^),(P- s 4^r-(v«+
*-1 im1 '*
< -2^7 + «*®L) < u>m+jv%* (2-9)
Al
and the fact that «/0^ = v, for k = 0 and tj = T we get
°) llj, Q
+ 2II ® l|5, r> 0(1(» lib, 0, + II1 III ?. Qt} (2.10)
578
VH. SYSTEMS OF EQUATIONS
This inequality has the same character as inequality (7.8) of Chapter III for a
fixed k. From it, by estimating the right side in the same way as in § 7 of Chap¬
ter III (see formulas (7.9)—(7.11)), we establish the uniform (with respect to U)
boundedness of die integral » and hence also of the integral as
long as the height does not exceed a certain number r (see the type of condi¬
tion (7.13)). By repeating the argument for the cylinders (?r>2r « 0 x (r, 2r),
Q2r jr = fix (2 r, 3r) and so oo, we exhaust the whole cylinder Qf in a finite number
of steps and find that the integral |t>|Qy does not exceed a certain constant, which
can easily be calculated from the quantities n, p, fij, r, q, mes 12 and T.
After this we return to inequality (2.8) with an arbitrary k > 0. Its right side
for any M > 0 does not exceed
0)11= a + 2!|®ll,.r. (?,,<*>(!®II?,F, <?,(*• + » <V‘>>
while the supremum of the left side with respect to M € [0, <*>) is
v, | max || a -|- J dxdt\
> v,-I «<*%,_•
Therefore
v* I «“><*. 0) ||| n
+ 2 II © I1?1 r- Q,|(*)(|lt’lly,7, II * II* r. 0,( (*))• (2.11)
For k>k = maxxeQti(*, 0) + 1 we obtain from (2.11) the inequality
p(|W + ll*l|r- %m) (2.12)
which essentially coincides with inequality (7.8) of Chapter III, from which in §7
we deduced the boundedness from above of the function u entering into it. In our
case this implies the boundedness of v in Qtl and an estimate for vrai ma*Qtl v
in terms of k and the known parameters from conditions (1.2), (2.1) and (2.2), as
long as tj does not exceed r. Repeating the argument for the cylinders Qr 2p
Q27 and so on, we obtain the desired estimate of vrai ma.n.QyV and hence also
of vrai maxQy |u|. Thus the following theorem has been proved.
Theorem 2.1. Suppose u(x, t) is a generalized solution from V^(Qy) of
the system (1.1) that is equal ta zero on Sy and is bounded for t = 0,^ and sup-
1) The assumption chat u(x, t) is equal to zero on St is not essential; it can be re¬
placed by the assumption that u|sj- is bounded.
§3. AN ESTIMATE OF |u|(^ 579
pose conditions (1.2), (2.1) and (2.2) are fulfilled for the system (1.1). Then
u(z, t) is a bounded function in Q j, and vrai maxp j-juj is estimated from above
by a constant depending only on n, vrai max|u(x, 0)|, T, mesQ and the para¬
meters v, fip q and r from conditions (1.2) and (2.1).
Under conditions (1.2), (2.1) and (2.2) an arbitrary generalized solution u of
system (1.1) belonging to (by virtue of Theorem 1.1, any solution ftom
V2(Qt)) will be a bounded function in any subdomain Q' C Qf that is separated
by a positive distance from diac part T'cTj. on which u is not bounded. We
formulate this in the form of a theorem.
Theorem 2.2. Suppose conditions (1.2), (2.1) and (2.2) are fulfilled for the
system (1.1). Then for any generalized solution u of it from the quan¬
tity vraimaxQ' |u|, where Q' is a subdomain of Qj. that is separated from Tj. by
a positive distance d, does not exceed a certain constant determined only by n,
v, fi, fjj, q and r from conditions (1.2) and (2.1), the norm |u||2i£r and d. If
also vraimax^gQ |u(x, 0)j = Mq < oo, then vraimaxjj'^g ** finite and is
estimated from above by a constant determined by M0 and the same quantities as
above, only this time the role of d will be played by the distance from Q' to S.
Theorem 2.2 is proved in basically the same way as Theorem 2.1. The modi¬
fications of the arguments connected with the necessity of introducing a cutting
function are analogous to those given in the case of one equation (see § 8 of
Chapter III). We will not repeat them here, assuming that the reader can do this
independently.
§3- AN ESTIMATE OF |»|^>
Let us show that under die fulfilment of inequalities (1.2), (2.1) and (2.2) it
is possible to estimate not only maxQj, |u| but also the HSlder norms for each
component of u with some positive exponent a> 0. To this end it is sufficient
to prove that bounded generalized solutions of the system (1.1) from
belong to the class B|My Sj, —, 8^, y, f, 8, k). Suppose u(*, t) 6
K3,'°(<?r) and vraimaz^j-|u(x, «))=</< For simplicity we will assume that
0 < uH*, f) < 1, i * 1, ■ ■ •, N- This can easily be achieved by introducing in place
of u‘(x, t) in (1.1) die function u‘(x, t) = (u‘ + M)/2M. We assume that this has
already been done.
As the functions ^(u) for u(x, () we take
580
VII. SYSTEMS OF EQUATIONS
<p« (u)=10A/«' + S(u')s.
+ /-!
N
(p*_(u)= 10/V(1 — bO+SC"')’.
By virtue of Lemma 9.3 of Chapter II it is sufficient to verify that the functions
“’*(*> *) = t)) satisfy conditions 3) and 4) in the definition of the class
B|w(Pr> MvSv---,S6, y, 8, f, k).
For this purpose, in the identity (2.3) we put ij = 5/V$(*, t)e*^ and =
UA(*, «)$(*, t), and add the resulting identities. Here 4K*, t) is an arbitrary
bounded function from j.) and is the vector whose Ith component is
equal to 1 and whose remaining components are equal to zero. This gives us
U
J J u*, (5A/e"l® + U4®)rfA:d/
o a
t,
+ J J [(«„»,, + -4,0 + *,)„ (WeW® 4- uA1*),,
o u
-f (S,u^ 4- Au +f)h (SNeW +• u„<D)] dx dt — 0. (3.1)
We then put <tK*, t) = 2£2majc {«)*/(*, t) - k; Ol s 2w^>‘^(x, *)£2, where k is
some number, &(x, l) is a smooth function that is nonnegative in some strictly
interior cylinder (?(p, r) = Kpx (,Q - r, ,Q) and equal to zero for x 6 Kp, and
We subtract the equality obtained with ij = tg - r from the same equality with
, j = tg. As a result we will have for the first summand
j j [w* (*C]J dx J** — J K, [w“ <*'f dx dt.
a l’~x q(i>. t)
Taking this transformation into account, we pass to the limit for ft —»0. In
what follows we will write simply w(x, t) in place of w^x, t). As a result of
tbe limit passage we obtain
J \wW(x, t)t(x, t)]3dxf’
*'T
-4- J (2at + a, dx dt =
<?<P.TJ ’ 1 >
§ 3- AN ESTIMATE OF |u|(^. 581
QIP. *)
—(Atu+ f()[2«, «<*>£*-4-(10Ate<'> 4- 2u) wW?
-+ 2 (lOAfeif + 2a)«t*)Kx<| —
— (BiHxt + An-f- f)(l<MVew -|- 2n)«<*){?J dx dt. (3.2)
The left side of (3.2) is estimated from below by the expression
i f [«.!»><*, t)U*. t)]2dt(‘
V.«
-+- 2v J -f- j (w?1)^2] dx dt.
Q(p.
Here and above = 1* € Kfi: w(x, t) > i!. All of the summands of die right
side are estimated in terms of tbe second and third summands of die left side of
(3.2) and in terms of
J (wOf 1.1 dxdt, J (wW)2!C|| £,!</*<«, 1*'(Tp. t).
Q(fo x) 0(H, t>
where /, £
H(A. p,-r) = J [mes<4*iP(f)l» dt, 1+M,
U~1
r — 7^y(l +x), and k = ~L.
By virtue of (2.2) the parameters q and f are subject to conditions (3.3) of
Chapter II. The estimates are carried out in almost the same way as in die proof
of the boundedness of v(x, t) in §2.^ In a certain sense they ate even simpler,
since we already know that u(x, t) is bounded. As a result we obtain from (3.2)
the inequalities
]■ t^Ux.t^dx + v j («<*>S1 'dxdt
A*,p<M Qip.»)
< J (vPH&dx
At. p ('o'"')
r t £(•+*) 1
+v[ max (ICJ’ + IU) I (®csl)2 dx dt +jx' (*, p. t) (3.3)
|eiP.« Q(p.t) 1
1) They are also very similar to estimates in § 20 of Chapter HI* For this reason
our presentation of them is quite brief.
582
VII. SYSTEMS OF EQUATIONS
for p < Pq or r < rg and any k with constant y being determined only by n, U,
ft, the indices q and r and the norms of the known functions from conditions
(2.1). The same inequalities also hold for u> = w[_, Inequalities (3.3) indicate
die fulfilment of condition 3) from die definition of the class By
the same token the membership of u(*, <) in %^{Qy, • < <) is proved. By virtue of
Theorem 9.1 of Chapter II u will belong to Ha'a^{Q j.) with some a > 0 and the
norms |u|^! for any strictly interior subdomain Q' C Qf are estimated in terms
of known quantities and the distance from Q' to F j. in order to obtain an esti¬
mate of the Holder norm of u in the whole cylinder Qy it is necessary to obtain
inequalities (3.3) when ft > max^r) pjr w(x, t) for w = uil± and in the cylinders
Qip, r) intersecting the surface Tj.. Such inequalities actually hold and their
derivation is exactly die same as the above derivation of inequalities (3.3), since
®(x, 0 = *<*>(*. I'liHx, t)£Vi°(Qr)
for any .smooth function £(x, () that is equal to zero in a neighborhood of some
circular cylinder Qip, r) intersecting die boundary Ty, and this is all that one
needs to know about $(x, t) in order to derive inequalities (3.3).
Thus u(*, t) belongs to the class %ijN(QT |J FT, ■ - •) and, 'Consequently,
we have
Theorem 3«1. Suppose conditions (1.2), (2.1), and (2.2) are fulfilled for the system
(1.1). Then any bounded generalized solution uix, t) of the system (1.1) from
V\’°iQT) belongs to lp'^HQj.) with some a > 0 and the norm |u|^!, where Q'
is an arbitrary strictly interior subdomain of Q'j., is estimated from above by a
constant depending only on n, M = vraimax^^. |u|, N, the quantities v, p; p^. q
and r from conditions (1.2), (2.1) and (2.2) and the distance from Q' to Ty. If
also S satisfies condition (A) and u|sr 6 H^'^^iVj-), then u € Ha,a^2iQT)
and |o| ^ is estimated from above by a constant depending only on n, N, U, v,
I1’ Mj» 9> r from (1.2), (2.1), (2.2), /3, !u]f> and the constants ag and &Q in the
formulation of condition (A). The number a > 0 is determined in both cases by
the values n, N, v, p, ftj, f}, the norms of the knoum functions and the indices q
and r from conditions (2.1).
§4. ESTIMATES OF HIGHER NORMS
583
§4. ON ESTIMATES OF |uj(*>, AND OF OTHER
HIGHER NORMS OF THE SOLUTIONS
In §§ 2 and 3 we showed that, if the coefficients and free term of a system
of equations possess somewhat better properties than the properties enumerated in
§ X, then any generalized solution of die system from die class will be
a Holder continuous function of (*, t). A further improvement in die properties of
die coefficients and free term of the system creates a further increase in die smooth¬
ness of all of its generalized solutions, with the form of this dependence being
j ust the same as for one equation in Chapter IQ- We will dwell only oo an a priori
estimate for |ux|^> All of the remaining propositions are proved either in the
same way as for one equation or actually follow from the corresponding propositions
established by us in Chapters III and IV for one equation. An estimate of |ux|^,
for the whole cylinder QT will be proved in the nest section directly for'quasi-lin-
ear equations. Tbe proof given there naturally simplifies for the linear case.
