Proceedings of Symposia in
Pure Mathematics
Volume 4
Partial Differential
Equations

Partial Differential
Equations

This page intentionally left blank

Proceedings of Symposia in
Pure Mathematics
Volume 4
Partial Differential
Equations
Charles B. Morrey, Jr.
Editor
j\| American Mathematical Society
v Providence, Rhode Island
V/VDED%V

PROCEEDINGS OF THE
FOURTH SYMPOSIUM IN PURE MATHEMATICS
OF THE AMERICAN MATHEMATICAL SOCIETY
HELD AT THE UNIVERSITY OF CALIFORNIA
BERKELEY, CALIFORNIA APRIL 21-22, 1960
with the Support of the
UNITED STATES AIR FORCE
Under Contract Number AF 49 (638) - 746
Library of Congress Catalog Number 50-1183
International Standard Book Number 0-8218-1404-4
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10 9 8 7 6 5 4 3 96 95 94 93 92

CONTENTS
Extensions and Applications of the De Giorgi-Nash Results.
By Charles B. Morrey, Jr 1
Dirichlet's Principle in the Calculus of Variations.
By James Serrin 17
Associated Spaces, Interpolation Theorems and the Regularity of Solutions
of Differential Problems.
By N. Aronszajn 23
Lebesgue Spaces of Differentiate Functions and Distributions.
By A. P. Calderon 33
The Majorant Method.
By P. C. Rosenbloom . .51
A Priori Estimates for Elliptic and Parabolic Equations.
By Felix E. Browder 73
Differential Equations in Hilbert Spaces.
By Francois Treves 83
A Maximum Property of Cauchy's Problem in Three-Dimensional Space-
Time.
By H. F. Weinberger 91
Comments on Elliptic Partial Differential Equations.
By Louis Nirenberg 101
Some Unusual Boundary Value Problems.
By Martin Schechter 109
A New Proof and Generalizations of the Cauchy-Kowalewski Theorem to
Nonanalytic and to Non-normal Systems.
By Avner Friedman 115
Regularity of Continuations of Solutions.
By Fritz John 121
Some Local Properties of Elliptic Equations.
By David Gilbarg 127
Estimates at Infinity for Steady State Solutions of the Navier-Stokes
Equations.
By Robert Finn 143

VI
CONTENTS
Interior Estimates for Solutions of Elliptic Monge-Ampere Equations.
By Erhard Heinz 149
Zero Order A Priori Estimates for Solutions of Elliptic Differential Equations.
By H. O. Cordes ... 157
Index 167

EXTENSIONS AND APPLICATIONS OF THE
DE GIORGI-NASH RESULTS
BY
CHARLES B. MORREY, Jr.
1. Introduction. The results of De Giorgi [3] and Nash [14] which are
referred to in the title of this lecture are their a priori estimates for the
Holder continuity of the solutions of equations of the form
on some domain G. Here ut0 means dujdxP, x = (x1,- • •, xv), and repeated
Greek indices are summed from 1 to v, as will always be done in this paper,
and the aa^(x) are supposed to satisfy
(1.2) m|A|2 ^ aae(x)\a\p S M\X\2 (|A|2 = £ A«)> 0 < m ^ M
<x = l
for all x on G and all A. They showed by completely unrelated methods
that any solution of (1.1) satisfies a uniform Holder condition on any compact
set F <= G which depends only on F, G, m, M, and bounds for \u\ on G in
the case of Nash and the L% norm of u on G in the case of De Giorgi. Since
the bounds and Holder conditions on u do not depend on any continuity
properties of the aaP, the results carry over to cases where the aaP are merely
bounded and measurable, in which case the equations (1.1) must be written
in the integrated form
(1.3) f £,XVA = °> ZeCXG)
where C](G) denotes all functions of class C1 on G with compact support and
the function u belongs to the space H\(D) for each bounded D with D c: G;
here £ a£^ etc., denote d^/dx", d^jdx^x^ etc.; if £ e H\(Q), T >a ienotes its
strong derivative.
Since the notation Hlv{D) does not seem to be standard, we identify these
spaces with those denoted by Wl(D) by Browder and the Russians and
8?V{D) by the writer (see [2; 8]); in the case p = 2, functions e H\(D) if and
only if they are strongly differentiable in the sense of Friedrichs [4].
It is important to have such estimates in order to discuss nonlinear
equations and it is especially important to be able to have the conclusion for the
i

2
CHARLES B. MORREY, Jr.
equation (1.3) when the aa$ are merely bounded and measurable in order to
be able to conclude the differentiability of the solutions of minimum problems.
For instance, De Giorgi was able to show, using his results, that any function
minimizing an integral
(1.4) I(z,G) = jj(x,z,Vz)dx (Vz = (z,i,.-',z,J
among all functions z in H\{G) having the same boundary values in which /
is of class C£ (nth derivatives Holder continuous with exponent fx, 0 < ft < 1)
n ^ 2, is also of class C£ provided that/ satisfies the following conditions:
There exist numbers, m, M, and K such that
f(x, z, p) ~ f(p), p = (Pu ••,#„)
(1.5) m\p\* - K ^ f(p) < M\p\* + K,
m\\\* S fPaPp(p)Xa\^ Jf|A|*
for all p and A. The existence of such a minimizing function was proved in
1943 by the writer (see [9] where much more general existence theorems were
proved).
The method of proof is as follows: First of all, it is straightforward to
show that if z minimizes I(z, G), then
(1.6) Jg (£ JPa + ifz)dx = 0, £ g Cl(G),
provided that/ satisfies (1.5) or any of the conditions below; in De Giorgi's
case, of course, fz — 0. Next select D c c G (i.e., D compact and D c= (?)
and choose D' with D c c D' c c G. Since each/Pa g L2(G), (1.6) holds
for all £ g H\0{G) and, of course, z e H\(G) if /(z, 6?) is finite, on account of
(1.5). So let £ g Hl0(D'), extend it to be zero in G — D'', and for small A,
define
(1.7) &(*) = *-![£(* - Aey) - £(*)], «*(*) = A"1^ + hey) - z(x)]
where eY denotes the unit vector in the xr direction. If this £a is substituted
for £ in (1.6) and if the obvious change of variables is made to get rid of terms
involving £(# — hey), and if one uses the integral form of the theorem of the
mean, one obtains
(1.8) J^ £ ,aaf(x)zKfflx = 0, I g H\0(Dr)
where the af satisfy (1.2) uniformly in h, af being given by
(1.9) aff(x) = tlfPaPI(l - t)Vz(x) + tVz(x + heY)]dt (a.e.).

THE DE GIORGI-NASH RESULTS
3
Now, equation (1.8) is just (1.3) where all we know is that the ajf are
measurable and satisfy (1.2). Moreover, from theorems on the Lebesgue integral,
it follows that the L^ norms of the zn on D' are bounded and, in fact, that
\\zh — 2,y||z>' -> 0. But from the De Giorgi result, we conclude that the zn
are equi-Holder continuous on D and hence tend uniformly to their limits
z tV on D which are therefore Holder continuous. Once this is known, the
higher differentiability results follow from previously known results (see
[11], for instance).
The corresponding difference-quotient device was used in 1912 by Lich-
tenstein [6] to show that any C2 solution of a minimum problem (1.4) in
which v = 2 and / analytic was of class C3 on interior domains and hence
analytic by Bernstein's theorem [1]. In 1929, E. Hopf [5] was able to obtain
the same conclusion by assuming only that zeC^ for some \iy 0 < fx < 1.
In 1938, the writer [7] obtained the same conclusion in case z is merely
Lipschitz; v = 2 in these two latter cases. All of these results assume that
something is known a priori about z, but only in special cases had it been
shown previously that solutions z existed which had these properties. In
1943, the writer employed the spaces, now denoted by H*, to extend the
rather meager previous existence theory. These results applied to cases
where v was arbitrary and z could be a vector function; some of these results
were extended to more general integrals in [10]. For v = 2, the writer was
able to show that these solutions, known to exist, were also differentiable,
provided / satisfied the conditions (1.12) below with v — 2 and k = 1, but z
was allowed to be a vector function. In 1950, A. G. Sigalov [15] proved
corresponding results for integrals where / satisfies (1.11) below with any
k > 1/2 but v still = 2.
All of these results involved consideration of equations of the form
(1.10) f [£>a^ + bau + ea) + i{ffuta + du+ f)]dx = 0, £ eCj((?)
with rough coefficients, but the methods used were peculiar to the case
v = 2. The De Giorgi-Nash results, then, represented an important
breakthrough in this field. In 1959, the writer [12] was able to extend the De
Giorgi-Nash results to certain equations of the form (1.10). These results
did not lead immediately to further differentiability theorems for minimum
problems but have been a useful tool in the recent results on differentiability
obtained during this year by the writer and his student E. R. Buley which
are the principal concern of this paper.
Last fall (1959), Buley had obtained a priori bounds for the solutions z of
minimum problems in which / satisfies conditions (1.11) below with 1/2 < k
^ 1. In January (1960), J. Moser kindly communicated his simplification
of the proofs of the De Giorgi-Nash results [13]. This enabled Buley to
extend his a priori results to problems in which / satisfies either (1.11) or
(LIT) below for any k > 1/2. With the aid of several lemmas proved by

4
CHARLES B. MORREY, Jr.
the writer, Buley could then show that any solution z of such a problem is in
fact differentiate provided k ^ 1; he was unable to carry through the
difference-quotient procedure when 1/2 < k < 1. The conditions on /
required by Buley are the following (/ is assumed of class G" in all cases):
mVk - K ^ f(x, z, p) ^ MVk, 0 < m ^ M,
2 Ul + fU + ft + fz%] ^ MxV*-\ V = 1 + z2 + |p|2,
(1.11)
2 [ft, + /a 3£ ^i ^"2 (all (*, z, p)),
mF*-i|A|2 ^ /^(x, z, p^A, ^ Jf F*-i|A|* (all A);
or the alternative conditions
(1.11') same as (1.11) except/ = f(x,p), V = 1 + \p\2.
The results of Buley and their proofs are sketched in §2.
The writer was able to extend (essentially) the results of Buley to the cases
where 1/2 < k < 1 by considering a sequence of auxiliary problems in which
is considered a function zr which minimizes I(z, G) among all z in the
appropriate space for which another integral J(z, G) ^ K. This method is
considered in some generality and applied to these cases in §3. In §4, the
writer applies that method and some more refined estimates to extend Buley's
differentiability results to cases where / satisfies the conditions
mVk - K <> f(x, z, p) ^ MVk, 0 < m ^ M,
2 (/I + fly + /£) ^ MXV*\ 7=1 + b|2,
(1.12)
2 (fPa + flaz + Ua*v? * MXV^ (all (x, z, p, A)),
mF*-i|A|2 ^ /PaVUe S MF*-i|A|2, £ ^ „/2.
For example, if the aa^(x, z) satisfy (1.2) for all (x, y, A) and
f(x9 z, p) = a«*(x, z)papf},
then / satisfies (1.12) with k = 1 but not either (1.11) or (1.11'). But
f = Vk satisfies either set of conditions with the appropriate definition of V.
We use the following notations and make the following additional
conventions: All integrals are Lebesgue integrals. All domains are bounded
and if G is a domain, dG denotes its boundary. A domain G is of class C1 if
and only if each point P of dG is in a neighborhood jV which is the image
under a 1-1 map of class C1 of a sphere B(xo, R) (center xo, radius jB) in
which P corresponds to xo and Jf C\ dG corresponds to the part of B(xo, R)
where xv = xv0; usually we take x0 = 0. The classes Cm(G), Cm(G), Lip (G)
have their usual significance and Lipc((?), for instance, denotes those Lip-
schitz functions with support (closed) in G. The space Hl0(G) is the closure

THE DE GIORGI NASH RESULTS
5
with respect to the norm in #*((?) of the space Glc(G). If 9 is a vector, V9?
denotes its gradient and \<p\ its Euclidean length. If G is a domain Gp
denotes the set of points x such that B(x, p) <=■ G. We use the notation
—7 for weak convergence. If S is a set \S\ denotes its measure. We
shall denote any constant C which depends only on the bounds m, M, Ky M\
k, and v by C; it is not assumed that such constants are all the same. These
results will be presented in more detail in the Proceedings of the International
Conference on Partial Differential Equations and Continuum Mechanics held
in Madison, Wisconsin, in June 1960.
2. The results of Buley. We shall treat only the case (1.11) and shall
assume v > 2; the modifications necessary to handle the cases (1.11') and/
or v — 2 will be clear. On account of (1.11), it is clearly sufficient to restrict
ourselves to functions z e H\k(G). So we suppose that z* e H\k(G) and let
2 be a function in H\k(G) such that z — z* e H\k$(G) and z minimizes J
among all such functions. We indicate how to prove that z is differentiate
on domains D c c G.
First. (1.6) holds for all £ e H\k^(G). This is easily seen by
approximating in H\k(G) by £ eC]{G) and noting from (1.11) that the/„a and/z e Lr(G)
forr - 2kj{2k - 1).
Next, we apply the difference-quotient procedure described in the
introduction to obtain the equation
(2.1) ^ Ah[CAah\e + Kzh + P%ef) + ?(6XzM + chzh + PW)]d* = 0,
leHltjoiiy), q= 1/2,
where An and the other coefficients are given by (a.e.)
Ah(x) = P{1 + [z(x) + tAz]* + \p(x) + tAp\*}dt,
Az = z(x + hey) — z(x), etc.,
(2.2)
Atftf = fVaPfi[x + they, z(x) + tAz,p(x) + tAp]dt, p(x) = Vz(x),
Ahbah = JV^[same]*, ^APKV = JV^same]*,
Pa(x) - max 1 + [z(x) + £Az]2 + |p(s) + £A#|2
and the other coefficients are defined correspondingly. We note that the
coefficients a%0, b%, ch, e£y, and/£ are uniformly bounded and
Ah(x) = 1 if k = 1, zA -^#y in L2k(Df),

6 CHARLES B. MORREY, Jr.
(2.3) Ah -* F*-i in £*,<*_!>(£') (i > 1), PA -> F in L*(D'),
^Jaf-* V*-"W = F-<*-1>/%a,(,inLa/(t_1)(Z)') (* > 1),
with corresponding convergence for the other coefficients. If h ^ 0 and
small, we may set
(2.4) £ = 1?%*, ^GLipc(D')
in (2.1). If this is done, we obtain the result that
(2.5) J^ ri*Ah\Vzh\*dx S C j^ tf + \Vv\2)Ah(zl + P>)ek,
C = C(m, if, Z, ifi, fc)
by using the Schwarz inequality, the device 2 ab ^ €«2 -f €_162, etc. Since
for each r\ e Lipc (D') the right side of (2.5) is uniformly bounded for all
small h, and since there is such an rj = 1 on Z), and since -4a(#) ^ 1, we see
that zA —, £>y in H\(D), Aqhafzhfi-^ something which must be V(k~1)l2aafipyP
in L2(D), and for any £ e Hl2ky0(D), we see that £ t-^4| -> F<*-i>/*£ >a in L2(D).
Thus with the aid of Lemma 2.1 below, we conclude the following:
Second. The functions py, U = Vm, and V(k~1)l2pye H\(D) and satisfy
the equations
(2.6) | F*-i[£ ,a(aa^y,0 + 6apy + F^av) + £(&«p„fa + cpy + V^fy)]dx = 0.
Moreover
(2.7) f |V*7|%r < Jk2 f F*-i(|Vp|2 + b|2)fo < oo.
Lemma 2.1. Suppose F is of class C1 for all (t/1,- • •, up), suppose each
up g H\(0) for some A ^ 1, suppose U = F{u1,- • •, up), suppose U and the
Va e L^(G) for some /x ^ 1, where
(2.8) Va(x) = f ^«WK(4 a = 1,. • ., y.
p = i
TAen U e Hl(G) and U a(x) = Fa(#) (a.e.). TAe same conclusion holds if F
is convex if, in (2.8), we replace the Fv by the coefficients of any supporting
plane to F at any point x where F does not have a unique tangent plane at
[u(x),.--,up(x)]
This is proved by choosing representatives u'P of vP which are absolutely
continuous along almost all lines parallel to each coordinates axis (see [2])
and noting that U = F(ul, • •, up) has the same property.
Next we show:
Third. Suppose the function U = Vkl2 e L^D') for some D' c c: G and
some t > 1. Then id = UT e H\{D) for each D c a D' and

THE DE GIORGI-NASH RESULTS 7
(2.9) f \Vw\Hx <: CT*a~* f wUxiiD <= D'a, a > 0,
where C = C(ra, M, K, Mi, k, v).
If it were possible to substitute
(2.10) i = v*U*r-Zpy) v e Lipc(D'),
in equations (2.6), the Schwarz inequality and (2.7) would yield
(2.11) f 7?2C/2T-2|vC7|2dx S C3 f (v2 + \Vii\*)U*dx,
JD' JD'
where Cz is independent of t. But this implies (2.9). Unfortunately, the
£'s in (2.10) are not known to e H\k0(D'). So, for each L, we define UL as
the "sawed-ofF" function: Ul(x) = U(x) if U(x) < L and Ul(x) = L if
U(x) ^ L\ then we define £ by (2.10) with U replaced by Ul. These £ are
still not known to e Hl^D'). However, these £ are, for each L, in jET|0(Z)')
and also </r£ e H\Q(D') with 0V£ e L2(Df), where we define
(2.12) 0 = V^^bH\(D').
To see that such £ can be substituted in (2.6), we define
(2.13) A" =fPa, B=fz.
It follows easily from our second step that Aa and B e #}(D'), that
(2.14) A% = F*-V^ye + bapy + FV),
and hence that 0-Ma e #2(#') and 0"^^° e i2(^)- Then, using (2.14) and
a series of lemmas proved by the writer (to appear in the Proceedings of the
Madison Conference of June 1960), we conclude that if £ has compact support
in Df with £ e #|(D')> 0£ e H\(D'), and 0V£ g £2(£>'), then we may substitute
£ in (2.6) to obtain
f (Jta4-y -f ££,y)da; = f (£,^°v - £,y£)^
= J^ tttaA°y - £ty4%)cto = 0, A% = £,
since (1.6) holds. Making these substitutions leads to
(2.15) f V*U$-*[\\?U\* + (r - 1)|V«7L|2]&
Since the right side of (2.15) is bounded for all L, we may let L ~> oo to

8
CHARLES B. MORREY, Jr.
obtain (2.11) (in deriving (2.15), it is convenient to notice that VUl = 0
almost everywhere on the set where Ul(x) = L).
Fourth. /// is of class C^ with t ^ 2 and 0 < p < 1 (C00, analytic), then
the solution z is of class 0^(0°°, analytic) on each domain D <= <= G.
In order to prove this, it is sufficient to show that U is bounded on interior
domains D <= cz G. For then the py satisfy (2.6) and the coefficients
Vk~1aa^i etc., are all bounded. Then it follows from the writer's extension
[12] of the De Giorgi-Nash results that the py are Holder-continuous on such
domains. The higher differentiability follows from known results as
mentioned in the introduction.
In order to show that U is bounded, we modify Moser's procedure slightly.
We suppose that B0 = B(xq, 2R) a G, Bn = B(xo, Rn) where Rn =
jR(l + 2-"), and define
wn = Usn so that wn = wsn_u s = vj{v — 2).
Then, for each n, we apply the Sobolev lemma (see [12])
(2.16) |Jb «£_!«**J''* g Co jB [IV^^I2 + R-^wl^dx, C0 = G0(v),
with the result (2.9) with D = Bn, D' = Bn-i, a = 2~nR, which yields
(2.17) f [|V^„_X|2 + R^wl.^dx g 2ClS2"-2.4«i?-2 f wl_,dx,
Ci being the G of (2.9); note that R„ ^ R. If we let
Wn = v%dx,
(2.16) and (2.17) lead to the recurrence relation
(2.18) Wn Z KlKYWl-u K0 = 2ClS-2i?"2, Z, = 452.
From (2 18), we conclude that U is summable to any power on B(xq, R) and
that
\U(x)\2 S lim Wlnlsn = Ka0K{ [ U2dx, xeB(x0,R),
n->oo JB0
a = (1 - a-i)-i = „/2, j8 = v2/4.
3. Extension of the results of Buley to the case 1/2 < k < 1. We first
state an obvious theorem, which will aid in the interpretation of the results
of this seption, and a convenient definition.
Definition. If/ and z are such that/Pa and/* are summable over each
Dc eg and if z satisfies (1.6), we say that z is an extremal for the integral
I(z, G).