Thus, suppose the coefficients and free teims of the system satisfy, in addi¬
tion to (1.2), die conditions
IS- w mi- £-• «•»>
in which tbe numbers q and r are the same as in (2.2). As to the solution u, we
assume that it is bounded and its derivatives ux are elements of V^’°(QT). We
will show that an estimate of can be obtained on die basis of Theorems
2.2 and 3.1 established in this chapter. To this end let us demonstrate that the
vector function V = uxJ cao considered as a generalized solu¬
tion from 0{ a system of the form (1.1), the coefficients and free terms
of which satisfy the conditions of Theorems 2.2 and 3.1.
In the integral identity (1.4) we take ij(*, t) = «), .where f4(*, l) is
an arbitrary function from that together with its first derivatives with
respect to Jf is equal to zero on Sy, -and then we carry out an integration by parts
in the first, second, third and last terms, transferring the derivative d/dx^ from
die second factor to the first. After this the equalities obtained for i = 1, —, n
are added and the result is written as an identity for U and 3 = (f j, f
f U(x. t) S (x. t)dx+ j J [— US, + («„U,y + ^,U -f F() S^] dx dt
00 — 0)dx = 0,
0 (4.2)
584 VII. SYSTEMS OF EQUATIONS
in which die A. are die matrices with elements
„dk, ij .k.di . it.k , da!J .1
<1/ e=0ibj -\-Gi O/T 6tf»
I. j, k= I, ..., n; d, 1=1 N, (4.3)
and die F(- are the vectors with components
I, 1 n, d— 1 N.
The identity (4.2) is valid, as is easily seen, for any {unction S from j,).
It implies that V is a generalized solution from of the system
V,-~4t(at>V'i + *'U + F') = °' (4-4)
which is a special case of the systems of form (1.1). By vittue of assumptions
(1.2) and (4.1) and the already kaowa boundedness of |u(x, /)| the conditions of
Theorems 2.1 and 3.1 are fulfilled for die system (4.4), and thus their inferences
concerning the boundedness of |U|^ are valid. Thus we have proved
Theorem 4.1. Suppose conditions (1.2) and (4.1) ore fulfilled and suppose n
is a generalized solution of the system (1.1) from such that uxjc 6
y2’°(9r) for k m l, r n. Then ux € a > 0, with the norms
Q' = Q* x (e, T), ( > 0, being estimated "from above by a constant determined
only by the norm ||uj|2 qj., n, N, v and ft from condition (1.2), the norms of the
coefficients and the indices q and r from condition (4.1), t and the distance from
Q' to S. If it is also known that u(*, 0) € then ux 6 ffa,a/,2(fi x [q, 71),
0 < cl < fj, with the norm Q\ = O' x [0, f] being estimated from above
by a constant determined by fi, the norm |u(*, 0)|q+^ and the above-mentioned
quantities, except *.
The exponent a is determined in both cases by the same quantities as |ux|^,
except full 2>q , d, t and |u(*, 0)|^ *&.
After an estimate for |ux|^ has been obtained it is possible to consider the
system (1.1) as a collection of separate equations of the form uj - da.-ut /dx. =
. 'Jl
4> and to utilize further results of Chapters in and IV for their solutions o . We
will not formulate all that this leads to in connection with tbe system (1.1), in
particular, when problem (1.1), (1.3) is uniquely solvable in W2m,m(QT), m > 1,
or in Hl‘l/2(QT), l> 2, but only mention that the final results are the same as
for one equation.
§5. QUASI-LINEAR PARABOLIC SYSTEMS 585
§ 5. QUASI-LINEAR PARABOLIC SYSTEMS. ESTIMATES OF THE
NORMS I > I, IN TERMS OF max^ |u, a J
We will consider in §§ 5—7 quasi-linear parabolic systems of the form
Uf—atj(x. t. n) nXfX/ 4- a (x. t, «. uj = 0. (5.1)
in which u(x, t) = (bH*, t), u2(x, t),-.., u^(x, t)) is an unknown vector function
defined in Qj.. For systems of the form (5.1) a theorem will be proved, under cer¬
tain conditions on the smoothness of the known functions and restrictions on their
behavior for large values of the arguments, on the existence of a unique classical
solution of the first boundary value problem. As a preliminary we will obtain the
a priori estimates needed for a proof of this theorem.
Suppose a(x, t) is a classical solution of the system (5.1), i.e. u(*, *) be¬
longs to C2,K(?y) and satisfies the system of equations (3.1). If the estimates
max | u| = A}, max! u*! =
Qt Qt
are already known, then all further a priori estimates follow from the results
obtained in Chapters III and IV for one equation. Indeed, each of the equations
of the system (5.1) can be considered as a linear equation for determining uH.x, l):
*K,] + 2(V. 0=0, (5.1')
where
au (jc, t) = ah(x, t, u(x, <)), am(x. t)
~a>(x. t, U(x, t). „,(*. t))+—(X' -
(We assume that tbe functions in equations (5.1) are sufficiently smooth and, often
without saying so, we will assume that all of the functions and their derivatives
involved in the calculations are continuous in dieir arguments.)
Equality (5.1') represents an equation for with bounded coefficients and
bounded derivatives
dati da,, _ dot,
dxk dum *k ' dxk '
From this, as was shown in § 3 of Chapter V, there follows an estimate of
We then obtain an estimate of uKx, t) in the norm of +a/2(^)') on the basis
of a theorem of A. Friedman (Theorem 5.2 of Chapter IV) on linear equations,
since each of the equations (5.1) can be considered as a linear equation with re¬
586
VII. SYSTEMS OF EQUATIONS
spect to uKx, t) with Holder continuous coefficients.
Thus, by taking into consideration the conditions of the theorems referred to
above, we obtain the following theorem.
Theorem 5.1. Let a(x, t) be a solution of the system (5.1) belonging to
c2,1(<?r), and suppose the system (5.1) is parabolic on it, i.e. the inequality
a,,(x, t, u(x. *))£<!; v > 0, (5.2)
is fulfilled in Qy. Suppose that the functions a^ix, t, u) are differentiable with
respect to the x^ and v} in the domain $ = |(as, t) 6 Qy, |o| < M =maXgr|u(a, t)|,
|p| <M l = mat)|}, and that these same functions and their derivatives
with respect to the x^ and u* and the functions a\x, t, u, p) are continuous on
9L An upper bound of their moduli is denoted by Then for Q1 C Q J the norms
w‘ch some a > 0 are estimated only in terms of the quantities Uj, v
from condition (5.2) and the distance from Q' to Pj.. These same norms are esti¬
mated in terms of the quantities M^, U^, v, the distance from Q' to Sy and
0)1 n*-
If, in addition, the functions a..(x, t, u) and aKx, t, u, p) satisfy a Hdlder
condition in 9 in the arguments x, t, u, p with exponents fi, fi/2, fi, fi respec¬
tively, then the norms |u|^,+|S* are estimated in terms of M^, My, v, the H'dlder
norms of the functions and a1 in 5M, and either the distance from Q' to Tj. or
|u(*, 0)|« and the distance from Q' to Sy.
Estimates of (and along with them estimates of !ul^) j-^) f°r the
whole cylinder Qy are obtained as corollaries of Theorem 6.1 of Chapter IV in
[6Sq] and Letnma 3.1 of Chapter II if mai^ |u;| has been estimated beforehand.
In order to obtain this estimate we form a divided difference with respect to l
for the system (5.1) and write the result in the form
v'— "TTi [fli A, + A‘v + + B‘v*i+ Av ^1 ~ °’ <5‘3)
where
v = -^.[n(jr, /-f A/) —o(x, /)].
r dait [x, t + t/W, tii (Jr. t -+• &t) + (1 — x) a (x, 0] .
a‘r=I dx-
1 o
§ 5. QUASI-UNEAR PARABOLIC SYSTEMS
587
I
b'r — f d“l[x' * + Tu<*’ < + AQ + (l —r)n(x, /)!
i *?,
U 0
. . v i, . dan (x, t, u) &*,, ,
«'(x. /, U, « = «'(*, /. u, pH ijpi—p?p,i+-j~-p‘r
(5.3) can be considered as a linear system with respect to v with bounded coef¬
ficients. Bounds for-these coefficients will be known to us if we know the quan¬
tity v from (5.2), M = max^ |u|, M j = maxg^, ju^l and a bound Mj for
da,, da, i da! da* do1 I
V*- *, B).« (X./. a, PJ.-JJ5-. jjsr. dt I
in tbe domain Si. These bounds can be taken independently of A(. By Theorem
2.1 we can estimate max^^,|v| solely in terms of Jfj,, and maxj.^,|v|. Pass¬
ing to the limit as At —» 0, we obtain an estimate of maxQj, |uj in terms of Mj,
and maxr ^, |u(|. Thus for the system (5.1) we have the following result.
Theorem 5.2. Suppose all of the conditions of Theorem 5.1 are fulfilled and
suppose the functions
day d2aif dJay d%a,j dao</ da1 da1 da1
UT’ dumdu1'' dumdt ’ Aumdx* ’ dx^dt ' do" ’ dp? ’ ~dT
are continuous on Si. A bound [or them is denoted by My Then max^ ju J and
wit^ some a > 0 are estimated in terms of v"1, it/j, U2, My maxr ^ |u{|
and the norms in #1+£>(1+0)/2(0f the values of u on Ty.
If, in addition, it is known that the a*(x, t, u, p) satisfy a Holder condition
on 38 in x with exponent fi and the corresponding HSlder constant does not ex¬
ceed My then u 6 H2+^’^*^2(Qj.) and the norm is estimated in terms
of the same quantities and the norms in
f]2 +0,1 +/V2(p^) of the values of u on
P f. Furthermore, all of the indicated norms of u depend on the norms of the
functions defining the boundary S: in the first case they depend on the norms of
these functions in 02; in the second, on their norms in tf2+^.
Thus Theorems 5.1 and 5.2 reduce the problem concerning a priori estimates
of the solutions of systems of form (5.1) to the problem of obtaining estimates of
max9rl°l and “a*pJ.KI-
588
VII. SYSTEMS OF EQUATIONS
§6. AN ESTIMATE OF |Uje|
Hie examples in §3 of Chapter I show that in order to obtain estimates of
maxqj, |u J we mast have agreement in the orders of growth of the coefficients of
the equation with respect to |p| * (pf )2 for |p| —> In out case the condi¬
tion of parabolicity has the form
v(|b|)&*<<V*./. v> 0, (6.1)
and therefore a necessary requirement on the otder of growth of a with respect to
|p| is the following:
1 a(jc, t, U, p)|<Ji(|nl)(l+|pl)a-
However, as the example of the system
I
1(1 s=: cos mx and «s = sln/njc
given by E. Heinz shows, this restriction on a(*» t, u, p) is not in itself suffi¬
cient for deriving such an estimate. But if the tetms having quadratic growth with
respect to p have a special form, then an estimate becomes possible. Namely, we
will assume that die system has the form
— *!/(■*• 1‘ u)Ux(xy + */(JC’ *• U' “,)«,,+ *>(■*;*»«• «jr) = (6-2)
where the functions b. and b are subject to tbe conditions
|0,(jr. t, U, p)|<tl(|o|)(l+ip|). 1
|b(x. t, n. p)|<l«(|«|)-f-P(|p|. |«|)1(1 4-Ipl)8./
where Ptjp], |u|) —> 0 for |p| —> oo, while ((J/) is a sufficiently small number
determined only by the quantities M, u(U) and fi{M) from (6.1) and (6.3). Of the
a.y(x, t, u) we will require that they satisfy .inequality (1.2), be differentiable
with respect to die xk and ul and satisfy the inequality
Ve will assume that <t(x, t)j= 0. (This assumption is not essential; i.e. if the
boundary 5 and boundary values of u|s are sufficiently smooth, then we can ex¬
tend u| j with preservation of smoothness into Qy and by a subtraction achieve
§6. AN ESTIMATE OF 589
homogeneous boundary conditions.) Let us first estimate max^ |ux| or, what is
the same, |du/dn| on die surface Sy
Lemma 6.1. Let u(%, t) be a solution of the system (6.2) that belongs to
C2'KQt), is continuous together with its derivatives ux in Qy and is equal to
zero on Sj» Suppose inequalities (6.1) and (6.3) ore fulfilled for (x, i) 6 Qp
u = u(*, t) and arbitrary p. Then max 5^, ju^x, t)| is estimated by a constant
depending only on max {u(x, t)|, the quantities u(Sl)t ((if) and /’(|p|, M)
from (6.1) and (6.3), maXQ|ux(c, 0)| and tke boundary S, which is assumed to be¬
long to the class 02.
Proof. It is clear that under the conditions of the lemma
Kl =1^-1 •
1 ''St- I *1 lsr
Let us take a point (xQ, tQ) on Sy. and a number r such that
I du U»ts) I __ max max j A* I __
I 6a I N Sj. I dn |
If *0 = 0, then < maxg lu^l*, 0)|.
Suppose t0 > 0 and suppose for definiteness that dur(*g, (g)/dn <0. We
take the function
N
■of ss= ur -4- 2 (“,)2
1-1
(it is dear that du>'/dn\sT = du'/dhls?) and consider the equality
N
r(«)+2S^'(“)=o.
‘-i
where £ is the left side of the 2th equation of die system (6.2) — in case
Cg)/dn > 0 it. is necessary to consider the function
N
i-l
It can be written in the form
w'-ati(x. t, u)®^ + 2atj(x, t. n)<«^
+ ^ + cr== 0. (6.5)
where cf = 2bl(x, t, u, uI)«! + br(x, l, u, u^), while the bKx, t, a, p) are the com¬
ponents of b(*, t, u, p). Let us introduce in place of wT a function vr by means
of the equality wr «■ The function <f> will be chosen below, but in such a
590 VII. SYSTEMS OF EQUATIONS
way that 4>’(y) > 0. We substitute in (6.5)
w' = w'v'; W, = q>'v'; w' , = ty'v' -f- <p"vr vr .
I I X, X, i j T *t*j T xl *J
This gives us
t. u
■+■ <7" aiJUx,ttlxJ + biV'J,l + yrCf = 0.
Hence by virtue of (6.3) we will have
2
aiiV'xx ^~°llK‘0'r 4- -7-0; «'
1/ <p 1/ Xj Xj 1 (p' ,/ ,r(
< [n(I -f |nx|)|vi| _(_ nj)(e -f />(|ux|))J, (6.6)
where p *» //W), f = fW) and /’(|p|) =* />(jp|, J/) are quantities from conditions (6.1)
and (6.3).
Applying Cauchy’s inequality (2.1) of Chapter II, we estimate the right side
of (6.6) from above by the expression
Here t = c(M) from conditions (6.3). while P <* Inax|p[>o^>(|p|)'
By assumption e(U) is small, while P(|p|) —> 0 for |p| —»°°. We assume that
(Af-f l)e(Af)<~- (6.7)
and iUj.(*Q, 8q)| > Aq, where the value of Aq is determined by the inequality
(«+l)/>(4,^)<-| for k>k„. (6.8)
It then follows from (6.6) that at the point UQ, tg)
V — at/**,*/ - au^xj <C[(P'K|J + (<P')’ ')• (6.9)
where e is a known constant.
Let us select <t>{y) in such a way as to fulfill for y > 0 the conditions
— -|r v (M) —- cq)'> 0 and q>(0)= 0.
where iAM) is taken from (6.1). These conditions are satisfied by the function
•poo^—^ina+j')-
For this </>{y) inequality (6.9) implies
§6. AN ESTIMATE OF ma*^r|ax|
591
(6.10)
The function vr, as well as wr, is
r, is equal to zero on S^, and
dvr 1 1 dwr I c dur I
dn |jr~ <(' (v') dn |Sr ~ v (Al) da |Sr
achieves its maximum on Sj. at the same point (xQ, tQ) as dur/da.
Let us now construct a barrier function t{i(x), i.e. a function satisfying the
relations
with sufficiently large k and m and a twice continuously differentiable function
$(*) having the following properties:
1) it is greater than zero in 0;
2) I®*! > const > 0 in Q;
3) the surface $(*) * 0 contains the point
If tbe domain Cl is situated entirely on one side of the tangent plane to S at
die point xQ (let this be the plane *B = and suppose *B > ,xj®) in (I), then
as ® we can take $(*) * *„ - Otherwise it is necessary to transform Q so
diat it is situated with respect to xQ in the indicated manner.
For the function vr(x, e) + tji(x) we have
Consequently, its maximum occurs on Tf Moreover, this maximum is achieved
at all points of the interval I* = *0, t € [0, 711, since
— c 1. x£Q.
max [vr(x, 0) f >|> (.*)) = vr (jr0, 0)+>|>(*0)t
max t|i (x) = (j) (x0).
s
As such a function we can take, for example, die function t/rix) =
(vr 4- >!>), — aij(vr + <P)xixj < °-
and
Therefore
592
VII. SYSTEMS OF EQUATIONS
Olv'jx, Q+ $(*)] I
.... da
aad 1a particular
—isll
d° L« **u
This implies tbe desired estimate for the quantity
u I dul
jM0= max max |-3—
/-1 fit Sj- *
The lemma is proved.
Let us now estimate the modulus of continuity of the {unction u(*, t) in Qy.
Lemma 6.2. Let u(x, t) be a solution of the system (6.2) belonging to C2'l(Qy)
and let max^ _|u(*, t)| «* M. Suppose the functions
, dan dan
ati(x, t. u). bdx, t, u, p). b‘(x. t, «. p). —.
are continuous and satisfy inequalities (6.1), (6.3) and (6.4) for (x, t) € Qy, |u| <
U and arbitrary p. Then for any interior cylinder Q' C Qy the norms |u|^? with
some a > 0 are estimated only in terms of M, iAM), p(M), AM) and jP(|p|, M) from
(6.1), (6.3) and (6.4) and the distance from Q‘ to Fj..
If, in addition, S satisfies condition (A), then the norm lul^j. is estimated
in terms of not only the indicated quantities U, lAM), fi(M), AM) and i’(|p|, M)
but also |u|<?>, (3 and the constants a^ and 0q from condition (A).
The proof of this lemma is analogous to the proof of Theorem 3.1 on solutions
of linear systems and reduces to verifying that the solution belongs to the func^
tion class 82N(Qp My 5j, - - •, Sg, y, r, 8, k) with 8 now different ftom infinity,
r = 2(n + 2)/n and k = 2/n. In view of this we will not carry out the proof of the
lemma here.
Let us now proceed to the formulation and proof of the basic fact of the pre¬
sent section: an a priori estimate of ma*qj. |ux(x, t)|.
Theorem 6.1. Suppose the conditions of Lemma 6.2 are fulfilled. Then the
quantity ma Xq, |ux(», t)| in Q' =* O' x (0, T) is estimated, in terms of M = max^ |u|,
the constants v{M), ft(M) and AM) and the quantity />(|p|, M) from inequalities
(6.1), (6.3) and (6.4), •maxQ|ux(*, 0)| and the distance from Q' to Sy.
If, in addition, u| j » 0 and S 6 02, then maxq^, |u%| is estimated in terms
of the same quantities M, UM), p(M), AM), P(|p|, M) and maxQ|ux(x, 0)) and the
§6. AN ESTIMATE OF max^luj 593
norm in 02 of the functions defining the boundary S.
Let us first consider the interior estimates of |ux|. As a preliminary .we will
estimate \ux\2 dx dt. To this end, in the identity
| t, n)u‘x %f
+ bi*'x) n + H Ax dt—°- / 6 (o. n (6-ii)
which is valid for any r](x, t) from ^>0(<?r), we put t *= T and if = ule^v(2{x),
where v ».jo)2 = (uO2 and £ is a catting function for fi. If «|Sj, » 0, then
we take £ a 1. In (6.11) we carry out a summation over I from Ito N. This
gives us
j. J *•<» »?dx\r0
fi
+j «*• { \ at,vxlvxp +
+ &a?j + %% + b)lv*, +M}} **** -0.
From here, by making use of (6.1), (6.3) and (6.4) (with t sufficiently small in
(6.3)) and taking X sufficiently large, it is not difficult to obtain tbe estimate
N n
/ S S dt < c. (6.12)
Qr )-U-1
Let us put t) = in (6.11), where f (x, () together with its first deriv¬
atives is assumed to be equal to zero on Sj.. Then after the obvious transforma¬
tions by means of an integration by parts and a summation over k and I we arrive
at the identity
t
j f
I+^Lal„>tx
^ d*k *> ax„
— ulxt+biUx,+*') (Aa's + «**!*»)] <** dt = 0, (6.13)
where V = 2^=1 XJ=I (u^)2- In (6.13) we put f = 2PS£2, where £(x) is a cutting
function for a ball KpC 0 and s > 0. Tbe first three terms of die integrand yield
594 VII. SYSTEMS OF EQUATIONS
the basic positive tetms, while the remaining teems can be estimated in modulus
from above with the use of inequalities (6.1), (6.3) and (6.4). This leads to the
inequality
7TT / V'+'t?d*i
0
t
+» J J WxiV^ + sV'-WV^dxdt
t
<e» J f + +£)dxdt (6.14)
0 tt
with a certain constant c{s) depending on known constants and tbe number s. On
the other hand,
J VS+X* dX mm J VS+'t2 2 u‘Xiu‘Xk dX
«„ Kp *•'
.— j (u'-«')[i/3+w+(i +S)n^2U™a™,t
+ V,i-'u‘Ilt2t&Xt\ ^<max|«'-4|ci (s) J
•'*» k0
+ V’uU2 + V’"&]dx.
If we take as Uq the value of uKx, t) at the center of die ball Kfi, then by
virtue of Lemma 6.2
/j1 w »t i
Taking p sufficiently small, we obtain
/ j (KJ|u,,|2£’+r+,®rf*. (6.15)
™x|“'-“S|Oa. « > 0.
This inequality together with (6.14) implies fot sufficiently small p < p(s) that
‘ f V*"fdxt,+ % j \{\nxxfVS?+V"X*)dxdt
o /rp
i
<*(*>/ J{\+Vs+l)£tdxdt. (6.16)
Inequalities (6.16) are valid for s = 0, 1, 2, •«•- Together with the original ine¬
quality (6.12) they permit us to give successively with respect to s tbe estimates
§6. AN ESTIMATE OF max^juj 595
max f V**' dx dt ^ c (s, Q'). (6.17)
»€lo. n0J,
Let us teturn to the identity (6.11). We take rj = £xk in it, where £(*, t) is an
arbitrary smooth function that is equal to zero in the vicinity of S T, and we carry
out an integration by parts. As a result (6*11) takes the form
t
/ J + ~0, (6.18)
where
0 Q
*&,+(%+>) U*H+W>-
From here it. is seen that each of the functions v = ulx^, km l-l, ■
• can be considered as a solution from F*,0((?y.) of the equation
4(a'/^+/5,')==0-
By virtue of assumptions (6.1), (6.3) and (6.4) and inequality (6.17) with s ** n it.
is possible to assert that the conditions
||/}-% r Q/<IV ? = *+!. /- = oo. Q' = Q'X(0. T),
are fulfilled for the a., and It therefore follows on the basis of Theorem
7.1 of Chapter III that taax.Q„ |u^S >n Q" C Q' is estimated in terms of v, pi, ftj
and the distance from Q" to die lateral surface of the cylinder Q'. Thus the first
part of Theorem 6.1 can be considered proved.