THE DE GIORGI-NASH RESULTS
9
Theorem 3.1. /// satisfies the conditions (1.11') for some k > 1/2 or if f
satisfies (1.11) and is convex in (p,z), and z* e H\k{G), there is a unique
extremal zfor the integral I(z, G) such that z e H\k(G) and z — z* e H\k0(G).
For then I(z, G) is a convex functional.
It is clear that (1.6) holds for all £e#2*0(6t). And if we apply the
difference quotient procedure, we arrive again at (2.5); but this time, it is
not immediately evident that the right side of (2.5) is bounded for all small
h, although the result of replacing Ah by its limit F*-1, Zh by pY and Ph by V
is bounded. The trouble is that Ah ^ 1 and Zh is not uniformly in L%(Dr).
So we consider a sequence of problems of the type described in the
introduction where the finiteness of the second integral guarantees that we may let
h -> 0 in the difference-quotient procedure. We then study what happens
as the second integral is allowed to be arbitrarily large.
We begin with some general remarks about such problems. The second
integral will be denoted by J(z, G), where
(3.1) J(z, G) = f F(x, z, Vz)dx
where we shall assume for simplicity that F satisfies (1.11) with k replaced
by m.
Theorem 3.2. Suppose f satisfies (1.11) or (1.12) with some k (^v\2 if
(1.12)). Let mr denote the larger of k and m and suppose that z* eH\m>(G)
and that J(z*, G) ^ L. Then there is a function zL e Hlm>(G) with zL — z* e
Hlm\o(G) which minimizes I(z, G) among all such z for which J(z, G) ^ L.
If zl is not an extremal for J, there is a unique number ft ^ 0 such that zl is an
extremal for the integral I(z, G) -f fjJ(z, G); so
(3-2) jG K ,„(/,. + nFpJ + £(/, + pFJVx = 0, I e #U(<?).
Proof. The first statement is obvious from the lower semi-continuity of
both integrals (see [9] )with respect to weak convergence. If J(zl, G) < L
and £ e Lipc(G), it is easily seen that J(zL + A£, G) < L for all sufficiently
small A; in this case, (3.2) holds with p = 0. If J(zL, G) = L and zL is not
an extremal, there is a £i such that
f (li.uFPa + ZiFJdx = 1, lieLipc(G).
It follows by fairly straightforward arguments that
f (l.afpa + ifz)dx = 0 whenever f (J >aFPa + £Fz)dx = 0,
ieUpc(G)9
so that a number /x exists. Since I(z, G) ^ I(zl, G) whenever J(z, G) ^ L,

10 CHARLES B. MORREY, Jr.
it follows easily that \i ^ 0. It is clear that p is unique if zl is not an
extremal for J.
Theorem 3.3. Assume the hypotheses and notation of Theorem 3.2 and also
that m > k, G is of class C1 and J has no extremal with z — z* e H\mQ(G) for
which J(z, G) > Kq. Then, if zk —? zo (is K -> oo through a sequence of
values, zo is a minimizing function for I(z, G) with z0 — z* e H\k(G). There is
a sequence of K -> oo such that Kfi(K) -> 0.
Proof. Suppose z* minimizes I(z, G) among all z e H\k{G) such that
z — z* e H\kQ(G). Then, from our hypotheses, it follows that z* is the strong
limit in H\k(G) of functions in H\m(G) and J is continuous with respect to
strong convergence. The first statement follows easily. To prove the
second, we define
cp(K) = I(zK, G).
Then, clearly, cp is nonincreasing. Next if K > K0, we have
cp(K + AK) S I(zK + AJi, G) where AK = J(zK + A£i, 0) - J(zK, G)
for all A near 0. Since AK/X -> 1 as A -> 0, we see that <p'(K) = —fi(K) a.e.
Hence fju(K) is summable for K ^ K\ > K0 and the result follows.
We now apply these results to extend Buley's results as indicated:
First. We suppose that f satisfies (1.11) with 1/2 < k < 1, we define
F = VJ2, V = 1 + z2 -f \p\2, we assume z* e H\(G) and is the unique
(Theorem 3.1) extremal for J with those boundary values, Kq = J(z*, G), and
G is of class C. Then, for each K, the functions pKy, UK — V][2, \jsK =
VK~k)'2, and *l>KpKy e H\{D) for each D cz c G with
(3.3) f (ilk + VkK~l)\VpK\2dx ^ 2Ca-2Kfi(K) + Ca~2 f V\dx,
Dcz D'a, D' c= c= G.
To prove this, we apply the difference quotient procedure to equation
(3.2) to obtain
(3.4) j^ {£ >z,,a + Ah(a"h%,e + Hzh + eJP][)] + Ua(1^. + chzh + fl)}dx = 0,'
I e Hl0(D'), fi = ii{K), zh = zKh, etc.
and the coefficients are given by their formulas in (2.2) with z replaced by
zk. This time, the An, are bounded and
Ahaf -> p-V, Ahbah -> Vk~lba, Ahch -> Vk~lc (a.e.),

THE DE GIORGI-NASH RESULTS
11
(3.5) AhefPl = j* /vv[x + they,■■■]dt^ F*~ V* in Lr(D'),
AkftPl-* V-T in Lr(D')> r = 2A/(2A - 1) > 2, q = 1/2.
Setting £ = tj2za and proceeding as before leads to
(3.6) ^ (,M + Ah)\Vzh\Hx Z Ca-* ^ 0* + AM + Ph)dx.
For each fixed K, the right side is bounded and tends to a limit and so we
may conclude as before that the pKy e H\{D) and zh —7 pKy in H\(D) and we
may let h -> 0 in (3.6) and sum on y to obtain (3.3) and the other conclusions,
remembering the definition of J.
Second. For a subsequence of K -> 00, Kfi(K) -> 0 and zk -> zo in
H\K{G), z0 being a minimizing function for I with z0 — z* e H\kQ(G), and on
each domain D c c (?, 0^ -* 0O> t7t -r *70, 0^^ -* ^p^ in #|(Z>),
Pky ~7 Poy in H\k{D), and (3.3) holds in the limit.
The first statements follow from Theorems 3.2 and 3.3. Then, since
Kfi(K) -> 0 and Vk -> ^o in L2k(G), we may let K -> 00 on the right in
(3.3). From (3.3), we conclude that the H\(D) norms of i{jk, UKi and *I>kPky
are uniformly bounded. Also
jD \VpK\*dx = Jd V^Vi»\VpK\^dx
Accordingly the results follow.
Third. Suppose z*eH\k(G) and f satisfies the hypotheses (1.11) with
1/2 < k < 1, G being any bounded domain. Then there is a minimizing
function for I(z, G) with z — z* e H\k0(G) which has the differentiability
properties stated in §2.
To prove this, we let {Gn} be an expanding sequence of domains of class C"
having union G and let zo be a minimizing function for I(z, G) with zo — z*
eHlk0(G). On each Gn, we approximate strongly in H\k(Gn) by functions
z'np g C^GJ, and for each n and p, we let znp = lim znpK as in the second part.
Each znp is minimizing for I(z, Gn) with znp — z'np g H\k0{Gn) and satisfies
the interior boundedness conditions. Thus a subsequence of znp —v zn in
H\k(Gn) where zn — z0g H\kfi(Gn) and zw is minimizing. If, for each n, we
let Zn = zn on Gn and z0 on G — Gn, then Zn — z* g H\k0(G) and each Zn is
minimizing. Thus a subsequence —7 z in H\k{G), z — z* g H\kfi(G) and z is
minimizing and the limiting bound (3.3) holds for z. The remainder of the
development in §2 now goes through except that this time \fs — F(i-*)/2>
iftAa, i/jB, Aa, B and 0"*£ e H\(D), so that the former £'s can be substituted in
(2.6) as before.

12 CHARLES B. MORREY, Jr.
4. Extension to the integrands/ satisfying (1.12) with k > v/2. The case
k = vj2 can be treated by first showing that a minimizing function in this
case satisfies a " Dirichlet growth" condition
9(r) ^ [cp(a) + Kav](rlayy ft > 0, 0 ^ r ^ a,
cp(r) = [ f V»'2dxY/V, B(x0, r) c G.
• LJB(x0,r) J
This is omitted here but will appear in the Madison Proceedings.
If one attempts to carry through the procedure of §2, one finds that the
equations (2.1) and (2.6) must be altered by replacing &£, ch, ba, and c by
Pffil, Phch, F^6a, and Vc, respectively, if the b% ch, ba, and c are to be bounded.
The argument in the proof of the second part would require that Vk+1 be
summable. In order to carry through the difference quotient procedure, we
must use the device of the preceding section and in order to handle the
limiting equations, we need the following lemma:
Lemma 4.1. Suppose w e L2(Bb) (Br = B(xQ, r)), w e H\(Br) for 0 < r <
b, H g Lv(Bb) and satisfies
(4.1) ( j Hvdx\ /V ^ Ci^, 0 ^ r S b, fi > 0y H(x) ^ 0.
Suppose w satisfies the condition
(4.2) f \Vw\*dx ^ C2r2 f WwUx + C^a~2 f wHx
JBr JBr + a JBr + a
0 < a S r, r + a S b, t > 1.
It is assumed that these conditions hold on any spheres B(xo, b) and B(xo, r -f a)
c: G. Then there is a constant C4, depending only on /x, v, Ci, C2, C3 and an
upper bound for a, such that
f \Vw\Ux S Cn*ar* f
w2dx, 0 < a S r, r -f a ^ b,
A = 2 + 4/x-i, B(x0, r + a) <= G.
Proof. Let us assume first that B(x0, R) <= G so that w e Hl[B(x0, R)]
and there is a constant C5 such that
From the Sobolev lemma used in (2.16), we conclude that
}l/s
I \Vw\Ux g C5(R - r)~2 f wHx, 0 ^ r < R.
JBr jBr
3mma used in (2.16), we conclude tha
f w^dx\1/S S O0(v) f (\Vw\* + r~*w*)dx (s = v\(v - 2))
(4.3)
g C0(C5 + 1)(jB - r)-a f w2cte, if RI2 ^ r < R.

THE DE GIORGI-NASH RESULTS 13
Now, we assume 0<a^r,r + a<b and apply (4.2) with a replaced by
a/2, the Holder inequality, and (4.1) and (4.3) with r replaced by r + a/2
and R by r -f a to obtain
J \Vw\2dx <; 4C3r2a-2 \ wUx
JBr JBr + al2
+ C2T2Ci(r + a/2)^0(C5 4- l)-4a~2 f wHx
JBr+a
^ 4r2a-2[C3 4- C2GiG0(r + a)^(C5 + 1)] f wUx
JBr + a
^ C5a~2 J wUx
JBr + a
provided
±T2C2CiCo(r + a)** £ 1/2 and C5 ^ 8C3T2 + 1.
Accordingly, if .
(4.4) r + a ^ a, 4t2C2C,iCW = 1/2, 5(s0, r + a) c 6?,
we see that
(4.5) f \Vw\2dx ^ Cer^a-* f w2ix, C6 = 8C3 + 1.
JBr JBr+a
Now, suppose merely that 0 < a ^ r, B(xo, r -f a) <= 6r. If 0 < a ^ a,
we can cover l?r by a finite number of spheres B(xt, a/2) such that each
B(xt, a) <= Br+a and no point in Br+a is in more than K(v) such spheres.
Then, from (4.5),
f |Vw|2dz g 2 f |Vw2|*c ^ 4C6r2a-22 f ™2^
Jfir i jB(Xi,a/2) i JBix^a)
a~2 f
(4.6)
^ 4,S:Cr6T2a-2 I w2dz.
If a ^ a, (4.6) holds with a replaced by a. From this, we conclude that
(4.7) f |Vw|2dz ^ 4Z06T2a2a-2a-2 f w2^
J^r JBr + a
where a is an upper bound for a. The lemma follows from (4.7) and (4.4).
We conclude with the theorem:
Theorem 4.1. Suppose f satisfies (1.12) with k > v/2, G is any bounded
domain and z* e H\k(G). Then there is a function z e H\k(G) with z — z* e
Hlk0(G) which minimizes I(z, G) among all such functions and for which the
interior differentiability results of §§2 and 3 hold.

14 CHARLES B. MORREY, Jr.
Proof. We shall assume first that G and z* are of class C1. We shall
prove the theorem in that case and derive certain bounds on interior domains
which depend only on the H\k bounds for z and on m, M, etc. Then the
argument in the proof of the third statement in §3 can be used to prove the
theorem in general.
Define
F(xyp) = 2-i(i + l)-iF*+i, V = 1 + |p|a,
and let H be the unique extremal for J with H — z*ei?2jfc+2,o» an(i ^
Ko = J(H, G). For K > Ko, let z# and jjl(K) have their previous significance;
we shall restrict ourselves to members K of a sequence such that Kfi(K) -> 0.
Applying the difference quotient procedure to zr as in §3, we obtain
l
{£4^1A<2M + Ah(afzh# + Ptb°hzh + PleY)]
D'
+ U,APKzh<a + Phchzh + P,Ji)}dx = o,
(4-8)
Aihafh(x) = I FpapJ_p(x) + tAp]dt,
/:
Am = F*|>(a;) + <Ap]<ft.
This time, for each finite K,
Axh — VkK in L„ Ah -> F|-* in L„
p = 1 + 4-1, a = 1 + 2(k - l)-i,
(4.9)
Pa -> Fk in Z*+i, zft -> pKy in L2*+2.
Setting £ = ij22a and proceeding as before, we obtain
(4.10)
( v*[nAu + Ah]\Vzh\Hx
<
C jD [(Mia + Ah)\^z\ + vzAhPhzl]dx.
For each finite K, VkK+l is summable, so that the right side of (4.10) is
uniformly bounded in h and we may let h -> 0 and sum on y to obtain
f vHpkV'k + Vf)\VpK\Hx £ Ca-*2(k + l)i^
JD'
+ C f VkK(\Vv\2 + VKV2)dx, D c= D'a,D' cz c (?.
If we set f/z = F^/2 and replace D by jBr = B(x0, r) and Z)' = Br+a <= c (??
we obtain (by using the proper 77),
f |V£^|2<fo g (7 f H\U\dx + (7a-2 f I7^cte (#| - VK),
JBr jBrra jBr + a

THE DE GIORGI-NASH RESULTS
15
since Kfju(K) -> 0. Since k > v/2 and I{zk, G) tends to its minimum, it
follows easily from the Holder inequality that the Hk satisfy (4.1)
uniformly in K. Hence, from Lemma 4.1 with t = 1, we conclude that
f \VUK\2dx ^ Ctd-2 f U\dxy
JBr jBr+a
0 < a ^ r, Br+a c G,
(f U%dxY'8 S C0(C4 + 1)«"2 f U\dx
using the Sobolev lemma (2.16).
Thus we may let K -> oo to conclude that U = F*/2, i// = F<*-i>/a,
0-14° and ^BeHKD) with 4« and B e H\(D) and 4>-lVA" e L2(D).
Moreover, if we set £ = ri2U2£~2pyy we see that $ and ifst, e H\(D) and 0V£ e
L%(D) so that we can conclude as in §2 that these £ may be substituted into
equations (2.6). If this is done and we let wL = UTL~1U, we see that wL
satisfies the condition
[ \VwL\2dx ^ C2r2 f Vw\dx + C3r2a-2 f w|<fo.
From Lemma 4.1 with V = H2> it follows that
(4.11) f \VwL\2dx ^ G^a~2 f w\dx = C^a~2 [ U2£~2U2dx.
JBr JB,+a jBr+a
If we assume w = UT e L2(Br+a), we conclude, by letting L -> oo in (4.11),
that weH\(Br) and the limiting inequality in (4.11) holds. Then the
remainder of the proof in §2 carries over with changes only in the constants.
Bibliography
1. S. Bernstein, Sur la nature analytique des solutions des Equations aux derivies
partielles du second ordre, Math. Ann. vol. 59 (1904) pp. 20-76.
2. J. W. Calkin, Functions of several variables and absolute continuity. I, Duke Math.
J. vol. 6 (1940) pp. 170-185.
3. E. De Giorgi, Sulla differenziabilitd e V analyticita delle estremali degli integrali
multipli regolari, Mem. Accad. Sci. Torino CI. Sci. Fis. Mat. Nat. S. 3a, vol. 3 Parte I
(1957) pp. 25-43.
4. K. O. Friedrichs, On the identity of weak and strong extensions of differential
operators. Trans. Amer. Math. Soc. vol. 55 (1944) pp. 132-151.
5. E. Hopf, Zum analytischen Charakter der Losungen regularer zweidimensionaler
Variationsprobleme, Math. Z. vol. 30 (1929) pp. 404-413.
6. L. Lichtenstein, Vber den analytischen Charakter der Losungen zweidimensionaler
Variationsprobleme, Bull. Acad. Sci. Cracovia, CI. Sci. Mat. Nat. (A) (1912) pp. 915-941.
7. C. B. Morrey, Jr., On the solutions of quasi-linear elliptic partial differential equations.
Trans. Amer. Math. Soc. vol. 43 (1938) pp. 120-166.

16
CHARLES B. MORREY, Jr.
8. , Functions of several variables and absolute continuity. II, Duke Math.
J. vol. 6 (1940) pp. 187-215.
9. , Multiple integral problems in the calculus of variations and related topics,
Univ. California Publ. Math. New Ser. vol. 1 (1943) pp. 1-130.
10. , Quasi-convexity and the lower-semicontinuity of multiple integrals, Pacific
J. Math. vol. 2 (1952) pp. 25-53.
11. , Second order elliptic systems of differential equations, Ann. of Math.
Studies no. 33, Princeton University Press, 1954, pp. 101-159.
12. , Second order elliptic equations in several variables and Holder continuity,
Math. Z. vol. 72 (1959) pp. 140-164.
13. J. Moser, A new proof of De Giorgi's theorem concerning the regularity problem for
elliptic differential equations, Comm. Pure Appl. Math. vol. 13 (1960) pp. 457-468.
14. J. Nash, Continuity of the solutions of parabolic and elliptic equations, Amer. J.
Math. vol. 80 (1958) pp. 931-954.
15. A. G. Sigalov, Regular double integrals of the calculus of variations in
non-parametric form, Dokl. Akad. Nauk SSSR (N.S.) vol. 73 (1950) pp. 891-894 (Russian).
University of California,
Berkeley, California

DIRICHLET'S PRINCIPLE IN THE CALCULUS
OF VARIATIONS
BY
JAMES SERRIN
This paper deals with the simplest kind of multiple integral variational
problem, that of minimizing an integral of the form
IW = f(x> u> ux)dx.
Here jB is a bounded open region in En, and u = u(x) = u(x\, • •, xn) is a
real-valued function defined in jB and taking on given values on the boundary
dR of R. The function / = f(x, u, p) is assumed to be continuous for all
values of its arguments and to satisfy the conditions
/ ^ 0, / convex in p.
We shall be concerned mainly with the principle that, in some fairly general
sense, an extremal should furnish an absolute minimum to the integral. In
the first part of the paper we discuss some properties of the integral I[u]
which are fundamental to the general problem. The second and third parts
treat the role of an extremal in the minimum problem, and include at the
same time some existence and differentiability theorems. Most of the proofs
are omitted, and will be published elsewhere.
1. The first problem is to define the integral for some reasonably large
class of functions, in order that the eventual minimum might be as strong
as possible. For functions u which are continuously differentiate there is
of course no ambiguity in the meaning of I[u], but once we go beyond this
class difficulties immediately arise.
There are two accepted methods for extending the meaning of I[u], The
more usual assumes that u is of class H, that is, has locally summable strong
derivatives. Then the extended integral is defined by
(x) Ih\u\ = f(x, u, ux)dx, ueH,
where the integration is in the sense of Lebesgue. The subscript H emphasizes
the nature of the extension; in fact, we shall henceforth reserve the letter I to
denote the integral defined only for functions ueC1. The integral IH is
17

18
JAMES SERRIN
certainly beyond criticism, but nevertheless it suffers from several
disadvantages. For example, it is not really elementary, it is less strong in the
case of surface area than Lebesgue's definition, and, finally, in certain
circumstances its computation may be extremely difficult if not constructively
impossible.
The other definition is related to ideas of Lebesgue and Weierstrass.
Thus we set
(2) Il[u] = Minimum limit I[um], um e C1, um -> u.
Various limiting processes may be used in (2). The most interesting is
probably the following: We suppose that each function um is defined in a
closed subregion Rm of Ry and require that
Rm -> R, \u — um\dx -> 0,
as m-> oo. With this limit process (local convergence in L1), it is clear
that II is defined for functions u which are locally summable.
Unfortunately, the integral (2) also has its faults—for example, in certain
circumstances it may not furnish as small a value as i#.
I think that the difficulties in connection with the integrals Ih and II are
inherent in the problem; in any case, it is worthwhile to consider each integral
on its own merits. To begin with, we have the following theorem which
serves to justify the definition of Il>
Theorem 1.1. Suppose that either
(a)/ = /(p), (b)/—ooa*|p|—oo, or(c)feC*.
Then the functional II is an extension of I. That is, for any function ueC1
we have Il[u] = I[u].
Proof. We consider here only case (a).
For a function <£(#), let <f>h = <f>h(x) denote an integral average of <f> over a
sphere of radius h about x. Also let S be a closed subregion of jB, and choose
h such that 0 < 2h < d(S> dR). Consider now a sequence {um} of
continuously differentiate functions which converges to u in the sense defined
above. Then for xe S and for any m such that 8 cz Rm and d(Sy dRm) ^ h,
we have
f(Uhx) S f{Umhx) + e = f(umXh) + € ^ f(umx)h + e
by Jensen's inequality, where e = e(m, h) -> 0 as m -> oo. Integrating over
S and interchanging orders of integration on the right-hand side leads easily
to
I[uh, S] g f(umx)dx + e Meas S,

THE CALCULUS OF VARIATIONS
19
whence letting m -> oo yields
(3) I[uh,S] S liminf/[^m].
Thus we obtain
I[u] = lim lim I[uh,S] £ liminf/[wm],
S-+R A-*0
and it follows that I[u] ^ Ii\u\. On the other hand, it is clear from the
definition of IL that Il\u\ ^ I[u]. Thus I[u] = Il[u], and II is an extension
of I. Case (b) is proved by a somewhat analogous argument, while (c) is
demonstrated by a simple (but non-trivial) modification of Tonelli's proof of
lower semicontinuity under uniform convergence [11].
Theorem 1.1 shows that II is a lower semicontinuous extension of I
provided / satisfies certain mild regularity conditions. A similar result is
true for the integral Z#, namely:
Theorem 1.2. Suppose that any one of the conditions (a), (6), or (c) of
Theorem 1.1 holds. Then for each function ue H we have
Ih[u] ^ lim inf IH\um], um eH, um-> u.
Proof. As above we treat only the case (a). By the reasoning which led
to (3), we derive
(3') I[uhjS] g liminf/^m].
Then by Fatou's lemma,
/i/M S lim I[uh, S] ^ lim m{IH[um],
so that Ih is indeed lower semicontinuous.
By similar methods it is easy to show that iif = f(p)i then for u e H
IH[u] = IL[u] = lim I[uh, S].
This relation also holds in more general circumstances, but because of the
limited space I shall not dwell upon this. Finally, we note the following
analogue of Tonelli's celebrated theorem [10] on surface area.
Theorem 2. If f = f(\p\) =£ const arid Il[u] < oo, then u has partial
derivatives almost everywhere and
jRf(\ux\)dx ^ IL\ul
equality holding if and only if ue H.
(The conclusion of this theorem is to be understood in the sense that there
exists some function in the equivalence class of u which has partial derivatives
almost everywhere; moreover, the symbol ux denotes the partial derivatives

20
JAMES SERRIN
of any such equivalent function. Of course if u is continuous, as is the case in
Tonelli's theorem, then it is unnecessary to introduce equivalent functions.)
2. Turning now to the variational problem, we assume that / = f(x, p)
and that fv exists and is continuous; the conditions of Theorem 1 are,
however, not required. A function u defined in jR will be called admissible
if it continuously takes on assigned (continuous) values on dR. By an
extremal we mean a function of class C1 in jR satisfying the integral form of
the Euler-Lagrange equation.
Theorem 3. Let uo be an extremal taking on the given boundary values.
Then
I[u0] ^ Ih[u], I[u0] ^ IL[u]
for each admissible function u in H or in L1, respectively.
This result is a generalization of Theorem 1 of [9], and is proved by the
same method. Since the hypotheses of Theorem 3 do not restrict the
derivatives of Uo near the boundary of jB, it is evident that I[uo] might be
infinite; indeed from Theorem 3, I[uo] is finite if and only if there is an
admissible function for which either IH or IL is finite. Furthermore, from
Theorem 3 we infer that if the Euler-Lagrange equation of a variational
problem has an admissible solution, then there exists a minimizing function.
Recent advances in the theory of quasi-linear partial differential equations
[1; 3; 5; etc.] thus furnish one with a variety of existence theorems in the
calculus of variations. As one example, if the integrand f(x, y, p, q) of a
two-dimensional problem satisfies the conditions
JVV + JQQ
f2 + f2
(fw + /qq)2
^ K(l + ^2 + ?2)>
where k, K are positive constants, then the variational problem is well set.
The methods of Theorem 3 also yield differentiability theorems for problems
whose Euler-Lagrange equation is solvable (see [9, §4]).
3. There is some interest in extending Theorem 3 to apply when the
assigned boundary data are discontinuous. If the boundary of jR is
sufficiently smooth (class A2+x) we may consider summable boundary values
g(x) with the boundary condition
(4) lim \g(x) - u(x + nt)\ds = 0,
t->0 JR
where n is the inner unit normal to dR. The following result then holds.
Theorem 4. // the Euler-Lagrange equation in its differentiated form is

THE CALCULUS OF VARIATIONS
21
linear and uniformly elliptic, then the conclusion of Theorem 3 holds for the
boundary condition (4).
The proof makes use of smoothing, lower semicontinuity, and the Poisson
integral representation for solutions of linear elliptic equations. Brelot [2]
and de Vito [4] have proved similar theorems for the special case of the
Dirichlet integral.
When the Euler-Lagrange equation is non-linear, Theorem 4 does not
apply. In this case an analogue of Theorem 3 is known only when the points
of discontinuity of the boundary values are more or less sparsely distributed
on dR. This result will be omitted since it is rather a technical matter.
The final theorem, though of a somewhat different sort, is of interest in its
own right and is proved by the same means.
Theorem 5. Suppose that f = f(x, p) satisfies
2 Pifp, ^ *H°, IM ^ k\p\°-i
i = l
where k, K are positive constants, and a satisfies 1 < a ^ n. Let u be an
extremal with an isolated singularity at the origin, and suppose that for some
8 > 0
r0(r(°-")/<°--i>+5), a < n,
(5) u =
[0(|log rl1"6), a = n, as r -> 0.
Then u is bounded in the neighborhood of the origin.
The case a = 2 corresponds to quadratic variational problems, while the
limiting case a -> 1 is roughly equivalent to Finn's maximum principle [6]
for equations of divergence structure. Theorem 5 is proved by means of
differential inequalities as in [7]. In conclusion, we note that condition (5)
is best possible, as shown by the example/ = \p\a, u = r(«-»)/(«-D if a < n,
u — log r if a = n.
References
1. L. Bers and L. Nirenberg, On linear and nonlinear boundary value problems in the
plane, Convegno Internazionale sulle Equazioni alle Derivate Parziali, 1954, pp. 141-
167.
2. M. Brelot, $tude et extensions du principe de Dirichlett Ann. Inst. Fourier Grenoble,
vol. 5 (1955) pp. 371-419.
3. E. De Giorgi, Sulla differ-enziabilitd e Vanalyticitd delle extremali degli integrali
multipli regolari, Mem. Accad. Sci. Torino CI. Sci. Fis. Mat. Nat. S. 3a, vol. 3 (1957) pp.
25-43.
4. L. De Vito, Sulle funzioni ad integrale di Dirichlet finito, Ann. Scuola Norm. Sup.
Pisa vol. 12 (1958) pp. 55-127.
5. R. Finn, On equations of minimal surface type, Ann. of Math. vol. 60 (1952) pp.
397-416.

22
JAMES SERRIN
6. R. Finn, Isolated singularities of solutions of non-linear partial differential equations\
Trans. Amer. Math. Soc. vol. 75 (1953) pp. 385-404.
7. D. Gilbarg and J. Serrin, On isolated singularities of solutions of second order elliptic
differential equations, J. Analyse Math. vol. 4 (1956) pp. 309-340.
8. C. B. Morrey, Multiple integral problems in the calculus of variations and related
topics, Univ. California Publ. Math. New Ser. vol. 1 (1943) pp. 1-130.
9. J. Serrin, On a fundamental theorem of the calculus of variations, Acta Math. vol.
102 (1959) pp. 1-22.
10. L. Tonelli, ouila quadratura delta superficie, Rend. Accad. Naz. Lincei vol. 3
(1926) pp. 357-363, 445-450, 654-658.
11. , Sur la semi-continuite des integrales doubles du calcul des variations. Acta
Math. vol. 53 (1929) pp. 325-346.
University of Minnesota,
Minneapolis, Minnesota

ASSOCIATED SPACES, INTERPOLATION THEOREMS
AND THE REGULARITY OF SOLUTIONS OF
DIFFERENTIAL PROBLEMS1
BY
N. ARONSZAJN
Foreword. The paper is arranged to suit an impatient reader, interested
in partial differential equations.
The more patient, systematic reader could start with §§2, 3 and 4 before
reading §1. In §§2 and 3 we review (without proofs) the abstract background
concerning interpolation of norms and interpolation theorems (§2) and the
theory of pairings of vector spaces, of associated norms and associated spaces
(§3). In §4 we review briefly a few facts in the theory of Bessel potentials,
needed in §1 and which were not published as yet.
1. Regularity of solutions of differential problems. We consider a
differential system {A; Bi} where A is a linear differential operator in a domain
D, and {Bi} a system of boundary operators on 3D.2 We state the
corresponding boundary value problem as follows: For a given function / defined
in D find a function u satisfying
(1.1) Au =f in D, Btu = 0 on 3D.
We denote by {^4*; Bf} the corresponding adjoint system, and introduce
the spaces V and W of functions in G^D) satisfying the systems of boundary
conditions B{v = 0 and Bfw = 0 respectively. We then form the pairing
[V, W, jd Avwdx].3 We interpret the boundary value problem as follows.
The function / defines an antilinear functional on W:
(1.2) F(w).= f fwdx.
The solution u is then considered as a realization of the functional F in an
associated space V relative to a couple of associated norms,4 ||v|| and \\w\\
for the above pairing. (V is the completion of V with respect to ||i?||.)
1 Paper written under contract Nonr 85304 with Office of Naval Research.
2 The coefficients of A and Bi and the boundary 3D will be assumed of class C-1*1
where r is the highest degree of operators. In many instances, however, much less
regularity will be needed.
3 The general notion of "pairing" will be considered in §3.
4 The general notions of associated norms and associated spaces are considered in §3.
23

24
N. ARONSZAJN
If the norms are reflexive, in particular, quadratic, the realization is
obtainable if and only if F(w) is bounded relative to \\w\\. The weaker the
norm \\w\\y the smaller the associated space V, and hence the more regular is
the solution u. The functional F(w) need not be given in the form (1.2);
for instance, it may be an integral over the boundary where w and its
derivatives appear linearly. Thus, boundary value problems in a form
different from (1.1) can be treated.
From this point of view, it is important 1 ° to know as many couples of
associated norms as possible, and 2° to interpret the regularity properties of
the elements of the corresponding associated spaces in terms of usual notions,
such as the fact that these elements are functions defined in D, continuous
functions, differentiate functions, etc.
If the differential problem is well posed (i.e., A(V) and A*(W) are dense in
L2(D)) two couples of such associated norms are readily available. These
are {j|v||o, ||-4*w||o} and {||.4v||o, \\w\\o}9 where || ||0 is the L2-norm in D.
These norms are quadratic and by a quadratic interpolation (see §2) we get a
continuous scale of interpolated associated couples of norms {||v||W, ||w||M},
0 S t ^ 1.
Another class of associated norms is formed by the couples {|H|o,p,
M*H|o,p'} and also {ll^llo.p, IMIo.p'} where 1 ^ p ^ oo, l/p + Ijp' = 1,
|| || o,j> denoting the norm in L?>(D).
The interpretation of regularity properties of the corresponding associated
spaces for general differential problems presents essential difficulties.
However, for operators with constant coefficients, some information can be
obtained by using ideas and results of Hormander [5] and Treves [8].
Consider now an elliptic boundary value problem, i.e., a problem where the
two quadratic forms Jz> |^|2d# and Jd |^4*w|2d# are conditionally coercive
relative to the boundary conditions, Btv — 0 or B*w = 0 respectively.5
This implies that A is elliptic; we will assume in addition that A is of even
order 2m.
If such a problem is well posed then, by using the extension theorem and
the compensation method of the theory of Bessel potentials 6 and by applying
Lions' interpolation theorem (see §2) we prove that the interpolated norms
||w||(<) and \\w\\(0 are equivalent to the standard norms in D of order 2mt and
2ra(l — t) respectively. The corresponding associated spaces V^ and W^
are therefore spaces of Bessel potentials of order 2mt and 2ra(l — t) in D
satisfying the stable boundary conditions contained in the system B& = 0
or Bfw — 0, respectively.
The continuity and differentiability properties of functions in F<*), in D,
and on 3D are known (see [3]). The solution of the boundary value problem
5 Criteria for coerciveness are given in Aronszajn [2], Schechter [7], and Agmon [1].
6 To be published in Bessel potentials. Part II by N. Aronszajn and K. T. Smith.
A brief explanation is given in §4 of the present paper.

SOLUTIONS OF DIFFERENTIAL PROBLEMS 25
in form (1.1)—if it lies in VM—satisfies the stable boundary conditions in an
ordinary sense and the unstable ones in a generalized sense.
The associated space V corresponding to the norm ||^4v|| o,p is now the class
of potentials of order 2m of L* functions satisfying the boundary conditions
Btv = 0.
The results obtained by the preceding considerations can be considerably
strengthened by using the local character of the regularity for solutions of an
elliptic problem. It should be noted, however, that the localization principle
depends on the local regularity of A, Bi, and dD.
Remark 1. Our results overlap partly with the results obtained recently
by several authors. The theory of associated norms and of interpolation
would lead to much more complete results if the extension theorem and
compensation method were available for Bessel potentials of 2> functions;
interpolation theorems which seem to be adequate for our purposes have
already been obtained by E. Gagliardo [4] and by J. Lions.
Remark 2. We could treat the case of a system of equations for systems
of functions. However, for over-determined systems, the definition of the
space W and also the notion of a well posed problem should be suitably
changed.
Remark 3. The results could be extended to domains of polyhedral type
(for which 3D may present vertices, edges, etc.). The only item lacking at
present for such an extension is an algebraic criterion for coerciveness of
quadratic forms in such domains.
2. Quadratic interpolation. Let V be a complex vector space and ||v||
and || w ||i two norms defined on V. We say that the two norms are
compatible if for every sequence {vn} c V which is Cauchy in both norms,
lim ||vn|| = 0 if and only if lim ||vn||i = 0.
Let ||v|| be a quadratic norm on V (i.e., ||?;||2 is a quadratic hermitian form
on V). Denote by V the completion of V relative to ||v|| (V is then a Hilbert
space).
Theorem 1. The class of all quadratic norms \\v\\\ on V compatible with
\\v\\ is in one to one correspondence with the class of all positive definite self-
adjoint operators H on V satisfying 1° V <= 3)(H) = domain of H; 2° V is
dense in S)(fl) in the graph-norm (i.e., in the norm \\v\\2H = \\v\\2 -f ||#?;||2).
If H corresponds to \\v\\, then \\v\\i = \\Hv\\.
Consider now two compatible quadratic norms ||#||(0) and |H|(1) on V.
We construct the interpolated norms \\v\\W on V, 0 S t S 1, as follows: let
F<°) be the completion of V relative to |H|(0) and H the positive definite
self-adjoint operator in F(0) corresponding to ||v||i by Theorem 1. We
define then
(2.1) ||i>||<«> = ||#«0||<°>, O^^l.

26
N. ARONSZAJN
This construction is called quadratic interpolation. If we interchange
||?;|| <°> and ||?;||{l) the interpolated norms are the same except that t is changed
into 1 — t. The interpolated norms are all mutually compatible.
We can now state the interpolation theorem due to J. L. Lions.
Theorem 2 {Quadratic interpolation theorem). Let V and W be two vector
spaces. On each of them consider two compatible quadratic norms ||v||*0), ||^||(1)
and ||w||(0) and ||wj|(1) respectively. Let T be a linear mapping of V into W
which is bounded relative to the norms ||^||(0), ||^||(0) as well as the norms ||v||(1),
||w||(1) with respective bounds Mq and M\. Then T is bounded also relative
to the interpolated norms \v\{i\ \w\{t) with bound ^M\~tM\.
Remark. The quadratic interpolation of norms was introduced
systematically in connection with the theory of Bessel potentials and their
application to differential problems (see §1). The interpolation theorem was
communicated to the author by J. L. Lions at the end of 1958. Soon
afterwards, the author found a different proof of this theorem giving the bound
Jfo~^i- Since then, E. Gagliardo [4] found a whole class of interpolation
methods for general norms, each of these methods leading to a corresponding
interpolation theorem. Also several other interpolation theorems were
communicated to the author by J. L. Lions.
3. Pairings, associated norms, associated spaces. We consider a complex
vector space F. A norm ||i?|| defined on V is called reflexive if the completion
of V relative to it is a reflexive space. In particular every quadratic norm
is reflexive.7
Consider a system [T, W, (v, w}] composed of two vector spaces V and W
and a bilinear hermitian form <(v, w}.s Such a system is called a pairing of
V and W. The system [W, V, (w, v)*] with (w, v}* = (v, w} is the adjoint
pairing of W and V. A pairing is called proper if for every v # 0 the anti-
linear functional \V, w) is not identically 0, and for every w ^ 0 the linear
functional (v, w} is not identically 0. Unless otherwise stated we consider
only proper pairings.9
A norm ||v|| on V is admissible for the pairing if every linear functional
(v, w) is bounded in this norm ; a similar definition is given for an admissible
norm on W. For an admissible norm \\v\\ on V we form the conjugate norm
on W:
(3.1) ||^||c = sup \(v, wy\l\\v\\.
7 It is advantageous to extend the theory more generally to pseudo-norms, but for the
sake of brevity we do not consider this in the present paper.
8 The term "bilinear hermitian" means linear in the left-hand vector and anti-
linear (or conjugate linear) in the right-hand.
9 The use of improper pairings is connected with the consideration of pseudo-norms.

SOLUTIONS OF DIFFERENTIAL PROBLEMS 27
For an admissible norm given on W we define the conjugate norm on F
similarly.
The operation of conjugation has the following properties which are easily
obtained:
1. The conjugate norm is admissible.
2. Starting with a norm on V and repeating the operation twice we obtain a
norm \v\cc S \\v\\ (and similarly when starting with a norm on W).
3. \\w\\c — \\w\\ccc and consequently, after the second repetition the operation
will give nothing new.
4. If the norm \\v\\ is reflexive then so is the norm \\w\\c.
5. // ||w|| is quadratic then so also is \\w\\c.
Two admissible norms ||v|| and \\w\\ on F and W are called associated if one
is the conjugate of the other. It follows from the above properties that for
any admissible norm, ||v||, the two norms ||?;||cc and \\w\\c are associated.
\\v\\cc is the largest of all norms \\v\\' belonging to an associated couple
{IIv II' > IIw II'} and satisfying || v ||' ^ || v || for all v e V. For a couple of associated
norms {||v||, ||w||} we consider the completions of F and W with respect to
these norms. They form Banach spaces V and W. The bilinear form
(vy w} has a unique continuous extension to V, W so that we obtain a pairing
[F, W, <vy w}] for which the norms of F and W are associated. F, W are
called associated spaces corresponding to the original pairing and the couple of
associated norms.
Remark 1. The study of associated spaces was begun a number of years
ago in connection with the extension of the notion of reproducing kernels to
Banach spaces. Nothing has been published on the subject except a short
communication to the American Mathematical Society in 1951.10
Since the publication of Nirenberg's results in 1955 [6] on regularity of
solutions of elliptic equations, it became apparent that the notion of
associated spaces gives an interesting interpretation of these results, and others
which extend the results of Nirenberg. The notion also gives rise to new
results in this complex of problems, and the significance of the new results
was clarified with the appearance of J. L. Lions' interpolation theorem (see
§2).
In general, we use the abstract completion of normed spaces. However,
we will consider the elements of F as anti-linear functionals on W and hence
we can identify elements of different completions (relative to different
norms) as elements of the algebraic anti-dual of W.11 We proceed similarly
with completions of W considered as subspaces of the algebraic dual of F.
This identification allows a comparison between associated spaces relative
to different associated norms.
10 Communications AMS, Bull. Amer. Math. Soc, Abstracts Annual Meeting, 1951.
11 This is the vector space of all anti-linear functionals on W.

28
N. ARONSZAJN
Remark 2. If V and W are functional spaces (on some basic sets) relative
to the associated norms, it is important to consider the functional
completions of V and W, and in particular the perfect ones, whenever they exist.
In this way the elements of the completions can be identified as functions,
which is of interest in differential problems.
Theorem 1. // {||v||, ||w>||} and {\\v\\i, \\w\\i} are two couples of associated
norms corresponding to the pairing [V, W, (v, w}], then \\v\\, \\v\\i and \\w\\,
\\w\\i are compatible norms for V and W respectively.
For two norms ||v||i and ||v||2 on V we say that the first is weaker than the
second (or the second stronger than the first), || ||i -< || ||2 (or || ||2 >- || ||i)
if there exists a positive constant M such that ||v||i ^ JbT||v|| 2 for all v e V.
If || ||i -< || || 2 and || || 2 -< I! ||i, the two norm are equivalent and we write
|| || 1 ~ || || 2. We have the following basic theorem.
Theorem 2. For two couples of associated norms {\\v\\i, \\w\\i} and {IHI2,
||w||2} which correspond to the pairing [V, W, (v, w)>], the following properties
are all equivalent:
1. ||v||i < |H|2, 2. \\w\\i > \\w\\2, 3. V1 => V\ 4. Jpi cz W*.
If B is a complex normed space and B* its anti-conjugate, i.e., the space
of all the anti-linear continuous functionals on B, the two form a canonical
pairing [B*, B, <&*, 6>] where <&*, 6> = b*(b); the norm of B and the
conjugate norm on B* form a couple of associated norms. If Jf? is a Hilbert
space with scalar product (hi, A2) it gives rise to a canonical self-pairing
[AT, AT, (hl9 h2)].
We say that a couple of linear transformations S and T form an
isomorphism {£, T] of the pairing [V, W, (v, w}] into (or onto) [Vf, Wf, (v'9
w'}'] if S and T transform V and W into (or onto) V and W respectively
with preservation of the scalar product, (v, w} = (Svy Tw}f. For proper
pairings, S and T must be one-to-one.
For a pairing [V, W, (v, w}] with a couple of associated norms {||w||, ||^||},
there exists a canonical isomorphism {$, T} of this pairing into the canonical
pairing [W*, W, w*(w)], where W is taken with the norm \\w\\; T is the
identity and 8 is an isometric linear mapping with respect to the norm ||w||
on V and the conjugate norm on W*. If the norms are reflexive then S(V)
is dense in W*. If the norms are quadratic and V and W are complete
relative to them (i.e. they are Hilbert spaces), then there exists a canonical
isomorphism {8, T) of [F, W, <v, w}] onto the self-pairing [Wy Wy (wiy W2)]
with T = / and S an isometric isomorphism of V onto W.
The following theorem is immediate,
Theorem 3. Let {S, T} be an isomorphism of[Vy Wy (v, w}] into [V, W,
(vfy w'y\ and let \v\' and \w\' be associated norms for the second pairing such
that 8(V) and T(W) are dense in V and W respectively. Then the norms
||$?; ||' and \Tw\' are associated for the first pairing.