It remains for us to verify the analogous estimate in cylinders intersecting the
boundary. The quantity maxj. (u^.) = is already estimated. An estimate; of
lhJ2>(?r is given for all of. QT. We must prove estimate (6.17) for domains tlp
intersecting the boundary S. For s > 1 we obtain inequalities (6.14)-(6.17) in
the same way as above as long as we replace £, = Vs g2 in (6.13) by the function
4=1 iot y>K-
\ 0 for
This is admissible, since such a £ is equal to zero on all of the boundary of the
domain In the case s = 0 we take in (6.9)
i =
0 for V < M»,
596
VII. SYSTEMS OF EQUATIONS
Then after a series of transformations and estimates analogous to those made in
the proof of Theorem 4.1 of Chapter V for one equation we arrive at an estimate
for “®*«e[o, r] Jqp n ix *•
In this way one proves the second part of Theorem 6.1
§7. AN EXISTENCE THEOREM FOR QUASI-UNEAR SYSTEMS
the a priori estimates obtained in the preceding sections for a solution
u(*. t) of the system (5.1) permit us to prove die solvability of the first boundary
value problem
u, — atJ(x. t, u)u^+*i(*. t, u. <g«i,(
+b(x, t. h, ux) = 0, (7.1)
0^ = 0. u|,..0 = ifc,(.*)
in the cylinder Qj. = Ox(0, T).
As a preliminary we note that max^j, |u(x, t)| for solutions of problem (7.1)
is estimated in the same way as for one equation if
a,}{x. t, 0)1,1,>0 (7.2)
aod
b\x, t, u, pV > — C, IuP— c2. c, =sconst>0, t—1, 2. (7.3)
Namely,
max
9.
ix|uU. /)l< min e^UaxluCj;, 0) | 4-l/UST] = M. (7.4)
T ‘>»l Id V c — ci J
A priori estimates of max^ |u(*, i)| are also possible under certain other condi¬
tions on the coefficients.
The following theorem is valid.
Theorem 7.1. Suppose the following conditions are fulfilled for problem (7.1).
a) Inequalities (7.2) and (7.3) ore valid for (x, t) € Qj\Pf and arbitrary u.
b) For (*, t) e QT, -|u| < U, where M is defined by (7.4), and arbitrary p
the functions
au(x, t, u), &,(x, t, u. p>. b‘(x, t, u, p),
are continuous and satisfy inequalities (6.1), (6.3) a*d (6.4). ( This gives an a
priori estimate for the solution: max^ ju^l < Mj, where is determined by
§8. LINEAR PARABOLIC SYSTEMS
597
the quantities in conditions a) and b), 5 and maxQ jux(x, 0)|.)
c) For (*, t) € Qy, |u] < M and |p| < the first derivatives of the functions
a. Ax, t, u), 6,(*, t, u, p), bl(x, t, u, p) with respect to x, t, u, p and the derivatives
d*a../duldum, d2a../duldxk, d2ai./duldt, d2ai-/dxkdt are continuous.
d) <I>q(x) 6 H2 *^a) and satisfies the compatibility conditions
\-al}(x, 0. 0W0XiX/-bt(x, 0, 0.
+ bl{x, 0,0,^Ox)H =0, / = 1 N.
J **• ti
e) SeHW.
Then\in\the class\H2+^’1+^^2(QT) there existsia unique solution'n{x, t)[of pro¬
blem (7.1).
The proof of this theorem for systems is carried out in tbe same way as that
of the corresponding existence theorem in the case of one equation (Theorem 6.1
of Chapter V), by means of the method of continuity in a parameter and an appli¬
cation of the LerayrSchauder fixed point theorem.
§8. LINEAR PARABOLIC SYSTEMS OF GENERAL FORM
In 1938 I. G. Petrovsku in the paper [93c] introduced a very extensive class
of parabolic systems of linear differential equations, which were subjected to a
detailed analysis. It turned out that many of the characteristic properties of solu¬
tions of parabolic equations of second order carry over to these systems. After¬
wards PetrovskW’s definition was generalized in various directions and, in a num¬
ber of cases involving more general parabolic systems than those of Petrovsku,
certain complete results were obtained.
We will begin with Petrovsku’s definition of a parabolic equation of high
order. Let L(x, t, d/dx, d/dt) be a linear differential operator ^ of arbitrary cider
with complex coefficients depending on x and t in some domain ?CS>+j. It
is clear that at any point (>, t) e Q the function L(x, t, i£, p), where (j is an n-
dimensional vector with components , • •'» (n and p is a scalar complex para¬
meter, is a polynomial in the and p. Let b be some positive integer and let
the degree of the polynomial L(x, t, if A, p\2b) in A be equal to 2 br, where r is
1) In §§8—10 we will frequently use the term "differential operator** or even simply
"operator** in the sense of "differential expression.*’
598
YU. SYSTEMS OF EQUATIONS
a positive integer. We denote by the principal part of the polynomial L, i.e.
the sum of those terras of L for which
Lo(x, t, l$K pl™)=lUrL0(x. t, l\, p).
Definition 1. An operator L is said to. be parabolic (2b-parabolic) at a point
(x, t) if for any real £ the roots ps of lie polynomial Lq(x, t, if, p) in the vari¬
able p satisfy th.e condition
Re^< —6|||24 (fl > 0). (8.1)
An operator L is said to be uniformly parabolic in a domain Qj. if it is para¬
bolic at each point of this domain and inequality (8.1) is fulfilled at each point
(*, t) € Qj with one and the same number 8 > 0.
For equations of second order with real coefficients 6 = 1, we have
Lq{x, t, i£, p) = p + a^(x, and, by virtue of condition (2.5) of Chapter I,
ft
RtPs — Ps — — 2 0 !/£,< —vii p,
/./-I
so that S = v.
From the definition of parabolicity we see that a single differentiation with
respect to t of a parabolic operatot is equivalent in force to a 26-order differen¬
tiation with respect to tbe space variables x (one says that a differentiation with
respect to t has weight 2b). The weight must always be an even integer; other¬
wise the parabolicity condition (8.1) will not be fulfilled for any real f.
Let us explain this in more detail. We assume that R(i£, p) is a polynomial
that is homogeneous in the following sense:
R {ill", pXs) = XNR (11. p). (8.2)
Here q, s and N are positive integers, while K is any complex number. Let
and p0 be roots of the polynomial R, where is a teal vector and Re pfl < 0.
It follows from (8.2) that, along with and pfl, and P0AS will also be
roots of the polynomial R. If s/q ji 2b, then we can select A so that A/*1 is real,
while Rep0A* > 0, and in this case condition (8.1) will not be fulfilled for all of
the roots ps for all real f. Indeed, if. s/q £ 2b, then for any J 6®, 3/2) we
can find integers k and m such that
&<.£.< 9. (8.3)
§8. UNEAR PARABOLIC SYSTEMS 599
Let argp0 =drr 6 (tt/2, in/2) and X - eivk/1. Then it follows from inequality
(8.3) that
^2m 4" ^*j jt < arg (Pq%3) ^2fft -j- n,
i.e. Rep0A* >0.
But if s/q - 2b, then such a X cannot be selected, since Xs = (X*)^ > 0 it
X« is real.
Thus, it the polynomial R is homogeneous in the sense of (8.2), then a para¬
bolicity condition of type (8.1) can only be fulfilled for it in the event s *= 2bq.
Let us assume that condition (8.1) is fulfilled fot a homogeneous polynomial
R. Then by .virtue of what has just been proved
R (/|X. pi?*) =n XN'R (/£. p). (8.4)
It is not difficult to show that
N, = 2 br,
where r is an integer and the coefficients of pr and f ?^r are different from zero.
Indeed, putting f = 0 in (8.4), we obtain
R(0. j»JL“)->*.W'*(0. /»)■
If /Vj / 2tr, then this equality can only .be fulfilled if both sides of it are equal
to zero for all p.
But then _ „
R (0. p) = 0
for all p, in particular for Re p > 0, which contradicts condition (8.1). Ihis proves
that Nj » 2f>r and /?(0, p) - ypr, y 4 0. It can be shown In an analogous manner
that the coefficients of die (jbr (/ » 1, •«>, m) in the polynomial fi are different
from zero. From what has been proved it follows that the numbers b and r for a
parabolic operator L are uniquely determined: r is the degree of the polynomial
L(x, t, if, p) in p, while 2fcr is its degree in the variables
Let us proceed to Petrovsku’s definition of parabolic systems.
Definition 2. A matrix differential operator £(x, t, d/d%, d/dl) with elements
*> d/dx, d/dt) in) is said to be parabolic in tAe sense of
/. G. Petrovskii if:
600
VII. SYSTEMS OF EQUATIONS
1) the operator
L{X‘ w)
is 2b-parabolic in the sense of Definition 1;
2) the degree of the polynomials t, ifA, pA^) in A does not exceed
2hr. and
LkJ(x, t, t\, p) = bip'l -f- L'n) (jc, t, 1%, p),
where L■ is a polynomial not containing prl.
Thus the systems £u =. f of I. G. Petrovsku are solved with respect to the
leading derivatives with respect to t, i.e. the derivatives d'iJ/dt >.
An important (and the most-studied) special case of systems that are parabolic
in the sense of Petrovskii is furnished by (be systems in which all the r. = 1;
they have the form
J?v*=*£. + Jl(x, t. = (8.5)
where dl is an operator of order 2b.
By the principal part of the polynomial L, p) we will mean the
sum of all those terms of L^- for whicb the condition of homogeneity
4,0. t, <|x, t, 4. p)
is fulfilled, while the matrix £q composed of the Ljy will be called tbe principal
part of tbe matrix £. It is obvious that the polynomial
M*. *■ lt />) — det.^’oC*, t. i\. p)
is the principal part of the polynomial L.
A characteristic of Petrovskii. systems is the fact that the degree of homo¬
geneity of the polynomials ij., whicb is equal to 2br., does not depend on ft.
Other definitions of parabolicity have been proposed in which condition 1) of
Petrovskii’s definition is retained but the degree of homogeneity of the polynomials
is permitted to depend on ft as well as /. For hyperbolic systems such a
generalization was made even earlier by Leray, and for elliptic systems, by Douglis
and Nirenberg. In these definitions the order of the operators L^. is determined
by a sum + I., where the and <j> (ft = 1, —, n) are certain collections of
integers. For parabolic systems one of these generalizations of Petrovskii’s
definition was proposed by T. Shirota [109b].
§8. LINEAR PARABOLIC SYSTEMS
601
Definition }. A system of form (8.5) is said to be parabolic in the sense of
Shirota if:
1) the operator L = det£ is 2b-parabolic-,
2) there exist integers (k ~ 1, • <«, m) such that the order of the operator
A^j does not exceed the quantity s^ — s^ + 2b.
Although these systems contain only derivatives of first order with respect to
t, while Pettovsku systems contain derivatives of arbitrarily high order, die latter
are a subclass of the former in the following sense. If in a system that is para¬
bolic in the sense of JPetrovskii one introduces new unknown functions in place of
the dkUj/dth (k = 1,. * *, r. - l) in the case r. >1, then a system of form (8.5) is
obtained as a result, and it can be shown that it is parabolic in the sense of
Shirota.
An even wider class of parabolic systems has been introduced in the papers
[116c, e].
Definition 4. An operator £ will be called parabolic if:
1) the operator L * det£ is 2b-parabolic in the sense of Definition 1;
2) there exist integers s, and t^ (k = 1, •. ■, m) such that the degree of the
polynomial l>k.(x, t, i£\, pA2 ) does not exceed sk + t. (if sk + < 0, then
L“kj ” and, in addition, ^
2 (** + '*) ==2ir.
where r is the degree of the polynomial L(x, t, i(, p) in the variable p.
By the principal part of £ we will mean the matrix £Q whose elements are
the polynomials t, it;, p), i.e. the principal parts of the polynomials L^.
These are the sums of all those terms of the for which the condition of homo¬
geneity
t, i\l, pX24)==r*+(/4/U. t. I|, p)
is fulfilled. If the degree of die polynomial Lk-(x, t, ifX, p2i) is less than + tj,
then L^j = 0. It is obvious that the polynomial Lq -- det £q is the principal part
of the polynomial L.
Petrovskii and Shirota systems are subclasses of die systems just defined.
Petrovskii systems are those parabolic systems for which s. =* • • «•=» s » Q
and t. = 2br.. Further, if a system that is parabolic in the sense of Definition 4
has the form (8.5), then for the diagonal operators L®.
602
VH. SYSTEMS OF EQUATIONS
L°jj(x. t. liK pX.u)= pXu + A),(x. t, i|X)
-WWuix. t, ii, P).
from which
Sj + t)—2b, tj= 2b—Sf.
Consequently, this system is parabolic in the sense of Shirota.
In connection with Definition 4 there arises the following question. Suppose
a matrix differential operator £ is given such that the operator L « detS is para¬
bolic in the sense of Definition 1. Is it always possible to select the numbers
and t. so that die operator £ is parabolic in the sense of Definition 4, and will
they be unique? Let a^- be die degree of the polynomial l, i£h, pA2t) in A.
The numbers &j and tj oust satisfy the following requirements:
m (8.6)
(s* + h) — %br,
where b and r are numbers uniquely determined by die polynomial L.
L. R. Volevi£ [125*. c] showed that such numbers always exist but are not
uniquely defined. First of all, if. certain numbers and satisfy .conditions
(8.6), then these conditions will also be satisfied by .the numbers + a and
t - - a. We fix the constant a by the requirement that max(-s. =■ 0. But even under
this condition the numbers sk and tk are sometimes not determined uniquely from
(8.6).
A system of two heat equations is a simple example of a system fot which it.
is possible to give infinitely many collections of numbers and l^. For it
/n_i_*J 0 \
= o p + VJ-
The operator £ is obviously parabolic according to Petrovskii and we can put
Jj s «2 = 0, (j = tj » 2. But along with this we can also take
f, = 0, s2== — ft, fi = 2, = 2+
s, = — k. i2 —0. tx = 2 4- A, t2 — 2,
where k is a positive integer.
We note, finally, that Definition 4 is somewhat different in form from the de¬
finition given in [116 c], although it is completely equivalent to it. The definition
§8. LINEAR PARABOLIC SYSTEMS
603
given here is mate constructive. The possibility of such an improvement in the
definition (when applied co elliptic systems) was pointed out by L. R. Volevic
[123a],
In Definitions 2—4 there is one common feature, namely die condition imposed
oo the determinant. This condition has also been subjected to generalizations.
Definition 5. The system
■^--4-<d(x, t.-g^u — O
is said to be parabolic in the sense of G. E. 'Silov if for all real ( the roots ps
of the polynomial
L(x, t. I. |. /0 — det[/>£-+-d!(*. *. 1)1 (8.7)
have the property
— *151*+*!.
where c and k are positive constants.
This class of systems is more extensive than die class of systems that are
parabolic in the sense of T. Shirota. These systems have been investigated in
the papers t10-110] for the case when the coefficients do not depend on t. The
Cauchy problem has beeu solved for them. The basic results are cited in the mono¬
graph [25a] (Appendix 1).
Let us cite in addition the definition of 2b-parabolic systems introduced by
S. D. Eidel'man [25a], In these systems differentiation with respect to different
Xfc has in general a different weight 1/2b^ with respect to differentiation with
respect to t. By b is meant the vector (fcj, * - ■, ba).
Definition 6. An operator S. (x, t, d/dx, d/dt) is said to be 2b-parabolic if
there exist positive numbers b^ (k = 1, - «•., n) such that:
1) the functions L^ix, *,.»£, •«■, i£n\l/2in, pA) are linear combina¬
tions of powers of A, with the maximal power not exceeding r. and
L*j(x. t, i%. p) — b{pr/ + Ltj(x, t. 1%. P).
where L1^. is a polynomial not containing p^;
2) the polynomial
Zo(jc, t; 4, p) = detjg,0(x, t, p).
where is the matrix composed of the principal parts Lj*. of the polynomials
which are the sums of all those terms of th.e L^- for which
604
VH. SYSTEMS OF EQUATIONS
di(x, t, <6,31*1 pl} = lrJLlj(x, t. 1%, p),
possesses the following property: for all real £ its roots ps satisfy the inequality
...+C")' 6>°-
As was pointed out in [2 5 a], .it. is possible to construct a fundamental matrix
(fundamental solution) for these systems and with its help to investigate die Cauchy
problem in roughly the same way as for systems that are parabolic in the sense of
I. G. Petrovskii.
la conclusion we cite the definition of strongly parabolic systems.
Definition 7. A system of form (8.5) is said to be strongly parabolic if the
operator A is strongly elliptic [124a], j,e. its order is equal to 2b and its prin¬
cipal part Aq satisfies the condition
Re Wo (*, t, /|) tj. t|| > 6111» 1 rj Is, (8.7)
where m
ti=(-Hi. • • ■. n»). K. nl = 2 u.
It is obvious that strongly parabolic systems are parabolic in the sense of
Petrovskii,. Property (8.7) permits us to use many of the methods set forth in
Chapter III in investigating the first boundary value problem for strongly parabolic
systems.
§9. STATEMENT OF THE BOUNDARY VALUE PROBLEMS
AND THE CAUCHY PROBLEM FOR PARABOLIC SYSTEMS
We will consider general boundary value problems for systems that are para¬
bolic in the sense of Definition 4. The boundary conditions in them are given by
the equality
(5U)
where S is a matrix differential operator and ® is a vector arbitrarily prescribed
on the surface Sy
Let us determine the number of rows that die matrix $ must have (i.e. the
number of conditions that must be given on the boundary) and the algebraic condi¬
tions it must satisfy in order for condition (9.1) to provide us with a well-stated
boundary value problem.
§9. BOUNDARY PROBLEMS AND THE CAUCHY PROBLEM 605
We will first define the principal part Sfl of the matrix S.
We denote by (q = 1, • • •, N; / = X, • > <, m) the elements of the matrix S,
and by fi j the degree of the polynomial B^.(x, t, i(k, pk2b) in A. If B^ = 0,
we take any .integer as fi^. Let
0, = max(p?/ — tj),
so that
By the principal part of the polynomial B^. we will mean the sum of all
terms of it satisfying the homogeneity condition
t, i$X. t, /*. p).
The matrix with elements B^- is $0. Thus the choice of the principal part
of the matrix % depends on the numbers t. and can be ambiguous.
Let us fix arbitrary admissible numbers Sj, t- and We take any point
(xQ, tQ) 6 Sj., introduce local coordinates ty} in the space En(xj, • - •, *n) at the
point € 5, with the yn-axis being directed along the inward normal iA*0), and
consider in ‘the half space yn > 0 die problem
J?a[x0, tQ. ^, ^]u = 0(-oo<y1...,y„_„/<+oo, y„>0).
, . (9.2)
^0 (xo- 10. u =
' y/I
The coefficients of the operators and in this problem are "frozen” at the
point (xq, tg), while all minor terms are discarded. Following Ja. B. Lopatinskii
[77], .we assume that the following local condition is fulfilled: no matter what
point (*q, tg) G S j we take, problem (9.2) will always be uniquely solvable for
any smooth vector function <£ with a compact support in the class of functions ad¬
mitting the transformation (1.19) of Chapter IV (x' denotes the coordinates in the
tangent plane to S at the point xg).
Let us express this condition in algebraic form. Going over in (9.2) to the
coordinates fy|, we obtain the problem
d d
606
VII. SYSTEMS OF EQUATIONS
It is easy to see that the operator 8 is parabolic with die same constant $ in
condition (8.1).
Problem (9.3) can be reduced to a boundary -value problem for a system of or¬
dinary differential equations by taking die Fourier transform with respect to the
variables y‘ = (yj, • • >, y„_j) and then the Laplace transform with respect to t
11.19) of Chapter IV).
As a result of this transformation we obtain
Here 4-P is an (» - l)-dimensional vector. The necessity of the
third condition is proved in the same way as in the case of the corresponding con¬
dition in (1.23) of Chapter IV.
and p, we will consider on the half line z > 0 a problem of form (9.4) for a cer¬
tain system of ordinary differential equadons (the number of equations is denoted
as usual by m):
Let us determine the conditions under which this problem is uniquely solv¬
able for any numerical vector h. We denote by the space of the soludons of
system (9-5) satisfying conditions (9.7). This space is obviously finite dimensional
and die number of its dimensions is equal to the number of roots of the polynomial
having a negative real part or, equivalently, the number of roots of the polynomial
P{ir) having a positive imaginary part (with the multiplicity of the roots being
taken into account). We denote these toots by r*, and their number by r+. Further,
the components of any vector y(z) € satisfy the equation
Disregarding for tbe moment the dependence of tbe operators 0 and 8 on £
(9.6)
(9.7)
(9.5)
P(X)*= det <£“(*.),
(9.8)
§9- BOUNDARY PROBLEMS AND THE CAUCHY PROBLEM
607
where
P* (T) = IICr—T+),
since they ate lioear combinations of the exponential functions which ace
possibly .multiplied by polynomials (in die case of multiple roots r^). All this can
be easily shown by passing ftom the system (9.5) to an equivalent triangular sys¬
tem (see [50]).
Let »s(z) denote some vectors forming a basis in the space Then the
solution of problem (9.5)—(9.7) has the form
r*
W(z) = 2 CSMIS(Z),
s-1
where the cs are certain constants. We will find them from conditions (9.6):
S SMs-W(2)Lc*=a* (9=1 ad.
/-I i-l
We have obtained a linear system of equations for the c$. In order that they
be uniquely solvable for any. it is necessary and sufficient that
N = r* (9.9)
a°d det«,s=0. (9.10)
where SX is the matrix with elements
ttl
v-SM'sWwL* (9-11)
/-i v
By virtue of (9-9) the matrix St must have r+ rows.
Let us now give another equivalent formulation to condition (9.10). To this
end we will prove that every .vector w(z) g 5[Sq is representable in the form
w(z)=^(^-)v(«). (9-12)
where 3* is the adjoint of S’ (S’ = IP"1 det IP) and v(z) is a vector whose compo¬
nents satisfy equation (9-8) (it does not in general belong to $Q).
Indeed, as v(z) we can take the vector
v(z) = ^(~L)u(z), (9.13)
where u(z) is a vector whose components are solutions of the equations
VII. SYSTEMS OF EQUATIONS
P(tf) “*(*) = «*(*)
and have the same structure as the tti^(z), i.e. are linear combioations of the escpo-
. 4-
neotial functions e s multiplied by polynomials. From the elementary theory of
ordinary differential equations we know that functions u* with the required proper¬
ties can be found.
Let us show that the vector v(i^of (9.15) satisfies all of the above-mentioned
requirements. Letting the operator 9(d/dz) act on both sides of (9.13), we obtain
equality (9.12). In order to prove that the vector v(z) of (9.13) satisfies equation
(9.8) we let the operator ?(d/dz) act on both sides of (9.12). This gives us
/>(£)**(,)«<>.
From this equality we obtain the stronger equality (9.8) since «*(*) is expressed
. +
only in terms of the exponential functions elTs
Thus the desired representation (9.12) for any vector v(z) £ 5Pfl has been
proved.
The degree of the polynomial P+ is equal to r+, and therefore equation (9.8)
has r* linearly independent solutions. It is not difficult to see that we can take
as these solutions the functions
V.W- J iJg-rfr (s=l r').
Y+
where y+ is a simple contour in the complex r-plane enclosing all of the r*.
Hence every solution of equation (9.8) is representable in the form
v(z)— f - elx! dx,
J P (t)
while a solution of the system (9-5) that is decreasing for z —» + °° is by virtue
of (9.12) representable in the form
_ m
S Pki(ft) Q] (T) ~~ dx, (9.14)
y+ j-l ^ 'T>
where Q and Q. are polynomials of degree r+ -1. In particular, die basis vec¬
tors w (z) have the form
§9. BOUNDARY PROBLEMS AND THE CAUCHY PROBLEM 609
No matter what polynomials Qj(r) (/ = 1, •. <, m) of degree r+ - 1 we take there
exist numbers «s (* “ 1> • • *» r+) such that for any 4 = 1, - ", m
m *•¥
J It P*J (iT> Qj w dt = 2 aX
v+ /-i w »-i
m r+
J %P*j(ty%as0W-j~—dT. (9.16)
v3- )-1 j-i M
Let us now consider die functions (9.11). As a consequence of (9.15)
Tw-.
where
Dql — 2. AqhP*y
By virtue of condition (9*10), from tbe equadoa
r+
—0 <s=l r+> (9-17)
J-1
it follows that « 0 (? - 1, •' «, r\ Let
9 r+
JV*) = 2<VVtt).