SOLUTIONS OF DIFFERENTIAL PROBLEMS 29
Remark 3. The statement in §1 that the couples of norms {||v||o, ||-4*w||o}
and {||^v||o, \\w\\o} are associated follows immediately from Theorem 3 and
the assumption that the problem is well posed. The statement with 2>-
norms replacing L2-norms follows similarly, but requires the additional proof
of the fact that if A(V) and A*(W) are dense in L2(D), they are dense also
in Lp(D), 1 ^ p < oo.12
Theorem 4. Consider a pairing [V, W, <v, w}] with quadratic associated
norms \\v\\ and \\w\\. Let [V, W (v, w}] be the corresponding associated pairing
and {S, 1} its canonical isometric isomorphism on [W, ffl, (w±9 w^)]. If \v\\
and ||w;||i are quadratic norms compatible with ||v|| and \w\ respectively and if
H and G are the corresponding positive definite self-adjoint operators on V and
W,ld then ||v||i and \\w\\i are associated if and only if G = SH-^-S"1.
As a consequence of Theorem 4, and by using the interpolated norms
introduced in §2, we obtain
Theorem 5. Let (H|<°>, |H(0)} and (IN(1)> IMI(1)} be two coupl™ of
associated quadratic norms for [V, W, (v, w}]. Then the interpolated norms
||v||W, ||^||(f)> 0 ^ t ^ 1, are also associated.
4. Some facts about Bessel potentials. The aim of this section is to state
and explain briefly some theorems and methods connected with the theory
of Bessel potentials, which were used in §1, especially those which were not
published as yet.14
We recall (see [3]) the two definitions of the class Pa of Bessel potentials
of order a in Rn. The one is that u e Pa if u is the convolution Ga*f for
some/g L2(Rn), where
(4-1) G°W = 2<^-w/1Vi/.rW2) *<-.>/2(M)M<°-»>/2, a > o,
Kv being the modified Bessel function of third kind.
The second definition is that Pa is the perfect functional completion of
C™ relative to the norm \u\a. This norm in terms of the Fourier transform
u of u is
(4-2) hi; = f (i + |£|W(£)|2e
JRn
This expression shows that quadratic interpolation between \\u\\a and \\u\\p
gives \\u\\Y, where y is the interpolated order a(l — t) + fit. The norm \u\a
12 This fact can be proved assuming only boundedness of coefficients of A, but much
stronger regularity of 3D and B^ is required.
13 See Theorem 1, §2.
14 Some of these theorems will be included in Bessel potentials. Part II by N.
Aronszajn and K. T. Smith.

30
N. ARONSZAJN
is most convenient for the study of potentials in Rn, but for domains D c Rn
we introduce the standard norm \\u\\atD, at ^rst ^or u G C°°(D),
w = 0 |i|=»i J^ |i|=a*
(4*3) HliU ^ I I |A^I2^ + y d*-*ADiV),
:|a,D
where a* is the largest integer <a and d$,D{v), 0 ^ j8 g 1, is the Dirichlet
integral of order j8 in i>, which for /? = 1 is the usual Dirichlet integral, for
j8 = 0 is J/) |v|2efo* and for 0 < j8 < 1 is
1 f f N*) - v{y)\%
C(n,P)JDJD \x - y\n+2f>
<*■*> *,Mv) = 7^ 77 ~2? dxdy,
with a constant C(n, f3) depending on n and /J.
For D = i?\ w 6 Co', ||w||a)jR« is equivalent to \u\a.
The perfect functional completion of the class of u e C°°(D) for which
||w||a,z> < oo, with respect to this norm, is the class Pa(D) of potentials of
order a in D. Each u e Pa(D) is locally in Pa, its derivatives of orders ^a
exist in the ordinary sense except on sets of corresponding capacity 0 and the
expression (4.3) gives its standard norm of order a.
Extension theorem. For each y > 0 and under suitable regularity
assumptions on D there exists a linear mapping u —> u, transforming functions
u defined in D into functions u defined in Rn such that u is an extension of u.
For a <; y this mapping transforms Pa(D) into Pa continuously.
The largest class of domains D for which this theorem can be proved at
present are domains of polyhedral type and class O*'1.15'15' The theorem
shows that Pa(D) is the class of restrictions of Pa to D and that ||^||a,D is
equivalent to the natural norm of restrictions in the functional space Pa,
namely, the inf jbj|a,#n for all v e Pa having the restriction u in D.
It is clear how we must define the local class Pfoc(^w) for any differen-
tiable manifold ^#n of class C"*'1. If the manifold is compact, Pfoc(^w) is
already a functional space Pa{J(n) with a norm defined by ||w||ivtf" =
2& W^ki^k1^ u(Tklx)\\%Rn where <f>k is a partition of unity corresponding
to a finite covering \°Uk) of *J(n and Tk are the homeomorphisms of class
C0*'1 transforming %k into Rn. All norms obtained in this way are
equivalent.16 Thus for a bounded domain D of class C™"1 the classes
Pa(dD) are defined for all a g m + L17
15 I.e., such that dD is a topological manifold (may be disconnected) of dimension
n — 1 and that for each point of dD there exists a neighborhood °tt and a O*'1
homeomorphism of % into Rn transforming % n D onto a geometric polyhedron.
For unbounded D a suitable restriction of the behavior of D at oo should be included.
15' Added in proof. Quite recently, A. P. Calderon proved the extension theorem for
a large class of domains (lipschitzian graph-domains) and for integral orders.
16 For non-compact manifolds we define Pa(^J/!n) only when ^tfn is a Riemannian
manifold and the definition is much more involved.
17 For CW'L domains of polyhedral type the classes Pa(dD) can still be defined but in

SOLUTIONS OF DIFFERENTIAL PROBLEMS 31
General restriction theorem. Let u e Pa(D) in a O**1 domain D.
Then the normal derivatives dku\dvk exist on 3D in ordinary sense for 0 ^ k <
a - 1/2, dku\dvk e P°-k-V*(dD) and
T II °V \\a-k-l/2tdD
with a constant C independent of u.
Converse to the restriction theorem. Let {vjc} be a system of functions
on dD, 0 ^ k < a — 1/2, vjc e Pa~*_1/2(^D). There exists a linear mapping
{v/c} ->ue Pa(D) such that Vk = dku\dvk. In addition
\\u\\%d ^ Ci 2 \\vk\\l-k-i/2tenfor all /3 ^ a,
k
the constant C\ depending only on D, a, and n.
The last theorem contains actually more than the exact converse of the
restriction theorem. To give it its full strength we should define the norms
||vjfc||/3-*-i/2 when j8 — k — 1/2 is negative. This is achieved by first
considering functions w defined in Rn. If for t > 0, the potential Grw exists and
is L2, we put ||w||-T = J \GTw\2dx.ls We then define ||t;||_TaD for v on dD
in the same way as ||w||Ttaz) was defined, by using a partition of unity on dD.
We can now describe the compensation method. Let {Bi} be a system of
linear differential boundary operators of orders ^m and let y > m -f 1/2.
The method consists in finding a linear projection mapping T of Py(D) onto
the subspace of P*(D) where the boundary conditions BiU = 0 are satisfied.
We have T2 = T, but T is not necessarily the orthogonal projection in
Py(D). We require further that if mi is the order of Bi then the compensating
function u — Tu satisfy the following inequalities
||« - Tu\\lD S C J ||^||I_mi_1/2,aB, for j8 S y,
i
with a constant C independent of u and /?.
Such a projection T exists if the coefficients of the S^'s are sufficiently
regular.
Remark. The compensation method was introduced by K. T. Smith and
the author in 1955 in their work on regularity at the boundary of solutions of
general elliptic boundary value problems. It was also used by the author
in the study of conditional coerciveness. The publication of these results was
delayed pending completion of the work on Bessel potentials. In the
meantime the compensation method in an ad hoc form was found independently
a more complicated way. All theorems which we state for Cm>^ domains remain valid
for such domains of polyhedral type.
18 It should be noted that the negative norm does not define a functional space.

32
N. ARONSZAJN
by different authors working on the above mentioned problems and used by
them with success.
Bibliography
1. S. Agmon, The coerciveness problem for integro-differential forms, J. Analyse Math,
vol. 6 (1958) pp. 183-223.
2. N. Aronszajn, On coercive integro-differential quadratic forms, Conference on Partial
Differential Equations, University of Kansas, Technical Report 14, 1954, pp. 94-106.
3. N. Aronszajn and K. T. Smith, Theory of Bessel potentials. Part I, University of
Kansas, Technical Report 22, 1959.
4. E. Gagliardo, Interpolation d'espaces de Banach et applications, Notes I, II, III,
C. R. Acad. Sci. Paris vol. 248 (1959) pp. 1912-1914, 3388-3390, 3517-3518.
5. L. Hormander, On the theory of general partial differential operators, Acta Math,
vol. 94 (1955) pp. 161-248.
6. L. Nirenberg, Remarks on strongly elliptic partial differential equations, Comm.
Pure Appl. Math. vol. 8 (1955) pp. 648-674.
7. M. Schechter, Coerciveness of linear partial differential operators for functions
satisfying zero Dirichlet-type boundary data, Comm. Pure Appl. Math. vol. 11 (1958) pp.
153-174.
8. F. Treves, Relations de domination entre operateurs differentiels, Acta Math. vol. 101
(1959) p. 1.
University op Kansas,
Lawrence, Kansas

LEBESGUE SPACES OF DIFFERENTIABLE FUNCTIONS
AND DISTRIBUTIONS
BY
A. P. CALDERON
In this paper we summarize the properties of functions in L* with
derivatives in Lv. Some of the results presented here are known (see
references), others are new.
We shall use systematically the notion of Bessel potential introduced by
Aronszajn, K. T. Smith and the author (see [1; 2]).
Let x — (xi, X2,- • •, xn) be a point in w-dimensional Euclidean space
En, a = (ai, <X2, • • •, an) an w-tuple of non-negative integers. We write
|*| = \1 *?]1/2> *" = «!«*• ' •<*, (Wa = (dldx1)ai(dldx2)a*' "(dldxnfny
x + y = (xi + yi, X2 + t/2,- —, Xn + yn), |a| = ai -f a2 + • • • -f an.
Constants will be denoted by C. One or more subindices appended to C will
indicate quantities on which it depends.
The Fourier transform of a function or a distribution / will be denoted by
/". We shall use the same notation for points in En and its dual, where the
Fourier transforms of functions on En are defined.
Bessel potentials. The Bessel potential of order z (z being any complex
number) of a temperate distribution /, denoted by Jzf is defined by
(Jzfy = (1 + 47r2|z|2)-2/2/~.
This is meaningful for all complex z. Clearly the operations Jz form an
additive group.
For R(z) > 0 the function (1 -f 47r2|^|2)-2/2 is the Fourier transform of an
integrable function Gz(x) where (see [1]),
(1) Gz(x) = y(z)e~M e-l*l*lt + -) dt, R(z) < n + 1,
where y(z)~i = (27r)<»-i>/2r(|)r
Theorem 1. For R(z) > 0 and 1 ^ p ^ oo the operation Jz transforms
Lv(En) continuously into itself, with norm S 1 for real z. Furthermore as a
function of z the operator Jz on Lv(En) is analytic (see [6, Chapter 3]) for
R(z) > 0.
33
Pf^1-

34
A. P. CALDERCN
Proof. If z is real and 0 < z < n -f 1, then Gz(x) is positive and, since
its Fourier transform is 1 at 0, it follows that Gz(x) is integrable and has
integral equal to 1. Hence Jzf = Gz */ transforms Lv(En) into Lv(En) with
norm ^ 1. If z is complex z = u + iv then clearly |6r2(#)| ^ OzGu{x). To
verify that Jz is an analytic function of z, it is enough to show that the
increment quotient of y~1Gz with respect to z converges to (dldz)y~1G in L1.
Now one sees readily from (1) that this increment quotient is dominated by a
multiple of Gu+€(x) + Gu-€(x), if the increment of z does not exceed e/2 in
absolute value. The argument just given is valid for 0 < R(z) < n -f 1
which is the range of z for which (1) holds. For other values of z the desired
result is immediately obtained from the fact that the Jz form a group.
Theorem 2. Let v be real and 1 < p < oo. Then Jiv transforms Lv(En)
continuously into itself with norm tkCp(\v\ -f \)n. More generally, let
R(z) ^ |a| ; then (djdx)aJz transforms Lv(En) continuously into itself.
Proof. According to the theorem of Mihlin [8] if a function cp(x) is such
that |&|v|(0/&r)v<p| ^ A, for 0 ^ \y\ ^ n, then the operation T defined by
(TfY = <pf~ is bounded on Lv(En), 1 < p < oo, and has norm ^AGP. In our
case we have <p(x) = (1 + 4tt2|^|2)-^/2 and <p(x) = (2rrix)a(l + 4tt2|x|2)-2/2
and an elementary calculation yields the desired result.
Theorem 3. If A = 21 (^/^)2 then (1 - A)J* = J*"2.
Proof. This is an immediate consequence of the definition of Jz.
Theorem 4. If 0 < u < n then
YxPGu(xi\
< Cttafle-l*l/2b|-»+«+M-l«l.
If I ^ p < co and Anf = f(x + h) — f(x) then if q = p/(p — 1)
\\&kGu(x)\\p S Cup\h\"-»t*,
provided n\q < u < n\q + 1 and u < n.
Proof. By differentiating (1) we obtain
/ d\a I , r°° / m(n-te-l)/2
Ux) x"G^xi = Gue~lxl 2 M!^r J0 eH*U(' + 2) m
where 0 S r + s ^ |a|. Now for |#| ;> 1 the integrals on the right are
bounded and thus the righthand side is 0(e~^x^2). For \x\ < 1 the integrals
are dominated by
C
1 +
Jo
e-\x\ttn-u-l+s^t
C\x\~*+
and the desired result follows.

DIFFERENTIABLE FUNCTIONS AND DISTRIBUTIONS 35
To establish the second inequality, in calculating ||AftCrt»||j> we split the
domain of integration in two parts 0 g \x\ ^ 2\h\ and \x\ ^ 2\h\ and
observe that according to the first part of our theorem the mean value
theorem gives
\&kGu(x)\ ^ C\h\\x\-"+»-\ |*| £ 2|A|;
on the other hand
\&kGu(x)\ ^ C(\x\~n+u + \x + a|-ii-hi), |a;| ^ 2|A|;
integrating these inequalities the desired result follows.
Definition. Let u be a real number and 1 ^ p ^ oo. We define
Lvu(En) to be the image of £*(#„) under Ju. IifeI4(En) then/ = Jug for
some g e Lp(En). This g is unique; we define the norm ||/||p u of / e L*(En)
by
ll/lk. = IsrlU-
Theorem 5. (a) jTAe spaces L* are isomorphic with IP(E^. (b) 7/
1 < £) < oo, Jz is an isomorphism between L*(En) and L%(En) where v = u +
R(z). (c) 7/ z is real this isomorphism is an isometry. (d) If u < v then
L*=> L> and for feL* we hive \\f\\PtV > \\f\\PtU. (e) // 1 < p < oo then
(d/fiXi) maps LJ continuously into L%_v (f) The spaces L* are complete.
Proof. Parts (a), (c) and (f) are evident consequences of the definition.
Let us prove (b) first. Suppose that z = a -f ir\ then Jz = J°Jir and
JZL* = JzJuLp = Ju+°JirD>. But, according to Theorem 2, Jir and J~iT
both transform Lp continuously into itself. Thus J^LF = Lp and JZL* =
ju+ojj, = Lp^o For part ^d) we have L? = jvjjp = jujv-ulp. but
according to Theorem 1, since v - u > 0, JV-ULP a Lv and thus Lp <= JM7y* =
LJ. Now if/g Zf? then/ = Jvg where g e LP and ||/||pt, = ||jr||p. But then
/ = Jujv-ug, and according to Theorem 1 Jv~ug e Lp and (| J*"^!* ^ ||?||p.
Therefore feL* and ||/||p>ti = ||J-VL ^ M, = II/L-- Finally
insider the operation J-u+1(dJdxi) = ((djdxfiJ^J-*. According to Theorem 2,
(djdxfiJ1 transforms 7> into itself continuously. Consequently J1(^/^a;<)J~u
transforms L£ continuously into Lv and thus d\dxi = Ju~1(J1~udj8xi)
transforms continuously L% into i£_i.
Theorem 6. Let p > I, u ^ v and l/q = l/p — (u — v)/ti > 0; £Aen
LJ c L? arid £Ae inclusion map is continuous. Let \ > u — n/p > 0;
£Aew every function in L% coincides almost everywhere with a function f in
Lip(w — n/p). Furthermore
|A«/| = \f(x + h)-f(z)\ Z G^H/11,.,1 A|«-»/»; I/I ^ C^H/II,,..
Proof. The case u = v in the first part is evident. Suppose that u — v
> 0; then Ju~v maps 7>» continuously into 7X In fact, according to
Theorem 4
|j«-»/| = |<y„_*/i <; <7(|x|-»+«-» * |/|)

36
A. P. CALDERCN
and the assertion follows from a well known theorem of Soboleff (see [11]).
But JU~VLP = IjP_v. Therefore L\_v <= Lq and the inclusion map is
continuous. Now applying Jv to both sides of this inclusion we obtain the
desired result. For the second part assume first that u < n; then we define
/ = Gu* J~uf. Since/ e L* then J~uf e LP and since, according to Theorem
4, \Gu(x)\ ^ Cue-M\x\-n+* then GueL^,q = pj(p - 1). Thus/is a
continuous function. Now an application of the second inequality in Theorem 4
gives
|AA/| g |(A»G.)*J-V| S ||A»G„||,||J-V||, ^ ClAI—'i/H,...
The case u > n reduces to the preceding one as follows. Since u < 1 -f njp
we can find v such that 0 < u — v < njp and 0 < v < n, and then, since
L*c: L« where Ijq = \jp - (u - v)jn, if / e Lvu then also f e L% and
\\f\\q,v ^ Cpuv\\f\\p,u, and the previous argument applied to/as an element of
L% gives the desired result.
Remark. The inclusion L% <= L% is still valid if p = 1 provided Ijq >
1 — (u — v)jn > 0. In fact, if these inequalities hold, then Gu-v e Lfl and
thus Ju~vf = Gu-v*f maps L1 continuously into IX Hence JU~VL1 =
L\_v c L9, whence L\ = JVL\_V a JvLq = L9.
Theorem 7. Let u be a positive integer and 1 < p < 00. jTAen f e Lhuif
and only iff has derivatives (in the sense of distributions) of orders ^u in L?.
There is a constant C = CPtU such that
c-wu.** 2 |(i)7|| son/I,...
o^M^u II \^x/ lb
Let u be a negative integer and 1 < p < 00. jTAen f e L% if and only if
f = 2og|a|^w (dld%)a9a, ga£ Lp and there exists a constant C = Cpu and a
choice of the functions ga such that
c-i\\f\\p,» ^ I Mp ^ cii/ii,...
Proof. Suppose u is a positive integer and / e Lg. Then / = Jug with
g e L*> and (dldx)af = (djdx)aJug which according to Theorem 2 also belongs
to L*> if |a| ^ ?*. Further
I (Ifil HI (£)H - <7--|g|' - Cj-4fh'u
since ||g||p = ||/||p,i*. Conversely suppose that (djdx)afe Lv for all a,
0 ^ |a| ^ u. Let r ^ 0 be an integer such that u -f r = 2s is even. Then
according to Theorem 2 (d/dx)aJrf e L* for |a| g w + r = 2*. Hence
J-uf=J-2sJrf=(l.A)sJrfeLP and || J-/||p g O. 2|.|« 1(^/^)^11,
^ ^u2|a|^u ||(Wa/L according to Theorem 1. But \\J-uf\\P = \\f\\p,u.
Suppose now that u is a negative integer and that / = 2|«!^-m (d/&c)a<7a
with gaeLp. Then/eL£ according to Theorem 5 (d). Conversely, if

DIFFERENTIABLE FUNCTIONS AND DISTRIBUTIONS 37
/ g LI, let r ^ 0 be an integer such that — u + r = 2s is even. Then
f = J»g = J-**Jrg = J~2sh where g e IP and h = Jrg e L*\ thus / =
(1 - A)*, and since for |a| ^ r, ||(a/ax)«% ^ Cpfr||A||p,r = Cp.r||fir||*
= Cp»r||/||p,ti, the desired result follows.
Definition. Let u be a real number and r the largest integer < u. Then
AS is the class of functions f,feL* such that \\b$J-rf\\p £ C\ h\u~r. The
norm of/ in Ag is defined as \\f\\Ptr + C0, where G0 is the least constant for
which the preceding inequality holds.
Theorem 8. (a) If u is not an integer the condition ||A|J"r/p ^ C\ h\u"r
is equivalent with \\AhJ~rf\\p ^ G\h\u~r. Thus in this case A£ coincides with
Lip (u, p).
(b) If I < p < co then Jv maps A£ continuously onto AJ+„. If v ^ 0 and
v + (u — r) £ 1, then this holds for I ^ p <> co.
(c) If I < p < co and u < v then A£ ^> Lvu ^ A?, inclusions being
continuous here.
Proof. We use the following identities which can be readily verified.
(ii) Aft - 2*A2-*A = 1 f>'A!-'A>
(iii) 2*A2-*A - 2M&2-»h = - ^ 2^A22-yA.
^ tf + i
We write g = J~rf and 5 = u — r. Suppose that ||A|g||p ^ C\h\*.
Then applying (i) to g, taking norms and letting JV tend to infinity, we obtain
■l|A#||, S lf,C2-l(2>\h\Y Z Cs\h\> if s < 1.
On the other hand, since ||Af^||p ^ 21^^^ the converse follows.
Let us now consider the action of Jv on /. Suppose first that 0 < v < 1
and assume that 0 < v + s ^ 1. Then what we have to show is that
IIAf/VL S C\h\8 + V. Indeed we have
AtJ»g = At(Gv*g) = (AhGv)*(Ahg),
and taking norms, applying Theorem 4 and Theorem 8(a), HAfJ^grflp S
||A»G,||i||A»flF||, S C\K\'+*.
Let us assume now that 1 < s + v < 2. Then we have to show that
W&hJ^^gh ^ 0\h\'+v-K First let us observe that
UA^A^II, = ||(AAlG„) * (A^ll,