Equality (9.17) can be written in the form
m
J —0 <s = 1 '•")• (9.18)
v+ J-i y 'T)
This is equivalent to the fact that
J 0 (9‘19>
y+ /•!
for any polynomials Q^t) of degree r + - 1. A proof of (9.19) is obtained by let¬
ting the operator
act on both sides of (9.16), then summing the resultant equalities over k from 1
to m, putting 2=0 and taking into consideration (9.18).
Since the polynomials Qf are arbitrary, (9.19) is equivalent to the fact that
for any / and any polynomial Q(r) of degree r+- 1 we have
610
VII. SYSTEMS OF EQUATIONS
Y+
which is obviously equivalent to tbe equality
2 d'Dq, (it)=Dj «t) = (T)P" (x). (9.20)
where Py are certain polynomials.
Thus we have shown that conditions (9.17) and (9.20) are equivalent, so that
from (9.20) it follows that all d^ = 0. This property of the matrix 2) with elements
D^. can be formulated in die following manner: the rows of the matrix
Sb{lx) = Jl{H')S‘{lx')
are linearly independent modulo the polynomial P +(r)-
In order to fotmulate this condition for problem (9.4) we must ^know what the
number r* is equal to for the polynomial
M&. H> P) = lx< P)-
The answer to this question is given by the following theorem.
Theorem 9.1. If the polynomial Lg satisfies condition (8.1), then for any
real £j, • •., and my complex p satisfying the conditions
Rep> — 6,|C|“ (0<a,<6), |p|+|t|>0. (9.21)
the polynomial Lg(i£, ir, p) as a function of r has br roots with positive, and
br roots with negative, imaginary part.
Thus under condition (9.21) r* <• br.
Theorem 9.1 was first proved by T. Ja. Zagorskii [I2$] under the assumption
that the polynomial L has real coefficients; it has been proved without this as¬
sumption in [2Sa] and [H6t],
We can now formulate the desired algebraic condition for problem (9-3) and,
consequently, for the original problem (9.2). We will fotmulate it straight off for
problem (9-2). Let f = (fj, • < •, £n) be an arbitrary real vector. It can be uniquely
represented in the form
| = £-f tv,
where £ is a vector lying in the tangent plane to the surface S at die point xQ,
v is the unit inward normal at *0 and r is a real parameter. Consider the poly¬
nomial
§ 9. BOUNDARY PROBLEMS AND THE CAUCHY PROBLEM
611
io(x0. t0. <(E+tv). p)
as a {unction of r ontiie whole complex plane. By .virtue of Theorem 9,1, under
condition (9.21) this polynomial has br roots with a positive imaginary part (we
denote them by r*(x0, tQ, £, p)) and br toots with a negative imaginary part. Let
br
L*(x0, <0, E. P. T) = jJ[t— t+(jc0, c. p)[ (9.22)
Ve will assume that tbe matrices £„ and %g satisfy the following condition,
which is called die *' complementing condition:” for any point (*0, eg) G Sj. the
rows of the matrix
25o(*#. t(5-+-tv). P) -#0(*o. *(£+tv), p)
are linearly independent modulo the polynomial L* as a polynomial in r if the
vector £ and the number p satisfy condition (9.21) for some 8^ 6 (0, 5).
For systems of form (8.5) that are parabolic in die sense of Petrovskii. (for
them r ~ m) the complementing condition is formulated in most papers in exactly
the same way as in the paper [77] of Ja. B. Lopatiusku: the rank of the matrix
J .00 (*0. t*. <«+TV. p)^ol(xo, 4,. 4-tv, p)o#(T)dx, (9.23)
Y+
where
tg* T»- >r>,
t»£*A € being the unit piatrix of order m, is equal to bm at each point (xg, tg) e
Sj. and /or any tangential £ and any p satisfying (9.21).
These two formulations are equivalent [l15c],
As was mentioned above, the choice of the principal part of the matrix
$ is dictated by the numbers t. and can be ambiguous. In this connection it can
turn out that for one choice of the numbers s^ and t., and consequently of the
principal parts of the matrices £ and % the complementarity condition holds,
while for another choice it does not.
As an example we consider in (he half space >0 of (he space £j(*j, x2, t)
a parabolic system decomposing into two separate heat equations:
.*«. It, ,) = ^0(C /x. p) = (P+l+%* ,+p + t.)-
For this system 4 •* 1, r m 2 and, as we saw in § 8; the following collections
of numbers and t. ate possible:
612
VII. SYSTEMS OF EQUATIONS
s, = — M, s2 = 0. = 2-|-*, f2 = 2 (A > 0).
s, = 0, s2 = — k, tt = 2, <j = 2-|-A (*>0).
Let
it. P)
-g :)■
For this operator the numbers Oj, and <r^ take the following values: if = 2 + k,
t2**2(k> 0), then - - 2 - k, a2 = - 2; while if tj * 2, <2 “ 2 + k (k > 0), then
tfj = - 2, <r2 “ ~ I* I* *s not difficult to verify that in the first case the complemen¬
tarity condition is fulfilled. In the second case
and since the rows of this matrix, and hence also of the matrix $0£0, are liaearly
dependent, the complementarity condition does not hold. In particular, it is not
fulfilled if we regard the operator £ as an operator that is parabolic in the sense
of Petrovskii (sj = s2 = 0, <j = <2 = 2).
We now turn to the problem of assigning initial conditions for parabolic sys¬
tems. For systems parabolic in die sense of Petrovskii the initial conditions are
assigned in the following way: if the highest order of the differentiation with re¬
spect to ( of a function u. is equal to r., then all derivatives of this function
with respect to t of lower order are given for t = 0. But in the general case, as
will be seen from what follows, the assignment of initial conditions in this way is
impossible.
We will assign them in the form
G{x' 1&' irH,_o==<P(*)’ (9.24)
where £ is a matrix differential operator with elements C^a »1,.«•Af ; k = 1, • • •, m)
and tf>{x) is an arbitrary vector function.
We define the principal part of the matrix <2 in exactly the same way as for
the matrix S. Let denote tbe degree of the polynomial Colt*’ P*26) io
A (and any integer for C^ = 0), and let pa » maij (y^ - «t). The sum C°ak of
all terms of the polynomial C^ satisfying the homogeneity condition
Ca*(*. '5?-. P^B) — i-Pa+lICaii(x. Z|. p), (9.25)
will be called the principal part of C ajc, while the matrix £q composed of the
C® k will be called the principal part of die matrix €.
§9. BOUNDARY PROBLEMS AND THE CAUCHY PROBLEM
613
Let us determine what the number N ^ must be and what conditions die matrix
£ must satisfy if. we are to be able to determine uniquely the value of an deriva¬
tive of any function u. fot t = 0 from the system
£’u = i (9.26)
and the initial condition (9.24) by merely differentiation and solving linear alge¬
braic systems (assuming, of course, that these functions and die coefficients
of the operators £ and £ are sufficiently smooth). We shall also require
that the condition have die same formulation for all parabolic systems and that it
involve only die principal parts of die matrices £ and 6, i.e. £Q and £g, as in
the definition of parabolicity and in the complementarity condition. Consequently,
the same properties must be possessed by the system
4)u==;f (5U7)
and the initial conditions
<*•(*• •&■• (9-28)
We will substitute into the left sides of (9.27) and (9.28) all possible vector
functions u(t) not depending on x. For them (9.27) and (9.28) go over into
^0(jc, 0, 0, j)u(() = f, (9.29)
C0 (x. 0, 4-) “ (01 = <P- (9.30)
In order for it to be possible to uniquely determine d^u.(t)/dl^\> 0)
from these equalities it is necessary that (9.29) and (9.30) constitute a Cauchy
problem for die system of ordinary differential equations (9.29) that is uniquely
solvable for any f and <j> and any fixed * € SI.
We have
6tiJ3’0(x, 0, 0, p) — l(x, 0, 0, p)=vWV.
in which by virtue of die condition of parabolicity (1.1) y(») ^ 0.
From the theory of linear systems of ordinary differential equations (see, for
example, [50]) we know that the number of arbitrary constants in die general sol¬
ution of the system (9.29) is equal to r, i.e. the degree of the determinant L.
Consequently, among die conditions (9.3Q) there must be r independent ones, and
since we wish to assign the vector <f> arbitrarily, the matrix € must have precisely
r rows. Thus
614
VII. SYSTEMS OF EQUATIONS
Nt = r.
From this it follows in particular that one cannot generally assign the initial data
for systems that are parabolic in the sense of Definition 4 in the same way as for
Petrovsku systems, since the system may .contain very high derivatives of the un¬
known functions with respect to t. This occurs, for example, in the case of a
parabolic system with
(p 4-6* pw \
(*== 1. r~2. j, = 0, s2 = ~8, tx = 2, <2=I0).
For the purpose of deriving an algebraic condition that will serve as a crite¬
rion for the unique solvability of problem (9.29), (9.30) for any f and <j> it is suf¬
ficient to consider this problem fot f = 0. Reasoning in exactly the same way as
in die derivation of the complementarity condition we see that in order for the ini¬
tial data (9.24) to possess all of the desired properties it is necessary that the
following condition be fulfilled: The rows of the matrix
3>(x. p) = eQ(x. 0. p)J&o(*. 0. 0. p)
are linearly independent modulo the polynomial
Pr = lkL{X' °'
for any x € 0.
As was shown in the paper [H«*], this condition is also sufficient for the
matrix & to have die desited properties. In addition, die following fact was
proved in [U6c], The formulated condition (we will call it the complementing
condition for die matrix C) permits one to uniquely determine, for a given parabolic
system and collections of numbers and for it, the possible numbers pa,
which turn out to be negative (pa< - 2b). Furthermore, this condition for a cer¬
tain wide class of domains Q (in particular, for all domains homeomorphic to a
ball) permits one to determine the matrix ^(j, 0, p) to within unimportant alge¬
braic transformations. For other Q die construction of the matrix €g is impos¬
sible and it is then necessary to give die initial conditions not in the form (9.24)
but in a certain different form ([U6c]> Appendix 4).
The complementing condition does not involve
e'a(x. lb p)~6o(x, p)—Gt(x, 0, p),
which also contains principal terms. This matrix, like the minor terms, can be
§ 10. BASIC RESULTS ON SOLVABILITY
615
arbitrary, and die only thing that we require of it is die homogeneity condition
(9.25) fot its elements.
Thus the general boundary value problem fot parabolic systems consists in
finding a vector u = (u1, • • - , um) for which
J?(x. t. £)«-f <(*. 06Qr).
9 (* ’ *' Ix • 4") “ |sr=°* (9'31)
where the matrices 5S and (? satisfy the complementing conditions .formulated
above. The operator £ is uniquely determined by this condition to within <2q,
minor terms and unimportant algebraic transformations. Tbe operator % can be
very arbitrary, but, as will be shown below, a dropping of the complementing con¬
dition for the matrix 25q can lead to a lack of correctness in the corresponding
boundary value problem.
§ 10. BASIC RESULTS ON THE SOLVABILITY OF THE CAUCHY PROBLEM
AND OF THE GENERAL BOUNDARY VALUE PROBLEMS
FOR PARABOLIC SYSTEMS
The algebraic conditions formulated in § 9 fot the matrices SB and £ are suf¬
ficient for the solvability of problem (9.31)- This was established in the book
[l 16c] with the use of die method presented in §§5—9 of Chapter IV.
For a formulation of the results we require the function spaces
and f), which in many respects are analogous to the spaces
and r,-*-(er) repeatedly used above. The norms in them are defined in the fol¬
lowing way. For any integral / > 0 let
and for nonintegral / > 0 let
2 W!'"
2»r+j-[/| '*• vr
. .(JL) _ l-lbr-t
<.“))%*= 2
vr 0 < l~25rl-s < 2fr Qt
616 VII. SYSTEMS OF EQUATIONS
where lor a < l ate die same as in § 1 of Chapter I.
la the space we introduce the norm
,<-<<+!<<•
The norm
2bm
is the norm in the space W^>m,m(Qy). Its principal part is denoted in die follow¬
ing way:
<«’£= 2 li°wU-
'2t>r+s-2t>m *•
The testrictions of die function u € W^m,m(Qy) and its derivatives
D'Pji (ibr + s < 2bm) to the surface Sj. belong so the space
where k = 2bm - 2 hr- s.
In order to define the norm in we must vary the definition given
in § 3 of Chapter II of the norm in Wlq’^2(Sj) in exactly the same way as the norm
of h1>1/2(qt) was varied in die definition of j) (i.e. the number 2 is
replaced every where by 26).
The spaces ffm'm(QT), f *~l/,!'inHs~l/qiST\ where I is a
nonintegral positive number and m and s are integral positive numbers, are de¬
fined in exactly the same way as in § 4 of Chapter ID (with obvious variations).
For example, j.) is the space of functions from satisfying
die null initial conditions
5-L- (— &])•
We now proceed to a formulation of the basic results of [llfic].
Theorem 10.1. Suppose the coefficients of the operators L^., B^j and
belong to the classes Hl'Sk'(l~Sk)/2b(QT), Hl~C'g’<'l~a*)/2b(ST) and Hl'paia)
respectively and S 6 where I is a nonintegral number satisfying the
condition
I > max (0. oir).
Then problem (9.31) has a unique solution, in the class of vector functions
with u/€tf'+V<i+,/)/Z6((>r), for any f> € Hl ~S>’<'l~Si)/2b(QT), 6
§ 10. BASIC RESULTS ON SOLVABILITY 617
anc2 <£a € satisfying the compatibility conditions
that are necessary for the existence of a solution from the indicated class. The
solution is subject to the inequality
+11®. +2, r !<'-■)). do.i)
G-l ' n-l /
The relation indicated in the theorem between the smoothness of the known
functions and the smoothness of the solution is exact, while the restrictions im¬
posed on the coefficients are necessary in order for each term of any equation of
the system to belong to the same class as that containing the free terra (the same
remark holds for the boundary and initial conditions).
The compatibility conditions for problem (9.31) consist in die fact that the
functions d^ui/dt*1 |(=g (ft > 0), which can be calculated from die system and die
initial data, must satisfy for *6 S the boundary conditions and relations obtained
as a result of differentiating the boundary conditions with respect to t. In the
theorem it is required that the vectors f, <f> satisfy the compatibility conditions
of minimal order, i.e. only those conditions which f = £u, $ = !Buj t- and 4* -
<ai|J=Q satisfy for u* 6 y).
We now fotmulate an analogous theorem for the Cauchy problem in tbe domain
D<r>-
"+1 = f }
'• I (10.2)
” <P- J
Theorem 10.2. Let I > 0 be a nonintegral number. If the coefficients of the
operators L^. and C^ belong to the classes and Hl~p*{En)
respectively, then for any f € s/’*i-s/^2^(£)(r)) and <f>a 6 Hl~Pa(En) problem
(10.2) has a unique solution in the class of vector functions u = (u1, •. ^, um) with
ui S H *^^(Dj^), and this solution is subject to the inequality
% i“/|V/1)<cCl, |//|K>+,|I |<paCPo))- (io-3>
The relation between the smoothness of the given functions and that of the
solutions of problems (9.31) (10.2) is determined by the numbers s-f t^ pa
and oy In §8 we saw that for certain systems the numbers s. and t. are uniquely
1) Since Sj <0, pa < 0 it follows that I - Sj > 0, I - pa > 0.
618
VII. SYSTEMS OF EQUATIONS
determined. For such systems Theorems 10.1 and 10.2 hold for any admissible s.
and t..
We consider as an example the Cauchy problem for a system of two heat equa¬
tions, which decomposes into two separate problems:
—• — A«* = /*, =
«‘lw=»IW' b!Lo=<pj(*)-
As was deteimioed in §8, there exist for this system infinitely many collec¬
tions of numbers a^, «2, tj, in particular,
S, = 0, Sj == k, t, = 2, =
where k is an arbitrary nonnegative integer. Here pj = - 2, p, = - 2 — k. Accord-
ing to Theorem 10.2, for any /* 6 ), f2 6 €
Hl*2(En\ there exists a solution of die problem with die follow¬
ing properties: u
i e Hinnm/ityty, *2 e It is seen that
the ambiguity in the definition of the numbers s. and tj is connected with die
fact that we can solve die Cauchy problem separately for the functions and
u2 in different classes: and
The basic argument in the proof of Theorems 10.1 and 10.2 is exactly the
same as in the proof of Theorems 5-1—5.3 of Chapter IV. But it. differs in certain
essential details, for example, in the function spaces in which problems (9.31)
and (10.2) are investigated. We will therefore present the basic steps in the proof
of Theorem 10.1. It is cen ered on a study of the following problem with null
initial conditions: find the vector function u = (a1, • ■ • , um) with uJ 6
where r < T, such that
JS’ix, t, -S-, 4)u = *.
' d 1 (10.41
*(*• *'W'
for fi 6 si’<( (Q,i, ^ cr«>/'24 (5^). This problem can also
be written in the form
Aa = h.
where A is an operator acting from the space
§ 10. BASIC RESULTS ON SOLVABILITY
619
into the space
according to the formula
f . n Hl-*r * V> (Qx) x fl «<«-»,) (5 )
/•l <?—l O T
i4u==(^’n. j^u |5t).
while h = (f, $) € §2^-
Theorem 10.3. Suppose the coefficients of the operators £ and 58 and the
boundary S satisfy the conditions of Theorem 10.1. If t < < T, where rQ is
a certain number depending on S., IB and the various characteristics of O and 5,
then problem (10.4) is uniquely solvable in the class and the solution is sub¬
ject to the inequality
+|, £-«>). (,0.5,
To prove this theorem we construct an operator R which acts from into
■is bounded and is such that for any h 6 v 6
= h + Hi,
RAh*=v + Wv,
where T and If are contraction operators if r < j-q. The operator R is given by
formula (7.8) of Chapter IV, in which the functions t/*) are replaced by vectors
v^\x, t) defined in terms of the solutions of the following model problems: the
Cauchy problem with null initial conditions
M-h- £)■■
= !. /'(=//
(OK.).
u>£h!+1i- (/j1;],)
and die boundary value problem in the domain RST^
(10.6)
(—
’ l Or
w)"k-»=0*
(10.7)
The operators £.q and in these problems are obtained from the principal
parts of £ and IB by freezing the coefficients and, if necessary, carrying out an
orthogonal transformation of the space variables x.
620
VII. SYSTEMS OF EQUATIONS
The solutions of problems (10.6) and (10.7) are expressed in terms of volume
and surface potentials, the kernels of which are elements of two fundamental mat¬
rices of the parabolic system S.Q{d/dx, d/dt) u = 0: a matrix TXx, t) with m rows
and m columns satisfying the system
is the unit matrix of order m) and a matrix §(*, 0 with br rows and m col¬
umns, which is a solution of a boundary value problem in the half space *„ > 0
with a ^-function in the boundary condition:
w)^(x' t) = °>
*0 ('£. 4r) 9 (x, /)|^.0 = b(x\ t) g*4'’. (10.8)
The following estimates are valid for the elements V, -, G- of the matrices
r and g:
i / JL\
I DtiAr„ | < « -1)» *) exp / cj.
(*•■/—> <#). (10.9)
itfoioj <cts(t - t)-g(-■*»-««-v^»») exp [»-»j
— 1. ..., m\ q ■= 1 br, *„ > 0). (10.10)
A considerable part of the difficulties arising in the investigation of parabolic
equations of high order and especially of parabolic systems is connected with the
proof of these estimates. Estimates (10.9) and (10.10) permit us to estimate the
volume and surface potentials (T*f ) and (§ *2$) in roughly die same way as in
Chapter IV, and to prove in this way estimate (10.5) for the solution of problem
(10.7) and an analogous estimate for the solution of problem (10.6). Tbe constants
in these estimates depend on r but they remain bounded for finite r.
A proof of the boundedness of the operator R, estimates of the norms of the
operators T and W, and a derivation of Theorem 10.1 from Theorem 10.3 are made
in the same way as in Chapter IV.
Theorem 10.1 is also valid for noncylindrical Qy, with the method of proof
given here being retained in die case of noncylindrical Qj. without any essential
§ 10. BASIC RESULTS ON SOLVABILITY
621
changes [ 11 *«].
We will now formulate a theorem on the solvability of problems (9.31) i° tbe
class of vector functions u(*, t) with v) € R^+t/’ (i+W26 (Qy) lot integral I + t.
and (I + t.)/2b. From, the latter condition it. is seen that we must confine ourselves
to only those systems of which t. - 2bti for integral (it is easily seen that in
this case ~ 26s'. for integral sp.
Theorem 10.4. Let I = 2bk, k > 0, be an integer, with I > <rmax- Let sk =
2bs‘ic, tk - 2bt'k. Suppose that S 6 Ci+‘max, the coefficients of the operators
Lkj have bounded (continuous) derivatives for 2 bp + v <1 - sk, and
the coefficients of the operators B s- and C^ belong to the classes
1/9-*, «-*s-^ md Hl-Pa-2b/<,H(Q) (( u „ arhitrarily
small positive number). Suppose further that the numbers
l — os
26
are nonintegral. Then problem (9.31) has a unique solution in the class of vector
functions for which IJ € (Qy) for any
f' € “r <DS f ' 2* t1-"'-?) (ST),
and <fia 6 B^~Pdr^^^(Sl) satisfying the compatibility conditions that are neces¬
sary for the existence of a solution from the class indicated above.
The following inequality holds:
+ 2 ll**l
" (S).
• (10.12)
The spaces B* Pa 2b^(Cl) differ from IP* Pa 24/,,(Jl) only for integral I -
pa— 2b/q and q £ 2 (see in this connection § 2 of Chapter 11). In the case b > 1
the number Z - pa - 2b/q can be integral (or q £2 (in contrast to the case con¬
sidered in Chapter IV).
The restriction on the numbers (10.11) is an analog of the condition q / 3/2
in Theorem 9-1 of Chapter IV.
The requirements in Theorem 10.4 on the coefficients in the minor terms of
the operators Lk-, Band are excessive. As a matter of fact it is sufficient
622
VII. SYSTEMS OF EQUATIONS
to require of the coefficients in the minor terms of the operator L^-, for example,
that € Wy **' 11 sk)/li (Qj.) for vJ 6 (Qj,). As in §9 of
Chapter IV, ic is possible to show that for this to be die case it is sufficient that
each coefficient in the minor terms of die operator Land possibly certain of
its derivatives, have norms (9.1) of Chapter IV, which are finite and tend to zero
for r —» 0, with a certain r > 1 depending on which derivative of which function
the coefficient is with. The same remarks apply to the operators Bs - and C^. It
is also possible to require less of (he boundary S.
We have always assumed that the matrix SB satisfies tbe complementarity
condition. But if this condition is violated at even one point (£, p) 4 0, with
Im - - 0 and Rep > 0, then estimates (10.1) and (10.12) cannot hold with the
constants c being independent of n. Consider in the domain ft* ^ the problem
•3“o (-Jj-. -§f) U (•*. t) = f (x, t),
(■as- * 4r\wLa_o “1® <*■t]-
in which the operators £q, Sq, (2q with constant coefficients contain only princi¬
pal terms. We assume that the complementarity condition is violated for some real
C0 and some p0 with Rep„ > 0. In this case the problem
-^o (<£j. Po) «(*«) = 0,
ICT • " <*«> I *= 0. (10.14)
n“°
•Ww5
has a solution different from zero. It is not difficult to see that for any A > 0 the
vector u^(x, t) = (u^,.,., u"), where
a{(x, t) =* A ''"'ieptxUV *Vu' (Xx„)
is a solution of problem (10.14) for f«0,$>0 and
<f (X) = (<p(V, cpQ.