38
A. P. CALDERCN
and applying again Theorem 4 and Theorem 8(a) we obtain
\\khlAh2J»g\\p ^ HA^IIiIIAa^IIp ^ C|Ai|*|A2|'.
Now let us set h = (xi, 0, • • •, 0), x\ > 0 and apply (iii) to Jvg. We get
! M
- y v
2N 2M
=r- A8-VV - — A2- Vp0
#1 #1
2z
i jy + i
X2~yA'
< — 2 2j(2"i*i),+r,
■ + ]
2*i ^
which means that as N and M tend to infinity the first side of the inequality
tends to 0. In other words (1/|h\)&hJvg tends to a limit I in L* as h =
(2-Nx±9 0, • • •, 0) tends to zero. Suppose cp is a function in C00 with compact
support. Then
J W\ ^hJvg)(pdx = j(JVg) m A-h(Pdx-
If we let h = (2~Nx\, 0, • • •, 0) tend to zero and pass to the limit we obtain
(pldx = - I -£ (Jvg)dx.
Hence I = (dldxi)Jvg. Let us now apply (ii) to Jvg, with hi = (x±9 0, • • •, 0).
If we take norms and let N tend to infinity we obtain
Let now A2 be such that |A2| = |&i|; then from the inequality above it
follows that
^A,2A^-^2(^)^|
S C\hx
s+v-l
Thus
and
Aa»(^)HI = G^s+V'1 + jh\ ^h^JVg^ = ci^i'4"-1
a-y*
^ C\h\
s+v-l
Similar inequalities hold for (djdx^g. On the other hand ||A^Jrt,gr||3, is a
bounded function of A, so from
Wg\\p£C\K\'+',
we get
Wg\\, s c\h\»-\
This combined with the inequalities for (djdxi)Jvg gives
IIAf/VLi = \\J-^tJvg\\v = W-l9\\v ^ C\h\»-\

DIFFERENTIABLE FUNCTIONS AND DISTRIBUTIONS 39
Consequently if 0 < v < 1 and fe Ag then JvfeAl+v. That the mapping
Jv from Aj + t, is continuous can be shown by accounting for the constants in
the proof given above. Finally if v is an integer, it follows from the very
definition of Ag that Jv is an isomorphism between A£ and Ag+v. Combining
this with the case 0 < v < 1 the general case follows. This completes the
proof of part (b).
^Now suppose feLp; then g = Jlfe L{ and (d/dxjg e U with \{d\dx^g\v
= Cp||/!l v Now let cp eC00 and have compact support. Then
l(p(Ahg)dx = \(k-hcp)gdx = - ^h \ dt\ \(p(x-th)^-dx\
it/:
K OSL
dxi
and Holder's inequality gives
(x — thi)hidt
ft
\g(x)dz
dxt
Since this holds for an arbitrary cp we obtain
dxt
*C\h\M,.i = 01*11/11,
JuAg =
where q = p/(p — 1)
\\^9\\pS \h\2
thus
||Afjy||^2||AAjy||^(7|A|||/||p)
and the first part of (c) follows, if u = 0. In the general case Ag
JULP = Lqu. The second part is proved by observing that if v > 0 then
A£ c L* where r ^ 0 is the largest integer < v; thus Ap c L? an(j the general
case follows as above.
Theorem 9. Letfe Z& = C\uLl and g e L%, 1 ^ p < oo, q = pj(p - 1).
Set (f> 9} = SfQdx; then |</, g}\ ^ ||/||p, 1*11(7 Ik,-** omd the bilinear functional
</> 9s? can be extended continuously to L*©Lq_u. Every continuous linear
functional on L% is of the form 1(f) = </, g} with g e Lp_u.
Proof. If u > 0 then one sees readily that </, Jugy = <e/M/, <7>- If
feLl then J~UE LI and
<f,9> = <JuJ~uf>9> = <J~uf,Ju9>-
Thus
\<f>9>\ = \<J~uf>Ju9>\ ^ \\J~ufh\\Ju9h = ll/lk-IMk—
Now for the extended linear functional we also have (J~uf, Ju9} = </> ?) f°r
all w. Let Z(/) be a continuous linear functional over Lp. Then l(Juh),
h 6 I>, is a continuous linear functional over 2> and therefore has the form
l(J"h) = (h,g} with g e IX Now if J"M^ = g then £ = JMg and l(Juh) =
<A, Jwg> = (Juh, g} = </, <7> = l(/) and 9 e Ltu.
Now we will prove an interpolation theorem of the Riesz-Thorin type.

40
A. P. CALDERCN
Definition. Let En and Em be Euclidean spaces. A linear operation A
defined on C™(En) (i.e., on the class of infinitely differentiable functions on
En with compact support) and having values in the class of temperate
distributions on Em is said to be of type 6 = (£1? £2> £3* £4) if Afe Ll(Em) with
V = 1/^2, u = £4 and
M/lli/^gCll/llv^, C<oo.
Theorem 10. Let Abe a linear operation as above and suppose that it is
simultaneously of types Qx = (£}, f J, f J, fj) arid 02 = (£?, ff, f§, £1) wAerc
0 < £{ < 1 /or i, j = 1, 2. TAen ^ is afco o/aK types 0 = A0i -f (1 - A)02,
0 ^ A ^ 1. jTAe theorem is also valid if one replaces En (or Em) by an
arbitrary (localizable) measure space and one replaces the restriction 0 < £\ < 1,
i = 1, 2 (or 0 < £| < 1, i = 1, 2) 6y £3 = 0, i = 1, 2 (or fj = 0). ^feo
C™(En) should be replaced by the class of simple measurable functions (or the
temperate distributions on Emy by the class of functions integrable on sets of
finite measure).
Proof. We shall restrict ourselves to the case of Euclidean spaces. The
argument employed here applies without change to the other cases.
Let £ be largest of £} and £?. Then L)£ <= L1^, i = 1, 2, and A can be
extended to LAJ* continuously with respect to both the norms as an operator
of type ■61 and 62. Thus we may assume that A is defined on L)£. Let us
denote by J\ and J\ Bessel integration on En and Em respectively. Let
Mz), 1%(Z) be linear functions of z with real coefficients and such that h(0) =
£h h(l) = £I> h(®) = ~£1> '2(1) = — £!• Let Kx be the operator on
functions on En defined by (K\fY = e~s^f^ and K2 be analogously defined for
functions on Em. Let
Bz = Jfz)K2AJYz)KY.
Then Bz is well defined on simple functions on En. For, if/is simple, then
KJe L*J? and the same holds for J^Z)KJ. But then AJ^z)KJe L*(Em)
with p = l/£* and u = fj, and K^AJ^KJe Ll(Em) and finally Bzf e
Ln(Em) with p = 1/^2- Further, Theorem 1 shows that Bzf, as an element
of Lg, is an analytic function of z for all z. Hence if g is a simple function
on Em then
gBzfdx
is an entire function of z. Now using Theorems 2 and 5 we find that this
function is 0(\z\ + l)w+m in 0 ^ R(z) ^ 1. Now for R(z) = 0 we have
from Theorems 2 and 5
WI|i/« g c(M + i)B+ml|/l|i/{!,
and a similar inequality holds for R(z) = 1. Hence by a theorem of E.

DIFFERENTIABLE FUNCTIONS AND DISTRIBUTIONS 41
Stein [12], it follows that for 0 g A ^ 1
iivii/* ^ emu*
where £2 = Af| + (1 - A)f|, & = A£ + (1 - A)ff. More specifically,
writing & = A£J + (1 — A)£f, t = 1, 2, 3, 4, we have
HJ-^A/^/H^ * C||/||1/{1.
Now letting € tend to 0 in #2 and taking € in K\ so small that fl/Hi/^ ^
2||ifi/||i/fl, and writing g = J^Kifwe obtain
\\M\m,U = y-^gh/t, ^ MWKifWv^ ^ 2CBJ-«V||1/fl = 2C\\g\\1/(vh,
and this establishes the desired result if g is of the form J^Kif. For g e
C£(En) the desired result is obtained by a passage to the limit.
Now we shall consider functions in L*(En), u > 0, and their restrictions
to lower dimensional subspaces. If f e L*(En) then / = Gu* g with g e LP.
We shall say that / is defined at a point x e En if the integral Gu * g is
absolutely convergent at x. The value of the integral at x will be by
definition the value of/ at x.
Theorem 11. Let Ek be a subspace of En andfeLl(En). Then (a) if
1 < p S 2, and u > (n — Jc)lp, the function f coincides almost everywhere on
Ek with a function in L^(Ek)y where u — v = (n — k)\p. The restriction off
to Ek induces a continuous mapping of L*(En) into Lvv(Ek). (b) If I < p < 00
then f coincides almost everywhere on Ek with a function in L\ where \\q =
(njh)(\jp — (u — v)/n) provided (n — k)jp < u — v < n\p, v ^ 0. Here
again, restriction of f to Ek induces a continuous mapping from L*(En) into
Proof. We shall first consider part (a). It is easy to see that in this case
it is enough to prove the assertion for k = n — 1. We shall designate
points in En-i by x = (x±, x2i • • •, xn-i) and points in En by (x,t) = (xi, x2,
• • •, xn-i, t). Let us consider the function
[1 + 47r2(|z|2 + *2)]-"/2(l + 47r2|z|2)»/2, v = u - l/p, 0 < u < 2.
We shall derive estimates for the Fourier transform H(x, t) of this function
with respect to t, and the transform K(x, t) with respect to all variables.
It will turn out that K(x, t) is integrable on En, so that taking transforms
with respect to x and t successively will be justified. We write
[1 + 4tt2(|o;|2 + *2)]-*/2(l + 47r2|z|2)»/2
and taking Fourier transform with respect to t we obtain
H(x, t) = (1 + 47r2|x|2)i/2«G[|«|(i + 47r2|a;|2)1/2],

42 A. P. CALDERON
where here, and throughout the proof of part (a), q = pj(p — 1) and G is
the Fourier transform of (1 + 47r2«s2)~w/2, that is
(2) G(s) - y{u)e~s e~st{t + *2/2)-«/^ft, 5 > 0.
Thus
#i(:M) - \t\v*H{x,t) = «i/«ff(«),
where $ = |f|(l + 47r2|x|2)1/2. By diiferentiation of (2) one obtains the
following estimate for the derivatives of s1^G(s)
\~[sl^G(s)]\ g Crs8-ye-sfz, 0 < u < 2.
where 8 is a positive number.
Let us now differentiate Hi(x, t). By induction we find that
(rjj^Hilx, «) = 2 Cv*\t\i ^ [51/^(5)],
where j — \a\ = |y|, |a| -f i — Z -f \y\ = 0 and the G are constants
depending on a, i, j and I. Since |x| |£| ^ 5 combining this with the estimate for the
derivatives of sl/^G(s) we obtain
(3)
|(1 + ±7TW)\y\i*U^yH^x,^ <, Cysse-S^.
Now we will derive some needed estimates for the Fourier transform K\(x, t)
of Hi(x, t) with respect to x. Let H(x, A) = gx^G(g) where now a =
2tt(A2 + \x\2)V2. Since H(x, A) -> 0 as A -> oo we have
H(x,il) = 2 f°°^-[c7VffO(c7)]ArfA.
By arguing as we did when estimating (djdx)yH\ we find that
|(i)v^[ffl/<,G(ff)]| - ^v^-1"1-1^2.
Differentiating the last integral, taking Fourier transforms with respect to x
and denoting by K the transform of H we obtain
\x\M\K(x,/jl)\ g CY rXdX f o«-lvl-ic-^/2da;
£ CYe-» P° XdX f o*-\v\-ie-°/*dx.
Jo J«»-i
Introducing polar coordinates in the inner integral first, and then in the

(4)
DIFFERENTIABLE FUNCTIONS AND DISTRIBUTIONS 43
resulting double integral we obtain
x\\v\\K(x, fi)\ ^ Cyer» J J (27rr)3-lvl-ie-(W2)r(r cos 0)n-«(r sin 6)rd6dr
and setting y = 0 and \y\ = w,
|#(z,/x)| < Oe-*; |»|»|X(a;>f*)| ^ Cfe-".
Now, evidently we have H\(x,t) = H(x\t\,\t\j2iT) whence K\(x, t) =
|£|-w+1JfjL(a;/|£|, |f|/2tt) and by the preceding inequalities we have
(5)
|JTi(a:,*)| ^ Ce-l'l/2"|<|-«+iinf[l,|j|-»].
Now Zi(x, 0 = \t\v*K(x, t) and thus (5) shows that K(x, t) e L*{En). In
what follows we shall be dealing with functions defined either in En or in
En-i; the operation jR will denote restriction to En-\ of a function defined on
En- Let g be a bounded function on En-i with compact support and
h(x, t) = g(x) * Ki(x, t). Let €^ be a sequence such that €^ = ±1, but
otherwise arbitrary, and consider the sum
M
M
N N
If we estimate the norm of this function in LQ(En-i) (for any given value of
t) by applying Mihiin's theorem and (3) we find that this norm does not
exceed C||^||a, independently of the values of N and M and of the choice of
the sequence €j. From this it follows by a well-known property of Rade-
macher functions that
[t^? "I1/2!!
2 i*(*,2«oi»
S <%|L
where £ is regarded as a parameter and norms are norms in L<*{En-\). Hence
we obtain
|[f
NM)|2^1/2'
t
"| i/2||2 II « r
I — 00 */ ^
•f oo /*2m + 1
2m
|ft(*,0|g
2m
2 II + °°
dt\\
\h(x,t)\*
t
*2 + °o
*
iff/2
Ik/2
= P f \h(x92n)\
\\J 1 — oo
f f |*(*,2"*)H A^C^IIflFifft.
Ji II -"£ Ik/2
2<ft
9/2

44
A. P. CALDERCN
On the other hand, if g = sup*| h(x, t)\ then (5) implies (see for instance [13,
Chapter 2, Lemma 3]) that \\g\\q ^ C||gr||g. Consequently
(6)
X-i J-00
|*0M)|«
(ft ^ 2
Lrif*
|*(M)|'
dt
dx,
and by Holder's inequality this last expression does not exceed
il<r!i"«-!
If
00 A(^j/]_2
dt
q/2
* <%iij-
But the left-hand side of (6) is precisely \\g(x) * K(x, t)\\* (g * K being
considered here as an element of L<*(En)). Thus we have ||gr * K\\q S G\\g\\q
where the norms are in L<*(En) and L^(En-i) respectively. If / is now a
bounded function with compact support in En we obtain by duality || R(f *
K)\\p ^ C||/||p» where R is the restriction operator and the norms are in
Lv(En-i) and Lv(En) respectively. Let now Ji denote, as before, integration
in En_l9 and J integration in En. Then J^~llpR(f * K) = RJ\-l^(f * K)
= RJuf and what we have just shown implies that \\RJuf\\p,u-i/p ^ C||/||p
= C\\Juf\\Ptu. Let now f e Lv(En), f ^ 0, and let fm be an increasing
sequence of bounded functions with compact support such that fm->f in
Lv(En). Then Jufm is continuous and, since Ju is a positive operation, it
increases with m. Thus RJufm is well defined, and according to the
inequality just established, RJufm converges monotonically to a limit in
Ll_ljp(En_1). At every point xeEn_1 where this limit is finite RJuf is
Hence RJuf is finite almost everywhere on
This result extends immediately to general
finite and coincides with it.
En_1 and belongs to L^_xlv.
functions in/e Lv(En).
Finally let us consider the case u ^ 2. Let r be the largest integer such
that r < u — I/p. Let/ = Jug, g eC™ ; then, since u — r < 2, if \y\ <; r,
K£)''
< c\
p,u—r—l/p
(!)>
p,u-r
£ c|/|k«.
If we take (d/dx)* to be differentiation with respect to coordinates of En-i
only, then R(d\dxy = (djdx)yRt and the preceding inequality implies that
&*
=S C\\f\\P,u.
\\p,u-r-l/p
If r = 2« is even, this implies that
||(l - tyRf \\p,u-»-vp = l|*/lk.-i/* ^ CI/11,.
If r = 2s — 1, then for \y\ ^ r
3
a* (ex) Ef
< c
p,u~2s-l/p
(s)>
p,u-l/p
£ o\\f\\v,u,

DIFFERENTIABLE FUNCTIONS AND DISTRIBUTIONS 45
and again from this it follows that
I(i - A)**/!!,,.-*-!/, = ||*/||*,-i/, n tf||/||„.„ = c\\j«g\\P.
The case of general / e L*(En) is obtained by a passage to the limit.
Let us now prove part (b) of the theorem. Suppose / e Off; let v be a
non-negative integer and u and p by such that (n — k)jp < u — v < n/pt
where k is the dimension of a given subspace Ejc of En. Let jB denote
restriction of functions to 2?*, and let (d/dx)*, \y\ ^ v, involve only
differentiation with respect to coordinates of 2£*. Then if (3/3x)vf = Ju~vh we
have ||% = \\(dl8x)yf\\ptU-v ^ G\\f\\PtU-v+\y\ ^ C\\f\\ptU. On the other hand
if \jq = (n/k)(llp — (u — v)/n) then, by a theorem of Il'in [8],
(I)H"IK^/IL"|J8/""'*I,"0B*1'
where the last norm is in Lv(En) and the preceding ones in L*(En-i) Now
since || A||p ^ C1/||jmi we nave
\m
<
\Q
c\\fh...
Since this holds for all y, \y\ ^ v, this gives
ll*/ll... ^ C\\f\\p,u.
We apply now Theorem 10 and pass, by interpolation, to general
non-negative v. Thus the continuity of the mapping jB as asserted in (b) is established.
The fact that for general/e L^(En), Rf coincides almost everywhere with a
function in L%(Ek) follows by a passage to the limit as in case (a).
Definition. Given a domain D in En we will say that the function /
denned in D belongs to L%(D), where fc is a non-negative integer, if/ e L%{D)
and has derivatives of order ^k in LP(D). The norm of fe L$(D) will be
denned as the sum of the norms of (d/dx)"/, 0 ^ |a| ^ k, in LP(D).
Definition. A domain D in En will be said to have a regular boundary
if there exists a finite open covering Ui of the boundary 3D, finitely many
finite cones y\ and a positive number e such that
(i) every point of 3D is the center of a sphere of radius e entirely contained
in one of the sets Ui;
(ii) every point of Ui n D is the vertex of a translate of y< entirely
contained in D.
Theorem 12. Let D be a domain in En with regular boundary. Then
there exists an extension operator $ which maps L%(D) continuously into
Ll(En)for all p, 1 < p < oo. More explicitly, £ is such that if f e L%(D) then
SfeLi{En) and Sf = f in D. Furthermore \\ff\\Ptk ^ CPtktD\\f\\ where \\f\\
is the norm off in Lv(D).

46
A. P. CALDERCN
Proof. We first construct a partition of unity by means of infinitely
diflFerentiable functions 17, r^ such that (i) they have bounded derivatives of
orders ^ k, (ii) y\i is supported by U{, (iii) the support of 17 is disjoint from
3D. This construction is standard, so we omit further details. We
introduce polar coordinates in En. If p is a non-negative number and v a unit
vector, then p and v are the polar coordinates of the point pv. Then
(?)
where a\ = (ai!)(oc2!)- • -(an!). Assume the cone y* has vertex at the origin
\x\ = 0, and let <f>i(x) be a function supported by — yi which is infinitely
differentiate in jrcj 5^ 0, non-negative and which coincides with a
homogeneous function of degree — n + k in a neighborhood of \x\ = 0. Then
clearly (dldp)k[pn~l<f)i(pv)] — i/ii vanishes near the origin and is therefore an
infinitely differentiable function. Let now/e L%(D) and let us extend/and
all its derivatives to be zero outside D. Define accordingly
\yi{pv)pn-l\j-\ f(x - pv)dPdv
- Mpv)f(x ~ pv)pn~Updv
where dv denotes the area element of the sphere \x\ = 1 and d is a* constant
to be determined later. We will show that f{ e Ll(En), that \\fi\\p>k ^
CpjtM" where ii/ll is the norm of / in Lk(D)> and that &=/ in Ui n D.
Once this has been established it will follow that
<f = i(*)/(*) + 2^(*)/«(*)
gives the desired extension.
Thus we merely have to show that ft has the properties stated. Assume
for the moment that the given function fe L%(D) is continuous and has
continuous derivatives of all orders in D. Then if x e Ut n D the function
f(x — pv) is k times continuously differentiable on the support of <pi(pv).
Hence the first integral in (7) can be integrated by parts k times with respect
to p alter which the right-hand side of (7) becomes
Cif(x) j(£}k~\p"-Vt(pv)l=odv,
and the integral here does not vanish if fa is not identically zero, which this
makes it possible to select d so that/ = ft in Ut n D. If/is now an
arbitrary function in L%(D) there exists a sequence of infinitely differentiable
functions such that/m ->/ almost everywhere in D and {djdx)afm converges
to (d/dxYf in the mean of order p on every compact subset of D, for [a| ^ k.

DIFFERENTIABLE FUNCTIONS AND DISTRIBUTIONS 47
This implies that $ifm -> £%f in the mean of order p on every compact
subset of Ui n D. Since Sijm = fm in Ui n D, this implies that S\j — f
almost everywhere in Ui n D.
Let us now examine the first integral in (7) more closely. Carrying out
the differentiation with respect to p indicated we can rewrite it as
(8)
where ?>a = (- l-)*(fc!/a!)i^i(pv) and /„ = (dldx)af in D and fa = 0
otherwise. Consider the convolution <pa * gr. Since qpa is infinitely differentiate
in |a:| ^ 0 and coincides with a homogeneous function of degree — n + k
in a neighborhood of the origin, ya* g can be differentiated k — 1 times by
merely differentiating <pa under the integral sign and thus, by Young's
theorem, we find that for |j8| = k — 1
11/ d\P II 1117 d\0 1 II 11/ 8 \a II
life) (9-HL=life) H HI*life) 4W1, -C|I?IU-
If g is continuously differentiable we could differentiate k times under the
integral sign by differentiating g once and then <pa, k — 1 times—that is, if
If in the integral representing the last convolution we exclude a sphere of
radius e with center at the singularity of (d/dx)^(pai integrate by parts, and
let e tend to zero writing (d / dxt)(d I dx)& = (d/dx)$, we obtain
(l)^° *g)
m
+ Cg
where Cg is the limit of the integrated term and G is a constant. Naturally
the convolution on the right is to be interpreted as a principal value integral
since (dldx)$(pa coincides with a homogeneous function of degree — n near the
origin. Thus we can write
(I) <*■ *g) = L«g(x -my)dy+L><g{x - yii) ^y)dy+cg{x
where h(y) is a homogeneous function of degree — n which is infinitely
differentiable in \y\ =£ 0 of mean value zero on \y\ = 1 (otherwise the

48
A. P. CALDER6N
principal value integral would diverge). Thus the first integral in the last
inequality represents a function whose norm in L* is dominated by C^Hp.
Now this combined with an application of Young's theorem to the second
integral yields
|(l) {<p° * ?)H
^ C||fli|,.
This combined with the inequalities obtained previously for the lower order
derivatives of cpa * g gives
\\<pa * g\\p,k S Ca\\g\\p,
assuming that g is continuously differentiate. Now this restriction can be
removed by a passage to the limit. If we apply this result to (8) we find that
the first integral in (7) represents a function in LftEn) with norm dominated
by G2\\fa\\p £ 11/11 where ||/|| is the norm of/ in L%{D). The second
integral in (7) represents a convolution of/ with a function in (7®. Thus by
differentiation under the integral sign and by Young's theorem it follows that
its norm in L%(En) is dominated by C||/||. Consequently we find that
|| $if || p,k S C||/||> and the proof of the theorem is complete.
References
Some of the spaces discussed in this paper have been considered by other authors;
see for instance [1; 2 ; 3 ; 4 ; 5 ; 10]. Similar spaces of functions on a compact Rieman-
nian manifold are discussed in [9].
1. N. Aronszajn and K. T. Smith, Theory of Bessel potentials. I, Studies in eigenvalue
problems, Technical Report No. 22, University of Kansas, 1959.
2. A. P. Calderon, Singular integrals, notes on a course taught at the Massachusetts
Institute of Technology, 1959.
3. J. Deny and J. L. Lions, Les espaces du type de Beppo Levi, Ann. Inst. Fourier
Grenoble vol. 5 (1955) pp. 305-370.
4. E. Gagliardo, Caratterizzazioni delle trace sulla frontiera relative ad alcune classi di
funzioni in n variabili, Rend. Sem. Mat. Univ. Padova vol. 27 (1957) pp. 284-305.
5. E. Gagliardo, Proprietd di alcune classi di funzioni in piii variabili, Ricerche Mat.
vol. 7 (1958) pp. 102-137.
6. E. Hille, Functional analysis and semigroups, Amer. Math. Soc. Colloquium
Publications, vol. 31, 1948.
7. V. II'in, Generalization of an integral inequality, Uspehi Mat. Nauk vol. 11 (1956)
no. 4 (70) pp. 131-138.
8. S. G. Mihlin, On the multipliers of Fourier transforms, Dokl. Akad. Nauk SSSR
(N.S.) vol. 109 (1956) pp. 701-703.
9. R. T. Seeley, Singular integrals on compact manifolds, Amer. J. Math. vol. 81
(1959) pp. 658-690.
10. L. N. Slobodetzky, Spaces of S. L. Soboleff of fractional order, Dokl. Akad. Nauk
SSSR vol. 118 (1958) pp. 243-246.

DIFFERENTIABLE FUNCTIONS AND DISTRIBUTIONS 49
11. S. L. Soboleff, Sur un theoreme de Vanalyse fonctionnelle, C.R. (Doklady) Acad. Sci.
URSS vol. 20 (1938) pp. 5-9.
12. E. Stein, Interpolation of linear operators, Trans. Amer. Math. Soc. vol. 83 (1956)
pp. 482-492.
13. A. P. Calderon and A. Zygmond, On the existence of certain singular integrals,
Acta Math. vol. 88 (1952) pp. 85-139.
Massachusetts Institute of Technology,
Cambridge, Massachusetts

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THE MAJORANT METHOD1
BY
P. C. ROSENBLOOM
1. Introduction and summary of results. The classical Cauchy problem
can be reduced, by various normalizations, to the following one: Given m
functions /« of the 1 + n + m -f mn complex variables t, xi, • • •, xn, u\y- • •,
um,piti,- - -,pm,n analytic in a neighborhood of the origin in the space of
these variables, to find functions m,- ■, um, analytic in a neighborhood of
the origin of the space of the variables t, x\, • * •, xn such that
dm [ dm dum\
for i = 1, • • •, m and m = 0 for t = 0. The Cauchy-Kovalevski theorem
asserts that there exists a unique solution. It is proved by an expansion of
^i> • • •, um in multiple power series and by comparison with a differential
equation in which the formal power series solution has non-negative
coefficients. The calculations involve a morass of indices and subscripts and
are so complicated that it is difficult to extract explicit concrete results on
the size of the domain of existence, the growth of the solution, and its
dependence on the given functions /i, • - •, fm. A further minor
unpleasantness is that the estimates of the radius of convergence of the power series for
the solution approach 0 as m or n approaches infinity. The same remarks
apply to other problems, such as the Goursat problem and the general
Riquier problem, in which the majorant method is used.
In the present paper we shall develop a new version of the majorant
method in which (xi, • • •, xn) and (uif- • •, um) are treated as elements of
normed linear vector spaces. It turns out to be just as easy to carry out the
argument in general Banach spaces, so that the results are valid also in
infinite dimensional spaces, They can then be interpreted concretely to
obtain implications for new types of functional equations in classical analysis.
Others who have treated differential equations in infinite dimensional spaces
are Gateaux [3], Levy [10] and Miehal [12].
Let X and U be complex Banach spaces and let V = Ux be the space of
bounded linear transformations on X to U with the natural norm
||p|| = sup \\px\\.
iiarii Sz 1 '
1 A preliminary report of this work was presented at the International Congress in
1950 (see [16]). This work was in part supported by the Office of Naval Research.
51

52
P. C. ROSENBLOOM
Let Cn be the space of n complex variables with the norm
IWI = (2N2) •
If u is analytic on a domain in C± x X to U, then we define the partial
Frechet differential with respect to x, operating on the point y e X, by
ux(t,x)y = ^u(^x + A2/)|A=o.
Note that ux is analytic on a domain in Ci x X to F. Let W = (7i x X x
C/x F with the norm ||(«, a, w,p)|| = max(|«|, ||s||, ||m||, ||p||).
If D is a domain in X, we shall denote by E(r, D) the domain |t | < r, x e D,
in d x X, and by <D the domain |t| < l,xeD, \\u\\ < 1, ||p|j < 1, in W.
We can now formulate the Cauchy problem in the following manner: Given
/ analytic on <£) to U, to find a function u analytic on some domain E(r, Di),
0 <r ^ 1, flia subdomain of D, to U such that
(1) ut = f(t, x, u, ux) in E(r, Di) and u(Q, x) == 0.
Our method is to expand u(t, x) = 2i° un(x)tn in a power series in t with
coefficients analytic in x. We obtain a recursion formula for Un+i(x) in
terms of %(#), • • •, ^y(#) and their Frechet derivatives u^x), • • •, u'N(x). In
order to estimate the coefficients we use the following lemmas :
Lemma 1. // F(x) is analytic in D and
\\F(x)\\ ^ AS-*, a ^ 0,
for x g D, where 8 = S(x) is the distance from x to the boundary of D, then
\\F'(x)\\ ^ Ae(\ + a)8-(«+« for xeD.
Lemma 4. // P(x) is a polynomial of degree n on X to U and \\P(x)\\ ^ 1
for \\x\\ S 1, then
\\P'(x)\\ ^ en for \\x\\ ^ 1.
Lemma 1 is a fairly trivial generalization of known results for functions
analytic in the unit circle. Lemma 4 is an easy generalization of Bernstein's
inequality (see [1]). In the most important cases of Cn and Hilbert space
the factor e can be eliminated by more delicate arguments (see Kellogg [8]).
Michal [13] announced this sharper result in the general case, but there
seems to be a gap in his proof.
If we assume that ||w»(a)|| ^ A^S-^-1) for 1 <; n ^ N, and that 8(x) ^ A
in D, then we can obtain a corresponding estimate for Un+i{x) providing that
F(z) = 2? K?n *s the solution of a certain ordinary differential equation.
We thus obtain:
Corollary lb. Let f be analytic and \\f\\ ^ M in <& and suppose that

THE MAJORAT METHOD 53
\\f{t, x, 0, 0)|| ^ e ^ M in E(l, D). Then the formal power series solution
u(t, x) of (I) is analytic in the domain ^i(Ki):
and satisfies there the inequalities
NM)M^iog(,_ci;l;8(i))<./2.
\\ux(t,x)\\ ^ 1/2
where C\(x) = A(l + e(M — e)) + e2(M — e). In particular \u(t,x)\ ^
4e|£|, \\ux(t, x)\\ S 4ee|£|/S(#) in the domain ^(^i) '>
^ «l.l MX)
xeD, 2\t <- -4t ^*
11 (1 + eKi)A + e2i^i
The refinement of having an estimate involving e as well as M is useful
for certain purposes. By using these estimates on un we can show that the
successive approximations:
MM(t, x) = 0,
t*<* + 1>(*, x) = P/(t, x, w<*>(t, x), uik)(r, x))dr.
Jo j
converge in l)i(Ki). Such a theorem has been obtained by Titt [18].
We next apply these results to the study of the dependence of the solution
on /. The main result is:
Theorem 3. If f is analytic and ||/|| ^ M on the domain |A| < 1, (t, x,
u, p) 6 2), inC\ x W to U, and u{f) is the solution of (I), then u(f) is analytic
for |A| < 1, (t, x) e ®i(4Jf).
In order to formulate the implications of this theorem in the most pregnant
form, we introduce a convenient terminology. If D is a domain in a Banach
space X, let 3( A Y) be the Banach space of bounded analytic functions on
D to Y, taking as the norm
N(f, D) = sup ||/(z)||.
D
By 3(2), Y; M) we mean the sphere N(f, D) < M in 3(2), Y). Then we
have:
Corollary 3a. The solution u(f) of (1) is analytic on g(<2), U \ M) to
3f(©i(4Jf), J7)ondiV(M(/),®i(4Jf)) g 1/2. IfN(f^) < M,N(g,®) S rifeT,
0 < r, £Aew
W/ + 0) - *(/), ®i(4(l + r)M)) g tffa, 5))/(2erif),
W/ + ?) - w (/) - tt'(/)gr, SDi(4(l + r)Jlf)) £ 9tf(gr, ^)2/(8er2Jf 2).

54 P. C. ROSENBLOOM
The function u'{f)g is the solution of the problem
vt = /«(*, x, u(f), u(f)x)v + fp{t, x, u(f), u{f)x)vx + g(t, x, u(f), u(f)x),
v(0, z) = 0.
The last result gives an estimate for the error in perturbation methods,
which is important in a quantitative discussion of such approximation
procedures as Newton's method (see Kantorovich [7]).
For the case of linear differential equations the above results can be
sharpened considerably. Let
f(t, x, u, p) = A(t, x) + B(t, x)u + C(t, x)p,
where A, B, and C are analytic on E(l, D) to U, Uu, and Uv respectively.
We can prove
Corollary 4a. If N(A, E{\, D)) g a, N{B, E(l, D)) g b, N(C, E(l, D))
^ c, then u(f) is analytic in D* •
\t\ < S(x)l(S{x) + ec),
and satisfies there
\\u(t,x)\\ S 8(s)(a/&){(l - 2/20)-^ - 1},
where z0 - (S(x) -r ec)-1, z = |*|/8(s). If u{t, x) = 2J° un(x)tn, then
and
Xn ~ (ajb)zQnnbzo as n~>co.
Corollary 4b. Let A, B, and C be analytic and bounded in the "bi-
cylinder" \t\ < R, \\x\\ < S and let M(R,S; A), etc., be the suprema of their
norms there. Then u(f) is analytic for
R(S - Hs|l)
11 S - \\x\\ + eRM(R,S;C)
and in the domain
1 ' S - \\x\\ + eRM(R,S;C) P '
we have
a(R,S) =
where
RSM(R,S;B)
S - ||z|| + eRM{R,S;C)
In pa~Hcular, if A, B, and C are entire functions and C is a function of t alone,
then u{f) is an entire function. IfC is at most of degree 1 in x, then u(f) is an
entire function of x for sufficiently small t.

THE MAJORANT METHOD
55
The observation that the domain of analyticity of u(f) can be estimated in
terms of C alone was already made by Schauder [17]. John [6] has remarked
that if A and B are entire and C is constant, then u(f) is entire.
If we interpret our results in special infinite dimensional spaces, we obtain
new results for functional equations which are not immediately tractable by
means of the classical Cauchy-Kovalevski theorem. For example, let
<f>{$, x) be analytic in the bicylinder |£| < 1, \x\ < 1, in C^ and let \(f>(t, x)\
<; k < 1 there. We seek a function v(t, x) analytic in a neighborhood of
the origin in O2 such that
vt(t, x) = vxx(t, <f>{t, x)), v(0, x) = F(x)9
where F is a given function analytic in the unit circle. We apply Corollary
4a to the case where X — Ci, D is the unit circle, and U = 3(D, C\); and
find that there exists a unique solution analytic in the bicylinder and that
\v(t,x) ~ F(x)\ £ (1 - \x\)(a/b){(l - \t\)-»M-\x\) _ i},
where a = max |jF"(#)|, b = 4/(1 - 1c)2.
If <f>(ty x) = kxy then this equation approaches the heat equation as k -> 1,
and in this limiting case a solution analytic in a neighborhood of the origin
exists only if F is an entire function of order ^ 2.
The essential step in the above method was Lemma 1, so that similar
results can be expected corresponding to any theorem giving an estimate for
the growth of a derivative of a function in terms of that of the function
itself. Thus by use of Bernstein's theorem on entire functions of exponential
type we obtain
Theorem 5. Let Z be a complex Banach space with a real subspace X such
that every element ze Z can be represented uniquely in the form j = x + iyy
xyyeX. Let f be analytic in the.domain z e Z, \t\ < l,\\u\\ < 1, \\p\\ < 1 in
W = C\xZxUxV and satisfy
\\f(t,z,u,p)\\ ^K(t,u,p)exip(p\\y\\)
for \t\ < 1, ||tt|| < 1, ||pI < 1, z — x + iy e Z, and
\\f(t,x,u,p)\\ S M
for \t\ < 1, ||«|| < 1, ||p|| < 1, xeX. Then u = u(f) is analytic in the
domain
\t\ exp (p\\y\\) < to = g(fa)l(2eM + g(fo)),
where
^0 = [1 + P + (1 - P + P2)1'2]'1,
gffl = 0(1 - 0(1 - pj,).
We hope in a future paper to show how other results obtained by the
classical majorant method can be simplified, sharpened, and generalized.

56
P. C. ROSENBLOOM
In this introduction we have often stated our results in a simpler form,
omitting certain refinements which are introduced in the following text.
2. Proofs and details. Let / be analytic and ||/|| ^ M in 1), and let
i\\f(t9 x, 0, 0)|| ^ € in E (1, D). We shall try to determine the solution of (1)
n a domain whose size can be estimated in terms of M and e. We note that
e"is a measure of how good u = 0 is as an approximate solution of (1). A
good estimate of u in its domain of existence in terms of e will enable us to
treat problems concerning the behavior of the solution of (1) as a functional
of/.
The function / can be expanded in the form
f(t, x, \u, up) = 2/*iro(s, u9 p)tk\lfim,
where fkim is analytic in x for x e D and a homogeneous polynomial in u of
degree I and a homogeneous polynomial in p of degree m. (See Hille [5].)
If we substitute
00
(2) U(t, X) = ^ Un{x)tn
1
into (1) we obtain a recursion formula for un which shows that there is a
unique formal power series solution of (1) of the form (2). In order to derive
this recursion formula explicitly we need to remind the reader of a known
formalism for handling power series in Banach spaces.
If P(u) is a homogeneous polynomial of degree n on U, then there is a
symmetric n-linear function P(ui, • • •, un), the so-called polar form, such that
P(u) = P(uy u,- • • yu). In fact
from which we see that n\P(ui9- • •, un) is the coefficient of Ai- • • An in this
expansion. This yields the formula
P(y>ir-,y>n)= ,
n\(27r)n
(3)
(see Hille [5]).
Hence if
1 /»2tt /*27r /n \
— I ...jo p(2w.)
•exp j — ^ iffitp&i- • -ddn
|Al An-0
u
1

THE MAJORANT METHOD 57
then
P(u) = ^ P(ukv- ■ -, «*„)'*> N = ftK 4, £ 1, 1 ^ v ^ n,
1
and the coefficient of tN is
2 P(u>kv-',ukH)
over all sets of indices such that 2i K = N. Analogous remarks apply to
homogeneous polynomials in several variables.
Let
fklrnfa Uir • ',Ui,pl9> • -,pm)
be the polar form oifkim with respect to u and p. Then if
00 00
u = V ^*, p = 2 jM*,
1 1
we have
/(*, x, u, p) = ^ ^(^ *>!,--, uN, pu• • •, pjr)**
where
P^ = ^fklm(x, Uvv • • •, ^, pav • • • , paJ
the summation being taken over all sets of indices such that
k + vi + • • • + vi + cri + • • • + crm = JV,
fc £ 0, Z ^ 0, m ^ 0, vi ^ 1, ^ ^ 1.
It is convenient to write Pjv in the form
Pn = /Woo(s, 0, 0) + Qiv = 4>n{x) + QN,
and we note that
00
f(t, x, o, o) = 2 M*)*"-
On substituting (2) into (1) we obtain
m(x) = 4>0(x) = /(0, x, 0, 0),
(4)
uN(x) = JV-1PjV-i(^, ^i(^),- • •> ^iv-i(z), wi(x),- • •, u'N-i{x)).
Here if v is analytic on X to [7, then v'(x) denotes the linear transformation
defined by
v'(*)y = -qxv(x + ty)U=o
the Frechet derivative of v with respect to x, and is analytic on X to V.
These equations (4) show the uniqueness of the formal power series (2)
and therefore, a fortiori, of the analytic solution of (1). For the existence of

58
P. C. ROSENBLOOM
an analytic solution of (1), it suffices to show that the series defined by (2)
and (4) converges in some domain E(r, D{) cz C\ x X. For this we need
some lemmas in order to obtain estimates on the functions un(x), which are
clearly analytic in D.
Lemma 1. If f(x) is analytic for x e D and
\f{x)\\ <> A8-°, a ^ 0, forxeD,
where
then
h(x) = distance from x to the boundary of D,
\\f(x)\\ £ e(l + a^S-"-1 forxeD.
Proof. Let y e X, \\y\\ S 1, and let 0 < r < S(x) = 8. Then
f(x + Xy)dX
f'(x)y = -L f -
A2
Now h(x + Xy) ^ 8(x) - r, so that
\\f'(x)y\\ S A(§ - r)-r-i.
Set r = S/(a +1). We obtain
(S - r)~ar~l = S-^-^l + a)(l + l/a)° ^ e(l + cOS""-1.
It is eviden" how this lemma enables us to pass from estimates of uw • -,
un-i to estimates of their derivatives. The next step in applying the
recursion formula (4) depends on a generalization of Bernstein's inequality
(see [I j) to Banach spaces. In the most important case of finite or infinite
dimensional complex Euclidean space, or Hilbert space, sharper forms were
already obtained by Kellogg [8]. Our very simple argument may have some
interest; we remark that even in the finite dimensional case it is sometimes
convenient to work with norms other than the Euclidean norm.
Lemma 2. (See Polya-Szego [14, vol. I, p. 137].) // P(u) is a polynomial
of degree n on U to the Banach space Z, and if \P(u)\ ^ 1 for \\u\\ ^ 1, then
\\P(u)\\ < \\u\\» for H| ^ 1.
Proof Let z*eZ*, \\z*\\ S 1, g(X) = \nz*P(ul(\\\u\\)). Then g is a
polynomial in A of degree n and \g(X)\ ^ 1 for |A| = 1. Hence, by the
maximum modulus principle, |<7(||w||_1)| g 1. The lemma now follows
immediately since z* was arbitrary.
Lemma 3. (See Michal [13], Martin [11].) Under the hypotheses of
Lemma 2,
nn
\lP(Uir-,Un)\\ ^ — \\ui\\ - • • \\un\\ S en\\ui\\- "\\un\\.