We first assume that Re pQ > 0. In this case for any A
§ 10. BASIC RESULTS ON SOLVABILITY
623
r oa
2 j <(§ ce~kx , k > 0,
while
at m
m m
> s .w <*• ^)>S*‘y)=2 <eiCcJ“/(x”»(4+';) > °-
Therefore inequality (10.1) with a constant independent of ti cannot hold for a sol¬
ution of problem (10.13). For sufficiently large A not even the estimate
£ (“Hx- T))on <c S K. l^. (10.15)
for the functions ul holds, no matter what constant c we take and how small the
A
number T > 0 is. This means that problem (10.13) is incorrect.
It is not difficult to verify that when Rep0 = 0 inequality (10.1) for problem
(10.13) is violated in the case of large A by the vector
u*i*. him.
where £(t) is a smooth function satisfying the conditions 0 < £(<) < I and
Jo for /<-•
C(0>={ or
1 for
Vectors violating inequality (10.12) for problem (10.13) are constructed in an
analogous manner.
We have thus proved that the complementarity condition is necessary for the
validity of estimates (10.1) and (10.12) for die solutions of problem (10.13). The
passage to tbe more general problem (9.31) is made in tbe same way as for ellip¬
tic equations
Thus the complementarity condition distinguishes a class of well-stated prob¬
lems for equations and systems of parabolic type, for which Theorems 10.1 and
10.4 are valid. This condition is fulfilled, in particular, for all of the basic
boundary value problems for equations of second order that were considered in
the preceding chapters, and also for problem (5.15). It is also fulfilled in die case
of die first boundary value problem for strongly parabolic systems; as was shown
in inequality (10.12) with q = 2 is valid for tbe solutions of such problems.
624
VII. SYSTEMS OF EQUATIONS
Hie situation is different in the case of tbe first boundary value problem for sys¬
tems that are parabolic in the sense of Petrovskii, even when they are of first
order with respect to t. Consider in the domain the first boundary value
problem for a system with constant coefficients
|r + ^o(/3j) »» = «(*. t).
t)
d>-
~dxl
(10.16)
«l/=o = <?(*)•
The operator contains only terms of order 2b.
The problem corresponding to (10.14) has the form
pu -f t^-o (*£> M =
(10.17)
0.
If m = 1, then it is completely obvious that this problem has only the zero
solution. This is also true in the case m > 1, n = 1. Indeed, for n = 1
where AQ is a numerical matrix. Let T be a transformation takiag AQ into a
Jordan form:
a, A,
TA0T-
0
By virtue of the parabolicity condition a.j / 0.
We introduce the new vector
v (x) = Tu(x).
Multiplying (10.17) from the left by T, we obtain the following problem for y:
Pv + '4o-T3T==0-
dx41
dx"
= 0,
(10.18)
(10.19)
§ 10. BASIC RESULTS ON SOLVABILITY
625
Since
det \pE A'0 (it)24] = det \pE + A0 (ix)”] == j0( Ip + <f*f «*J.
each operator
AS) I d d\ d , „ d26
L \n'nr a-
is parabolic. From (10.18) and (10.19) it follows that the function tTm is a solu¬
tion of the problem
,b — 1 Tjm J —
S“U-„“ =0,
&X u=0
Therefore = 0 and the next to last equation in (10.18) has the form
Consequently v‘m ”1 = 0,.and so on. Thus problem (10.18), (10.19) and problem
(10.17) have only the zero solution.
However, this assertion does not hold for n > 1. Consider problem (10.17)
for a system of two equations, the matrix if) of which has the form
^o(£- <*)
((x-tYVrxU-lf+l-’-Kl-x -4d+WtJ«-,
—I n t% _ 2 2 I
^ _(r-J VT+7t*-l) (t+/Vl+*.ta_i)+5' -Hu-l4 +M/
(here £'2 = f ?, A > 0). This system is parabolic with 4 = 1 since
det(pE + J.0(iL ^)~(p+C*+t*_, + T*)*-0H-P+t,>*.
The complementarity condition is violated in this problem for £' = 0, p « ^Cn-i
and any £n_j > 0, since for the indicated values of p and £ problem (10.17)
with die matrix of (10.20) has the nonzero solution
S(je#) = (*B«-mxt*-i'«. 0).
Thus for n > 1 there exist systems parabolic in the sense of Petrovsku. for
which the first boundary value problem is poorly stated.
Let us return to the question of the solvability of the boundary value problem
(9.31) when Ac complementarity condition is fulfilled.
For systems that are parabolic according to Petrovskii some of the above-
626
VII. SYSTEMS OF EQUATIONS
mentioned results of [l 16c] were obtained in earlier papers by other authors. For
example, Theorem 10.4 is proved by a different method in the papers [2] of M. S.
Agranovic and M. I. Visik, where the general boundary value problems for systems
that ate parabolic according to Petrovskii are investigated in tbe spaces
with integral and nonintegrai I > 0. Briefly, this method is as follows. The case
when the coefficients of the operators £ and $ depend only on x is considered
first. These problems are solved by application of the Laplace transform with
respect to t. The passage to the general case is effected by successive steps in
small intervals < t <
Exact estimates of solutions of boundary value problems in the norms of
for (Petrovskii) parabolic systems of first order with respect to t (without
an existence theorem) were announced by L. N. Slobodeckii [U !•>].
In the case q- 2 exact estimates are obtained with tbe use of Parseval’s
equality, which removes the need for estimates (10.9) and (10.10).
Exact estimates in Holder norms have been established by A. Friedman [31a]
for solutions of (Petrovskii) parabolic systems of form (8.5), interior to the domain
in which these solutions are defined. Friedman proved these estimates by general¬
izing a method proposed by Douglis and Nirenberg for elliptic systems.
We proceed now to a brief account of the results, obtained by the methods of
potential theory, on tbe solvability of the Cauchy problem and of the boundary
value problems for systems parabolic in the sense of Petrovskii. Systems of form
(8.5) are the ones that have been most studied by these methods.
A fundamental matrix Z(x, t, t) for such systems was first constructed by
S. D. feidel'man [25b(ij] on the assumption that tbe principal coefficients of the
operator Q are sufficiently smooth functions. These results have been carried
over in [25b(ll)]f [135] and [132c] tD the case when the coefficients of Q satisfy
only a Holder condition. In [25b(ID] a fundamental matrix has also been constructed
for (Petrovskii) parabolic systems of general form. It is required in [25b(li)] that
the coefficients of the operator (2 satisfy a H'dlder condition only with respect to
the variables x and be continuous in the variable I. Further progress in this
direction has been achieved in the note [79], where it is required that the coeffi¬
cients of Q satisfy the more general (than a Hdlder condition) Diai condition.
If the coefficients of the operator 3 satisfy a Holder condition in Dn+1 with
respect to * with exponent a, and are continuous in t, then the elements Z..
of the matrix Z are subject to the following inequalities:
§ 10. BASIC RESULTS ON SOLVABILITY
\t?xZij{x, t, T)|<e(f —r)
627
n-j k
2b
exp
U-ll
•2b
26-1
ITT,
\D?Zu(x. I, t, t)-
k <2A;
■ D?Zu(x'. 6.
n+2b j.fi
1.1)|
Z4
exp
I x-
(t- t)
n> \
Tx"5rrr /
-4- expj — C
(*-tV
26
i
..ijrr
V <f-T)'
fi P G.
The differential properties of the functions Z;. are improved together
with those of the coefficients of the operator g. jf tj,e coefficients
have continuous derivatives D*°D* satisfying a Holder condition with respect to
x, then there exist derivatives D™°D”Z.j(*, £ l, r) (2hng + m = 2b + r, r *= 2i&0 + A)
and also [25 a].
The fundamental matrix Z has been used to investigate tbe Cauchy problem
*L + ^a = f.
” (10.21)
“U«ipw
in various function classes, including those is which the functions ate increasing
exponentially for |*| —» eo (see [25a; l iia, c]); in particular, exact estimates of
type (10.3) have been obtained for the solution of this problem. A detailed study
of the properties of the matrix Z has also made it possible to investigate the
stability and behavior for i —> + <*> of the solution of problem (10.21), to study the
Cauchy problem for operators with increasing coefficients and to prove theorems
of Liouville type for solutions of parabolic systems. All of these and many other
results, some of which have also been proved for (Petrovskii.) parabolic systems
of general form, are presented in the monograph [2 5a], which also contains an ex¬
tensive bibliography.
Substantially less complete results have been obtained by the methods of
potential theory fot boundary value problems. Problems of the form
628
vn. SYSTEMS OF EQUATIONS
have been considered by tbese methods with the assumptions that the order of the
operators B^., which can depend on q, is less than 2b (the order of the operator
2) and that the following condition of Ja.B. Lopadnskii is fulfilled: the rank of
tbe matrix (9.23) is equal to bn.
The general plan fot solving problem (10.22) in the indicated papers is as fol¬
lows. First the matrix Z is used to reduce this problem to the same problem but
with f = 0 and <f> = 0. The solution of this latter problem is then sought in the
form of a sum of potentials
bm
0-2 /*/Gjs(x, t, T)n'(i, X)dSh. (10.23)
j-1 0 s
where the G-s are kernels having the following properties.
1) For each & * 1, • • •>, bm the vector G*5* * * * * • ^ns) is a solution
of the homogeneous system
£)o»-o
in a domain Qy.
2) The principal parts of the G-s with respect to order of singularity are
functions G^(x - g, t - r) forming the solution of a boundary value problem of
type (10.8) in the half space *t/(£) > 0 bounded by the tangent plane at the point
(£, r) € Sy to the surface Sy. The functions are subject to inequalities
(10.10) in this half space:
\Duy,o%](x. oi
< c (, _ t)-T exp f CI£, (10.24)
V i** J
where is the order of the operators
The functions n*(£, r) in (10.23) are determined from a system of Volterra
integral equations obtained from the boundary condition. An exact and explicit
derivation of these integral equations has at present been given only under one
of the following assumptions: 1) the orders f3q of all operators B ^ are the same
and 2) n = 1, although the functions G.$ with the above-mentioned properties
have been constucted without these additional assumptions.
The method of potential theory was the first method to be used in investigat¬
ing boundary value problems of form (10.22). Beginning in 1956 T. Ja. ZagorskiT
published a series of papers and the monograph [1291 in which problem (10.22)
§ 10. BASIC RESULTS ON SOLVABILITY
629
is considered in bounded convex domains. But these papers and the monograph
contain important errors (see in this connection [116c]), S. D. Eidel man [25»]
obtained the exact estimates (10.24) for the kernels GW, constructed the func¬
tions G-s in nonconvex domains and proved the solvability of problem (10.22) un*
der the above-mentioned additional assumptions. Also, in the note [7-4] jfidel’man
and G. Ja. Lipko announced certain results on the solvability of boundary value
problems of form (10.22) in noncylindrical domains and in domains with infinite
boundaries.
Estimate (10.24) and all of the results following from it were obtained by
Ei del 'man for those systems for which the complementarity condition is fulfilled in
the case of the first boundary value problem. The example of (10.20) shows that
this class of systems does not coincide with the class of all (Petrovskii) parabolic
systems of the form (8.5). It should be noted that this restriction on the class of
systems is in no way connected with the representation of a solution in the form
(10.23) or with the determination of the densities /its. It is connected with the
method of proof of estimate (10.24). In the analogous estimate (10.10) for
more general systems has been proved by another method. Therefore all of the
subsequent arguments and final results of &idel'man are valid for any (Petrovskii).
parabolic system of form (8.5).
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