THE MAJORANT METHOD 59
Proof. Let \\ui\\ = • • • = \\un\\ = 1. Then ||2i % exp (iOk)\\ ^ n, so
that ||P(2i uk exP Wt))|| = n11- We obtain our conclusion immediately
from formula (3). (Polya-Szego [14, vol. I, p. 30, no. 167].)
Another proof of Lemma 3 sheds more light on the result. If U and Y are
arbitrary complex Banach spaces, let <xn = an(U, Y) be the least upper
bound of ||P'(w)|| as P ranges over all polynomials of degree n on U to Y such
that \\P(u)\\ g 1 for HI ^ 1. Evidently an(U, Y) = a„(l7,Ci) = an(U).
For the device of considering y*P(u)9y* e Y*, \\y*\\ ^ 1, shows that
an(U, Y) S oin(lJ). On the other hand, if Q(u) is a polynomial on U to C\
of degree n such that \Q{u)\ g 1 for \\u\\ ^ 1, then P(u) = Q(u)y, yeY,
\\y\\ = 1, is in the competition which defines aw(f7, Y). Michal [13]
announced a proof that an(U) = n for all U, but his proof contains a lacuna.
Lemma 4. For all U, n ^ ocn(U) ^ en. If U is a finite dimensional
complex Euclidean space or a Hilberi space, then an(U) = n.
Proof. The latter result was proved by Kellogg [8]. For the general
result, let P be a polynomial of degree n on U to C± such that |P(w)| S 1 for
H| S 1. Then
d 1 C2n
P'(u)ui = — P(u + Atti)L=o = x— P(u + Uireid)e-iddO.
va 2irr Jo
If H| S 1, ||wi|| ^ 1, then \P(u + ^ire'»)| S (1 + r)» (Lemma 2), so that
|P»^i| ^ (1 + r)"*-1.
If we choose r — l/n, we obtain the desired result.
Lemma 5. Under the hypotheses of Lemma 2,
||P(«i,---,«,,)|| ^ai(Uy:;an{U)\\M---h4 =|8„(t/)IKII---IKII-
n
Proof. PM(u) = P'M is a polynomial of degree n — 1, so that
||P(2)M|| ^ ||PU>»|| ^ afl-i(^) Sup ||P<1)M|| S <*n-l(U)an(U),
etc. But
Pfai, ••-,*») - (••■(P<»>(0)«1...K)/n!>
which implies the above result.
If ||ttn|| ^ an for all n ^ 0, then we say that J ^^n *s dominated by 2 <MW>
and write
Lemma 6. // u = 2i° %** ^ <M0 an^ ^M ^ a homogeneous polynomial
of degree n such that ||P(^)|| ^ I for \\u\\ ^ 1, then
P(u) < pn(U)W)*.

60 P. C. ROSENBLOOM
Proof. We have
1
where
y* = 2 P{<Ukv • • • > uk)
the summation being over all sets of indices such that k± -f • • • -f kn = k.
The result now follows from Lemma 5.
An obvious analogue of this lemma holds for polynomials in several
variables.
We are now ready to tackle our main problem. Suppose u is defined by
(2), where un(x) is analytic in D and
\\Un(x)\\ ^ XnS(x)~(n-l) for X E D.
That is, we suppose that
u(t, x) <? 8(x)F(t/8(x)) for xeD,
where
(5) F(z) = £ A„z».
1
Then
||<(*)|| ^ enXn8(x)-n forxeD,
so that
Ux{t, x) < e{tlS)F'{tlS).
Now by Cauchy's inequalities on the coefficients of a power series, we have
\\fkim(x,u9p)\\ ^ M iorxeD, \\u\\ ^ 1, ||p|| ^ 1,
||<£jv(z)|| ^ € for x g D.
Hence
f*im(x,u,ux) < Mpl(U)MV)(8F(tl8)Y[(etl8)Ff(tl8)r
and
f(t, x, u, ux) - f(t, x, 0, 0)
<M ^ t%(U)pm(V)(8F(tl8)y[(etl8)F'(tl8)]™.
l + m>Q
Let
00 00
(6) l(z) = 2MU)zl' «Mz) = ^MV)**.
0 0
(Note that p0(U) = j30(F) may be taken as 1.) Then
f(t, x, u, ux) - f(t, x, 0, 0) <§ M(\ - t)-^(8F(tlB))i>1((etl8)F'(tl8)) - 1}
< M(l - Atl8)-i{<{,(AF(tl8)Wi((etl8)F'(tl8)) - 1}

THE MAJORANT METHOD 61
for 8(x) ^ A < +00. Let w = F(z), z = £/S. We may formulate the
result of this computation as
Lemma 7. If u(t, x) <^ Si^/S), 8 = 8(x),for x e D and
00
U(t, X) = ^ Un(x)tn
1
— = /(£, x, u, ux), u(0, x) = 0 m D,
— - f(t, x, 0, 0) < M(\ - ^-ity^)^^') - 1}
using the notation of (6).
We thus obtain
Theorem 1. If w = F(z) = 2i° ^z*1, An ^ 0 (all n), is the solution of
the differential equation
(7) w' = e(l - Az)-1 + Jf(l - ^2)-i{0(^w?)0i(g2ti?') - 1},
w(0) = 0,
and
u(t, x) < 8F(tl8), for 8 = 8(x), xeD
and
du
— = f(t, x, u, ux), u(0, x) = 0 for xe D,
then
u(t, x) <? 8F{tj8) for xeD, 8(x) ^ A.
To complete the proof, we merely note that
f(t, x, 0, 0) < e(l - 0"1 < ^(1 - ^/S)"1
for xe D, 8(x) ^ A.
Corollary la. If u(t, x) is defined by (2) and (4) and 8(x) ^ A in D, £Aew
m(«, z) <? Si^/S) /or x g D.
// F is analytic for \z\ < p, then u exists and is analytic in ^(p, A); \t\ <
ph(x), 8(x) ^ A.
Proof. Let sN(t, x) = 2f un(x)tny where un(x) is denned by (4), and let
<jjv(£, x) be denned by
-^ = f(t, x, sN-i, (sN-i)x), <tn{0, x) = 0 tor xe D.

62
P. C. ROSENBLOOM
Then
aN(ty x) = SN(t, x) mod tN+1,
that is, the difference aN — sN = tN+1g(t, x), where g is a power series in t
with coefficients analytic in D. The result now follows from Theorem 1 as
soon as we have shown that ||wi(#)|| ^ Ai for x e D. But ||^i(a;)|| ^ e for
x e D, and Ai = e by (7), which starts the induction, and thus completes the
proof.
In order to treat (7) we make the following obvious remarks on majorants.
If (f>i(z, w, q) and <f>2(z, w, q) are power series with non-negative coefficients
and (f>i < <f>2, then the formal power series solutions of
w'v = <f>v(z, wv, zw'v), wv(Q) = 0, v = 1, 2,
satisfy w\ <t w^. If <f>(z, w) and ip(z) are power series with non-negative
coefficients, and 0(0, 0) = 0, 0u,(O, 0) = «oi < 1, 0(0) = 0 and
0(2) « 0(2, 0(2))
while 0i(z) is the formal power series solution of
0!(Z) = 0(2, 0!(Z)), 0x(O) = 0,
then 0 <^ 0i. If 0(2) is a power series with non-negative coefficients and
0(0) = 0, then 0 < 20'.
Let
(8) j8- sup pn(C/)i/», y = supj8„(F)i/»,
so that 1 ^ j8, y ^ e, and j8 = 1 if C/ is Euclidean. Then
0(w) <(l - pw)~\ 0i(w) <g(l - yw)"1.
If w(2) is the solution of (7), then
w'(z) <^ e(l - Az)-1 + Jf(l - Az)-i{{\ - $Aw)-i{\ - eyzw')-1 - 1}
<4 6(1 - Az)-1 + M(\ - Az)-i{(l - (PA + ey)zu/)-i ~ !}•
Hence 2w' <^ v, where v(z) is the solution of
(9) v = 2(1 - ^2)"i{6 + Jf((l - cw)-i - 1)}, c = jM + ey.
Let i = 2(1 - ^2)~i. Then
v = (2c)"1!! - (M - e)c£ - [(1 - (M - e)c£)2 - ±CeQV*}
(10)
= 2e£/{l - (M - e)ci -f [(1 - (if - £)c£)2 - 4ce£]1/2}.
We see that v is analytic for |£| < £1, where £1 is the smaller positive root of
the equation
(1 - (M - e)cQ* - 4c££ = 0,

THE MAJORANT METHOD 63
so that £1 = \jK, K = c(M^ + eW)*, and v(z) is analytic for \z\ < po =
(A + K)~K Now
(11) V(p0) = eU2lc(MU2 + 6l/2) ^ 1/(2C).
Furthermore, it follows from (10) that
v(z) ^ 2ez/{l - (A + (M - e)c)z} = 2ez/(l - Ciz), 0 ^ z ^ po,
where Ci = A + (M — e)c, so that
(12) F(z) = J*J i^z^f1^ ^ (2e/Cl) log ((1 - c^)"1)
for 0 g z ^ po.
We have thus proved
Corollary lb. // u(t, x) is defined by (2) and (4) and 8(x) g .4, £&en u is
analytic in the domain ^(iJTi):
xeD, \t\ < h(x)l{(\ + pKx)A + ey#i},
i.e., where K± = (ilf1/2 + e1/2)2 and fi and y are defined in (8). The coefficients
satisfy
\\Un(x)\\ £ 26Ci»-l/w
H*,*)|| g 8(^)^), ||lfc(*>*)|| g ezJ"(z) £ 6V(2),
wAere z = |£|/S(#).
Note that
\\u(t,x)\\ < 1/(2)8) g 1/2, \\ux(t,x)\\ < l/(2y) <£ 1/2.
For the sake of simplicity of the resulting formulae, we shall often work,
in the rest of this paper, with somewhat cruder estimates. Since poCi < 1,
we have
v(z) S 4ez for 0 g z g p0/2,
(13)
F(z) g 4ez for 0 g z g p0/2,
and therefore
(14) Ht,x)\\ £ 4e|*|, K(*,s)|| S 4e€|*|/8(s)
in the domain 3D2(i£i), where <Da(lf i) is x e D,
a\t\ < 8(*)/{(l + fiKjA + eytfi}.
We also assume that 8(x) ^ A in D.
We now apply these results to the determination of the solution of (1) by
successive approximations.

64 P. C. ROSENBLOOM
Theorem 2. Let u^(t, x) be a sequence of functions analytic in a
neighborhood of the origin such that
(15) *<*>(*, x) < S(x)F(tlS(x)) for k = 0,
(16) u<k + 1)(t, x) = f/(r, x, u(k)(r, x), U(k)(r, x))dr.
Then uW satisfies (15) for all k ^ 0 and is therefore analytic in ^(po, A). If
u is the solution of (I), then
(17) w<*> = u mod t* for k ^ 0,
(18) \\uM(t,x) - u(t,x)\\ ^ 28(x)fji*+iF(po)
in ^([Jipo, A), 0 ^ /x < 1.
Proof. The first assertion of the conclusion follows from Theorem 1.
We then observe that, using the notations of Corollary la, if uM = u mod tk
then uW = Sk mod tk, and therefore u(k+v = o-jt+i = sk+i mod tk+1 from
which (17) follows by induction. If
00
Fk(z) = ^ XnZn>
k + l
then
uto(t, x) - u(t, x) < 28{x)Fk{tlS{x)).
But
Fk(z) ^ (zlpo)k^F(p0),
which yields (18).
Next we study the dependence of the solution on the function /. The
main results flow easily from the following corollary to Theorem 1.
Theorem 3. Let f(X,t,x,u,p) be analytic and \\f\\ ^ M for |A| < 1,
(t, x,u, p) g T). Let u(X, t, x) be the solution of
3u
(19) — - /(A, t, x, u, ux), u(X, 0, x) = 0.
Then u is analytic for |A| < 1, (t, x) e T)i(4ilf).
Proof. Let 0 < a, y = (/x, x), v(t, y) = u(ap,, t, x). We consider y as a
point in X\. Note that if q e V±, the space of bounded linear transformations
on Xi to U, then q can be represented uniquely in the form q = (uo, p),
uoeU,peV with qy = fiuo + px, and we have max (\\uo\\, \\p\\) ^ ||?||
S \\u0\\ + ||p||.
Let F(t, y, v, q) = /(a/x, t, x, v, p). Then we can apply Theorem 1 to the
equation
dv
- = F(t, y, v, q), »(0, y) = 0.

THE MAJORANT METHOD 65
The domain E(lja, D) <= X\ plays the role of the domain D of Theorem 1.
If 81(1/) is the distance from y to the boundary of E(l/a9 D), then &i(y)
= min ((I I a) — |/x|, 8(x)). Then Theorem 1 shows that v is analytic in the
domain
4eyM\t\l{l - (1 + ^M)\t\) < 8i(y) = min ((1/a) - |/x|, 8(s)).
Therefore w(A, £, a:) is analytic for | A| < 1 — a8(x), (t, x) e <Di(4Jf). Theorem
3 now follows since a is arbitrarily small.
If D is a domain in the Banach space X, let %(D, Y) be the space of bounded
analytic functions on D to the space Y, taking as the norm
N(f, D) = sup ||/(a:)||.
D
It is known (see Hille and Phillips [5, p. 112]) that if/ is bounded on D to Y,
then / g g(Z), Y) if and only if f(xQ + Xx) is analytic at A = 0 for all xo e D.
all x e X. Let u(f) be the solution of (1).
Corollary 3a. The function u(f) is analytic on the sphere ^(T), U; M)
= {f\fe %(<$), U),N(f,<£>) < M} to 5(®!(4Jf), J7), am/ tf(ti(/), 3>i(4Jf))
g 1/(20) ^ 1/2. // N(fy ®) < Jf, i%, ®) ^ rilf, 0 < r, then
(20) tf(t*(/ + <?) - u(f), 3>i(4(l + r)i*f)) £ i%, ®)/(2rjBJf),
(21) #(**(/ + gr) - *(/) - u'(f)g, ®i(4(l + r)Jf)) g 9i%, WI^M*).
The function u'(f)g is the solution of the problem
dv
(22) - = fu(t9 x, u(f), u(f)x)v + fp(t, x, u(f), u{f)x)vx + g(t9 x, u(f), u(f)x)y
v(0, x) = 0.
Proof. If fe S(®, C7 ; if), g e 3f(®, J7; rJf), then set h = rMg/N(g, ®),
so that AT(A, S>) = rif, and let <£(A) - ^(/ + AA)(*, a;), (*, x) e $h(4(l + r)ilf).
Then <£ is analytic for |A| < 1, ||</>(A)|| ^ 1/(2)3). We apply Schwarz'lemma
to <£(A) -- </>(0) and then set A - N(g, ^)j{rM)9 which yields (20). We note
that, by Landau's theorem [9, p. 26], j|<£(0) + A<£'/0)|| ^ 5/(80), and then
apply Schwarz' lemma to <f>(\) — </>(0) — A</>'(0) to obtain (21).
Corollary 3b. Iffis analytic in 1) and N(f, T>) < M, and if v is analytic
and \\v\\ < 1, li^H < 1 in E(l, D), and if
then
— - f(t, x, v, vx)\
S e ^ M in E(ly P), v(Q, x) = 0,
N(v - u(f)y 5h(8if)) g €/(4j8if).
Proof. Let g(t, x) = vt — f(t, x9 v, vx), and apply (20).
Equation (22) is a linear differential equation. Linear equations are

66 P. C. ROSENBLOOM
important for their own sake. The general theory can be considerably
sharpened in the case of linear equations.
Let
(23) f(t, x, u, p) = A(t, x) + B(t, x)u + G(t, x)p,
where A, B, and G are analytic on E(l, D) to U, Uu, and Uv respectively.
We assume that
00 00 00
(24) A(t, x) = ^ An(x)t*, B(t, x) = 2 Bn{x)t», G(t, x) = 2 Cn(z)t*,
0 0 0
and that for x e D, n ^ 0, we have
(25) \\An{x)\\ S an, \\Bn(x)\\ ^ bn, \\Cn(x)\\ ^ cn,
and set
CO 00 00
a(z) = 2 anZn, b(z) = 2 bnZn, c(z) = 2 cnZn-
0 0 0
The recursion formula (4) reduces to
ui{x) = .40(#),
(26)
uN+i(x) = (N + l)-1^^) + % BN-k(x)uk(x) + 2 CW(#)t4(aO >•
If
\\uk(x)\\ g A*8(a;)-(*-D foraiGAU^iV,
then by Lemma 1 we have
r N N -\
\\uN+i(x)\\ ^ (N + lHa* + 2 fc*-*A*S-(*~i) + e2fcc^"^8"*f
for S(z) s J if
N N
(N + l)XN+l = *nA* + 2^-*^42y"*+1 + e 2 ic^-*A*il^-*.
i i
Thus if A(z) = 2r Awzw, we obtain the differential equation
A'(z) = a(4z) + Ab(Az)A(z) + ezc(Az)A'(z),
whose solution is
(27) A(z) = ifj(z) exp <f>(z)
where
rw Jo 1 - e^c(^C) Jo 1 - elc(AQ
We now let A —> 8(x) and obtain:

THE MAJORANT METHOD 67
Theorem 4. If u — u(f), where f is defined by (23), and f satisfies (24) and
(25), then u is analytic in the domain Di:
(29) e|'N|t|) < S(*)
and satisfies
u(t, x) < 8(x)A(tj8(x)),
(30)
ux(t, x) <§ e(t/8(x))Af(t/8(x))
where A is defined by (27) and (28) with A = S(#).
Corollary 4a. // ||4(*, z)|| ^ a, || JB(*, z)|| ^ 6, \\C(t, x)\\ ^ c in E(l, D)
then % Is analytic in the domain D\:
(31) |*| < 8(x)l(8(x) + ec), i.e., 8(x) > ec\t\j(l - |*|),
and satisfies there
(32) u(t, x) < 8(x)A(tl8(x)),
where
A(z) = (a/6){(l - 2/20)-^ - 1}, zo = («(*) + ec)-\
i.e.,
/or xe D. Note that A„ ~ (a/b)zQnnbzo as n~> 00.
Corollary 4b. Le£ ^ = w(/) where f is defined by (23). Let A, B, and C
be analytic and bounded for \t\ < R, \\x\\ < S, and let M(R, 8\A), etc., be
their suprema in this "bicylinder." Then u(f) is analytic for
1*1 *ls - N>
11 8 - \\x\\ + eRM(R,S;0)'
and in the domain
I'1 K8- \\x\\ + eRM(R,S;C)' 0 < p < 1,
we have
RSM{R,S;B)
where
a(R, S) =
S - \\x\\ + eRM(R,S;C)
In particular, if A, B, and C are entire functions and C is a function of t
alone, then u is an entire function. If C is at most of degree 1 in x, then u(f)
is an entire function of x for t sufficiently small.

68
P. C. ROSENBLOOM
The observation that the domain of analyticity of u(f) can be estimated in
terms of C alone was already made by Schauder [17]. John [6] has remarked
that if A and B are entire and G is constant, then u(f) is entire. The proof
of this corollary is obtained by making the substitutions t = Rt, x — Sy, and
applying the previous corollary.
Let us illustrate our results by a problem which does not seem to be
immediately tractable by means of the classical Cauchy-Kovalevski theorem.
Let </>(t, x) be analytic in the bicylinder \t\ < l,\x\ < 1, inCz — G\ x G\ and
\</>(t, x)\ ^ k < 1 there. We seek a function v analytic in the neighborhood
of the origin in (7 2 such that
dv
(35) — = vxx(t, W* *))> »(0, x) ee f(x),
where / is a given function analytic in the unit circle. Set v(t, x) = f(x)
-f u(t, x), so that
— = fiftt, x)) + uxx(t, 4>(t, x)), u(0, x) = 0.
Let X = C\ and U = 5(2), Gi), where D is the unit circle in G±, and set
Mt, x) = f*(4>(t, x)), B(t, x)u = uxx((/>(t, x)), C{t, x) = 0. If H| S 1, then
\\B(t,x)u\\ ^ max \uxx(z)\ ^ 4/(1 - k)2 = 6.
By Corollary 4a, there is a unique solution analytic in the bicylinder, and
\u(t,x)\ S (1 - H)(a/6){(1 - |*|)-*/a-W) - 1}
where a = max^i^ |/w(^)|. For example we might take (f>(t, x) = kx, 0 <
i < 1. As fc -> 1, equation (35) approaches the heat equation, and then there
exists a solution analytic in a neighborhood of the origin only if / is an
entire function of order ^ 2.
The essential step in the above method was Lemma 1, so that similar
results can be obtained whenever the coefficients un(x) belong to some
class of functions which is preserved by the recursion relations (4) and for
which there is an estimate, like that of Lemma 1, for the derivative of a
function in terms of the function itself. We shall illustrate this with the
following problem.
Let Z be a complex Banach space with a real subspace X such that every
element z e Z is representable uniquely in the form x -f iy, x,y e X. Let U
be as before, and let V = Uz. Suppose that f(t, z, u, p) is analytic for
\t\ < 1, ||tt|| < 1, \\p\\ < 1, and all z e Z, and that there is a p > 0, and a
K = K(ty u, p) such that
(36) \\f(t, z, u,p)\\ ^ K(t, u,p) exp (,>||2||) for all z e Z,
and further that
(37) \\f(t,x,u,p)\\ ^ M for \t\ < 1, \\u\\ < 1, ||p|| < l.xel

THE MAJORANT METHOD
69
We consider the equation
(i)' Yt =f(t>z>u>uz)> u(°>z) = °>
and try to obtain information about the solution u(f). For this purpose we
need a fairly trivial generalization of a known theorem of Bernstein (see
Boas [2, p. 206]).
First we remark that if IF is a Banach space, and L e Wz, then the
restriction of L to X, which we denote by PL, is in Wx, and P is a linear
transformation of Wz onto Wx with a bounded inverse. Let 17 = r\{Z, W)
= ||P_1||. Then 1 ^ 17 ^ 2. In most common cases 17 = 1. If z = x -f iy,
x, y e X, then we shall write # = $tz, y = !$z.
Lemma 8. // F(z) is an entire function on Z to W and
\\F(z)\\ <; KexV(P\\z\\) (allzeZ),
and
then
and
\\F(x)\\ S-M (allxeX),
\F(z)\\ ^ M exp (P\\3z\\)
\\F'(z)\\ ^ 1PM exp (p\\3z\\), allzeZ.
Proof. Let w* e IF*, ||u>*|| ^ 1, x, y e X, and set cf>(X) = w*F(x + Xy).
Then \</>(X)\ ^ K exp (p||sj| + p||y|||A|) for all A, and \tf>{X)\ ^ ilf for A real.
Hence |</>(A) |g M exp (p||y||3(A)) and in particular |^(i)| ^ Jf exp (p||y||).
Since w* is an arbitrary element of the unit sphere of W*, we obtain the
first inequality. Furthermore, |<£'(0)| = \w*F\x)y\\ S pM\\y\\. Let y0 be
a fixed element of X and apply the first part of the lemma to F'(z)yo = F\(z)
and the second part immediately follows.
Using the same notation as on pp. 56-58, we find that/fcjOT(z, u,p) is an entire
function in z of exponential type p and a homogeneous polynomial in u and p
of degrees I and m, respectively, and that
\\ftim(z,u,p)\\ £ M exp {p\\3z\\),
zeZ,\\u\\ £ 1, |b|| S 1.
From the recursion formulas (4) we obtain ||wi(z)|| ^ M exp (p||-3z||).
Suppose that ui(z),- • •, un(z) are entire functions of z = x + iy, x, y e X,
and that
\\uk(z)\\£Mkexp(kp\\y\\)9 k=lr.-,N, allzeZ.
Then
(N + 1)K + 1(*)|| *2Mexp{p\\y\\)el+'*
■K(*)ll-- -K(*)ll IK(*)II-HK(*)II
•exp (p||y|| + Z + ra + p||y||(vi + • • • + am)),

70 P. C. ROSENBLOOM
the summation being over all sets of indices such that
k + vi + • • • + vi + ori + • • • + am = iV, fc ^ 0, Z ^ 0,
m £ 0, v< £ 1, o-i ^ 1.
Hence
11^+1(2)11 £ MN+! exp ({N + l)p||y||), afl 2 e Z,
if
(tf + l)ifjv+i = if 2 (eMn)' "(eMVl)(r)eaiMai)- -(rjeamM0m).
Let
^(T) = 2Jf^,
i
so that
00
T0'(t) = ^ vMvTv.
1
Then 0 is the formal power series solution of the equation
0' (T) = Jf/(1 - r)(l - 4(r))(l - Wrf (t)), 0(0) = 0.
Therefore
r<f>' <4 Mrj{\ - T)(l - W)(l " W0')>
so that
r<f>' < (rje)-^(r)
where 0 is the solution of
(38) <?(0) = 0(1 - 0)(1 - pxfs) = veMrl(l - r), 0(0) = 0.
By the implicit function theorem there is a unique solution 0(t) analytic in a
neighborhood of the origin, and since its Taylor coefficients are positive, its
first singularity is on the positive real axis. Now g(ifj) has an analytic
inverse function on the interval 0 < 0 < 0o, where 0o is the smallest positive
root of gr'(0) = 0:
(39) 0o = (1 + p + [1 - p + p2]*/2)-1.
Let k = k(p, M) = g(ipo)l(rjeM). Then 0 is analytic for |t| < to =
fc/(l -f k). In this circle |0(t)| ^ 0o, so that
0(t) <^ 0ot/(to - t), 0(t) ^ 0ot/to,
and
0(t) <^ — log (t0/(t0 - t)), 0(t) <; 0oT/(?]eTo).
We have thus proved
Theorem 5. // / satisfies (36) and (37), JAew -a = u{f), the solution of
(1)' is analytic in the domain
(40) |*| exp (p\\y\\) < to = g(ipo)l(veM + g{if,0)),

THE MAJORANT METHOD
71
vjhere g(i/j) and «^o are defined by (38) and (39). In the domain (40) we have
00
u(t9 X) = ^ Un{z)tn
1
where un(z) is an entire function and
K(«)|| * (0oTO-» exp (»/>([ y||), all z e Z,
and
\\u(t,x)\\S^JtleXVr{pM)-
It would be easy to introduce refinements in terms of an estimate of
f(t, z, 0, 0).
By the same method we can obtain results on the Goursat problem:
To find u(t, t, x) analytic in the neighborhood of (0, 0, 0) in C\ x d x X
such that
—— = f(ty T, X, U9 lit, UT9 Ux, UtXy UTX, UXX) Utt, Utt),
u(t9 0, x) = u(09 t, x) = 0.
Here f(t, t, x, u\, u^ u^ p±y p2, ps, q> £>4, Pb) is assumed to be analytic on
the domain: \t\ < 1, |t| < l9 xeCl, \uj\ < 1, \pi\ < 1, ||?|| < 1 in the
space CixCixXxUxUxUx V x V x V x Vx x U x U to the space
U. O is a domain in X satisfying a condition of the form:
8(x) < K in £1
We define
A =/*4(0, 0,z,0, 0,..-,0).
and
B =/pB(0,0,x,0,0,...,0).
We can prove that a solution exists under either of the following
conditions :
(a) ||-41| ||B|| < k < 1/4 for x e fi, where k is a constant, or
(b) AB = BA in O, and the spectrum of AB is at a positive distance
from the ray A ^ 1/4.
In the case where U — Ci, so that A and 1? are scalars, the existence of a
solution under condition (a) was proved by Riquier (see [15, p. 39]), and
under condition (b) Gyunter [4]. The case where U = Cn, n > 1, seems to be
new.
We hope to give in a later paper full details of these results and also
applications to singular Cauchy problems for equations of the form
tk — = f(t9 x, u, ux).

72
P. C. ROSENBLOOM
It is clear that the present version of the majorant method is very flexible
and lends itself to the derivation of quite explicit results in a variety of
problems.
Bibliography
1. S. Bernstein, Leons sur les proprietes extremales et la meilleure approximation des
fonctions analytiques d'une variable reelle, Paris, Gauthier-Villars, 1926.
2. R. P. Boas, Entire functions. New York, Academic Press Inc., 1954.
3. R. Gateaux, Fonctions d'une infinite des variables independantes, Bull. Soc. Math.
France vol. 47 (1919) pp. 70-96.
4. N. M. Gyunter, On analytic solutions of the equation s = f(x, y, z, p, q, r, t), Mat.
Sb. vol. 32 (1924) pp. 26-42.
5. E. Hille and R. S. Phillips, Functional analysis and semigroups, Amer. Math. Soc.
Colloquium Publications, New York, 1957.
6. F. John, On linear partial differential equations with analytic coefficients, Comm. Pure
Appl. Math. vol. 2 (1949) pp. 209-253.
7. L. V. Kantorovich, Functional analysis and applied mathematics, Translated by
C. D. Benster, U.S. Dept. of Commerce, Nat. Bur. Standards, Los Angeles, NBS Rep.
1509 (1952).
8. O. D. Kellogg, On bounded polynomials in several variables, Math. Z. vol. 27 (1927)
pp. 55-64.
9. E. Landau, Darstellung und Begriindung einiger neuerer Ergebnisse der Funk-
tionentheorie, Berlin, Springer, 1929.
10. P. Levy, Sur les equations aux derivees fonctionelles, Journal de l'Ecole Poly-
technique, ser. 2, no. 17 (1913).
11. R. S. Martin, Contributions to the theory of junctionals, California Institute of
Technology Thesis, 1932.
12. A. D. Michal, On a non-linear total differential equation in normed linear spaces,
Acta Math. vol. 80 (1948) pp. 1-21.
13. A. D. Michal, On bounds of polynomials in hyperspheres and Frechet-Michal
derivatives in real and complex normed linear spaces, Math. Mag. vol. 27 (1954) pp. 119—
126.
14. G. Polya and G. Szego, Aufgaben und Lehrsdtze aus der Analysis, Berlin, Springer,
1925.
15. C. Riquier, La methode des fonctions majorantes et les systemes d'equations aux
dirivees partielles, Memorial des Sciences Mathematiques, Fasc. 32, Paris, Gauthier-
Villars, 1928.
16. P. C. Rosenbloom, Proceedings of the International Congress of Mathematicians,
vol. I, Cambridge, 1950, p. 442.
17. J. Schauder, Das Anfangswertproblem einer quasilinearen hyperbolischen Diffe-
rentialgleichung zweiter Ordnung in beliebieger Anzahl der unabhdngigen Verdnderlichen,
Fund. Math vol. 24 (1935) pp. 213-246.
18. E. W. Titt, An initial value problem for all hyperbolic partial differential equations
of the second order with three independent variables, Ann. of Math. vol. 40 (1939) pp. 862-
891.
University of Minnesota,
Minneapolis, Minnesota

A PRIORI ESTIMATES FOR ELLIPTIC AND
PARABOLIC EQUATIONS
BY
FELIX E. BROWDERi
During the past several years, an extensive study has been made by various
writers of the regularity of solutions of elliptic boundary value problems.
(A comprehensive bibliography appears at the end of this paper.) It is the
object of the present note to summarize some of the most recent results
obtained by the author concerning a priori estimates for elliptic and parabolic
boundary value problems and their application to the question of the
existence of solutions for these problems. We do not have the space here to
describe in detail the relation of these results to those obtained by other
participants in the present symposium, specifically Nirenberg and Schechter,
except to remark that our treatment of parabolic equations seems to be
unique within the framework of these roughly parallel lines of development.
The content of the results which we shall discuss may be divided into
three main parts:
(I) A priori estimates and existence theorems for general elliptic boundary
value problems.
(II) A priori estimates for mixed initial-boundary value problems for
general parabolic equations.
(III) Stronger a priori estimates for second-order parabolic equations and
some related existence theorems in the large for quasi-linear parabolic initial
value problems.
In connection with topic (III), the writer is very much indebted to Jurgen
Moser for several stimulating conversations in which the latter described his
important simplifications in the technique of obtaining a priori estimates of
the Di Giorgi-Nash type.
Section 1. Let A be a linear elliptic differential operator of order 2m with
complex-valued coefficients defined on a domain G of Euclidean n-space En.
We may write A in the form
(1.1) A = ^ aa(x)D*9
1 Sloan Foundation Fellow,
73

74
FELIX E. BROWDER
where, for each multi-index a = («i,- • •, an) of non-negative integers, Da
denotes the partial differential operator
PI (Dj)a>> D, = i-i A and \a\ = £ «j.
Let To be an open subset of the boundary T of G. We consider the
boundary value problem
(1.2) Au = f in G; BjU = 0 on To, lgj^w,
where Bj is a differential operator of order rj for each j, (rj < 2m), with
coefficients defined on I\/is a given function defined on G, and the unknown
function u is sought from some class of functions defined on G.
Writing Bj = 2i0l ^ ^j^^D®, we define the characteristic forms a(xy £) of
^4 and bj(x, Q oi Bj by
(1.3) a(*,£)= ^ a«(*)5«; M*,S) = 2 &J.*(*)5*.
|a| = 2m |/3| = ry
where £a = n* (£*)"'•
We suppose the boundary r of G smooth enough to have a unit normal
vector N(x) at each of its points.
Definition. The boundary value problem (1.2) is said to be regular at
the point xo of T if both of the following conditions are satisfied:
(i) For each unit tangent vector T to V at xo, the polynomial a(xo, T -f
AiV (xo)) in the single complex variable A has exactly m roots (counting
multiplicities) in the upper A half-plane.
(ii) If CXQtT is an oriented rectifiable Jordan curve in the upper A half-
plane containing the m roots described in (i) in its interior, then
(1.4)
Det | f A*-i&;(zo, T + XN(x0))[a(x0i T + A^o))]-1^!]
^ c > 0,
with a constant c independent of the unit tangent vector T.
The a priori estimates with which we are concerned in the present section
are framed in terms of two families of norms, the W1>* and C^h norms.
Definition. For each non-negative integer j and exponent p with
1 < p < oo, W^p(G) is the family of functions in Lv(G) all of whose
distribution derivatives of order <j lie in Lp(G). W*>p(G) is a reflexive Banach space
with respect to the norm
(1.5) Hir'.p = 2 \\Dauhp-
Definition. For each non-negative integer j and exponent h with
0 < h < 1, C^h(G) is the space of functions satisfying a uniform Holder

ESTIMATES FOR ELLIPTIC AND PARABOLIC EQUATIONS 75
condition with exponent h on G together with all derivatives of order <j.
C1>h(G) is a Banach space with respect to the norm
(1.6) H|c>'A = H|c' + 2 SUP {\Dau(x) - Dau(y)\-\x - y\~h).
\a\=j x,yeG
We shall say that u lies in W\$. (G U To) if each point x of G U To has a
neighborhood V such that the restriction of u to G n V lies in W^^(G n F).
A similar definition follows for (7^(0 u T0).
All of the theorems which we state below in §§1 and 2 are established in
[9; 10; 11]; most of them were announced in [5; 6; 7; 8]. The basic a priori
estimate in W2m>* was announced by the writer in the Proceedings of the
American Mathematical Society, 1956, p. 382.2 The definition of regularity
of a boundary value problem is due to Lopatinski [17].
We shall assume without explicit mention in the results which we state
below that the domain G on which our boundary value problem is defined is
uniformly smooth, to be precise, that it is uniformly regular of class C2m and
locally regular of class C4m. (The definition of uniform regularity is given
in [7].) We shall assume also that r0 is a smooth piece of T, as made precise
in [10]. With our regularity assumption on G, it follows that if u lies in
Wf™tP(G U T0), then the derivatives of u of order < 2m are defined in an
essentially unique way on T0 to lie in Lfoc (T0). We shall define our boundary
value problem (1.2) using these generalized derivatives, which by the Sobolev
imbedding theorem will be continuous derivatives for p > n (and for lower
p for the lower order derivatives).
Theorem 1. Let u be a function in W2™,pi(G U T0) for some p± > 1, with
u in Lv(G), and suppose that uis a solution of the boundary value problem (1.2)
for a function fin Lv(G) (1 < p < go). Suppose that the top-order coefficients
of A are uniformly continuous on G, A is uniformly elliptic on G, all coefficients
of A are uniformly bounded, and that Dabjt0 is uniformly continuous for \a\ ^
2m — rj. Suppose that the boundary value problem (1.2) is regular at each
point of To with a regularity constant c uniform over each subset of To whose
distance from T — To is bounded away from zero.
Then for every subdomain G\ of G with dist (Gi, T — To) ^ do > 0, u lies
in W2m>v(Gi), and we have an inequality
(1.7) \\u\\w2m,P(Gi) ^ k(do){\\u\\Lr>iG) + ||/|Up(o}
(k(do) independent of u).
Theorem 2. Suppose that in Theorem 1, / lies in C°>h(G), all of the
coefficients of A lie in C°>h(G), D^bj^eC^T) for \a\ ^ 2m - rh and G
uniformly regular of class C2m+h. Then u lies in C2m>h(Gi) and there holds an
inequality
2 All the estimates in the W2m>P spaces are based, of course, upon the well-known
singular integral theory in LP due to Calderon and Zygmund.

76
FELIX E. BROWDER
|K||ca-.*(G1) i Hdo){\\f\\c°'*(CF) + M|c°.*<G)}
(k(do) independent of u).
The information which one may extract from Theorems 1 and 2 about the
existence of solutions to elliptic boundary value problems is easiest to
formulate in operator-theoretic terms. We consider a number of operators
4o,p, ^4i,p, A0th, A\th all obtained by restricting the differential operator A to
a specific family of functions u satisfying variants of the boundary conditions
in (1.2). More specifically (if we let D(T) denote the domain of the operator
n
D(A0fP) = {u:ue W2m>*(G), BjU = 0 on T0 for all j,
D*u = 0 on T - T0 for |a| < 2m},
D(Ahp) = {u:ue W&*{G U T0), An e L>(Q),
BjU = 0 on To for all j},
D(A0fh) = {u: ueC2m>h(G), BjU = 0 on T0 for all j,
Dau = 0 on T - To for |a| < 2m},
D{A1Jk) = {u:ueCfch{G U T0), Au eC°.*(G),
BjU = 0 on To for all j}.
For u in the domain of any of these operators, Tu = Au. A0,p and A\tV
are operators in J>(6?), A0,h and Aifh in C°>h(G).
One of the simplest consequences of Theorems 1 and 2 is that when
r = To, we have AotP = A\tP and Aotn = Aitk. In this case, we set
Ap = A0,p = AlfP and Ah = A0,h = Aith.
Theorem 3. A0tP and AifP are closed operators in Lv(G). Aq^ and A\th
are closed operators in C°>h(G). If G is a bounded set, the ranges of all these
operators are closed. The operators Ao,p, Ao,h have finite-dimensional null
spaces. The co-dimension of the ranges of AitP and A±th are finite.
We suppose now that A' is another elliptic differential operator on G,
(B'lf- • •, B'm) another family of boundary differential operators. Let Aq^
A[q be defined in Lq(G) with respect to the portion ro of V and the family
of boundary operators (JBJ,-• •, JB^), with q = p(p — l)"1. Then the
boundary problem (A, Bj) is said to be adjoint to (A\ B]) on T0 if
(1.8) {Au,v) = (uyA'v)
for u 6 D(A0p), v e D(Af0q), with (/, g) the inner product between Lp and L9.
Theorem 4. Suppose that both of the boundary value problems (A, Bj) and
A\ jBj), which are mutually adjoint, satisfy the conditions of Theorem 1 on G.
Then the adjoint (Ao,p)* of AqiP (in the sense of complex Banach spaces) is
A[q9 while the adjoint of Alp is A'0q.

ESTIMATES FOR ELLIPTIC AND PARABOLIC EQUATIONS 77
A simple extension of Theorem 4, which is essentially a corollary, is the
following:
Theorem 5. Suppose that we are given r disjoint smooth open pieces r<*> of
T, and on each T(k) a pair of boundary problems (A, B(k)) and (A', Br^k)) which
are adjoint on TW. Suppose that each of these problems satisfies the conditions
of Theorem 1 on each T<*) and that T — [J^ TM has no interior. We define the
operators Ap and Arq in LP(G) and Lq(G) with the domains
D(Ap) = {u:ue Wf™*(G u \Jk r<*>), B^u = 0 on r<*> for all j and k,
Au g Lp(G)}9
D(A'q) = {u:ue W2m«(G), B?k)u = 0 on r<*> for all j and k,
D°u = 0 on T - To for |a| < 2m - 1}.
Then the operators Ap and Aq, considered as operators in LV(G) and Lq(G),
q = p(p — l)"1, are one another's adjoints. In particular, if the domain G is
bounded, the ranges of these operators are closed, so that by the Hahn-Banach
Theorem, f lies in the range of Av if and only if (f, v) = 0 for all v in the null-
space of Aq.
Section 2. Turning now to parabolic problems, we specialize our
hypotheses by assuming Dirichlet boundary conditions of order m, i.e., that
Bj — (d/dnY"1. All of the following results are valid for any set of Bj having
real top-order coefficients which when evaluated at any boundary point
yield a self-adjoint semi-bounded boundary problem for the constant
coefficient operator obtained by evaluating A at the same point.
Theorem 6. Let Ap be the realization of A under null Dirichlet boundary
conditions'in Lp(G) (An the realization of A under null Dirichlet boundary
conditions in C°>h(G)). Suppose that A is uniformly strongly elliptic and has
real-valued top-order coefficients, as well as satisfying the regularity conditions
of the hypothesis of Theorem 1 (or Theorem 2). Suppose further that either:
(a) G is bounded
or,
(b) The formal adjoint A' of the differential operator A also satisfies the same
regularity conditions.
Then, for all complex A with Re (A) ^ 0 with |A| ^ Xp ^ 0, the operator
(Ap -f A/)-1 exists on all of Lv(G) and satisfies the inequality
\\{AV + A/)-i|| S Jf,|A|-i.
(Similarly, (Ah -f A/)-1 will exist for all complex A with Re (A) ^ 0 with
|A| ^ Xh > 0 on all of C°>h(G) and satisfy the inequality
\\(Ah + A/)-i|| ^ Jf|A|-i.)
From Theorem 6, using the results of Kato [15] and Tanabe [33], which
generalize to time-variable operators the semi-group theory of Hille, Yosida,

78
FELIX E. BROWDER
and Phillips, we derive an existence theorem for the solution of the mixed
initial-boundary value problem:
Su
(2.1) —=-A(t)u\ u(t)-+g as£-^0; Bju = 0 on V for t > 0,
dt
where u(x, t) is defined on G x {t > 0}, lies in D(AP) or D(Ah), respectively, for
all t > 0, where A(t) is a uniformly elliptic differential operator on G for
each t, satisfying the conditions of Theorem 6 and with coefficients which are
continuously differentiate in t. The condition that u lies in D{AP) implies,
of course, that for t > 0, the Dirichlet data of order m of u vanishes on T.
By the Sobolev imbedding theorem, for (m -f \)p > n, the data is taken on
in the classical sense.
We close this section with some brief remarks on the proof of Theorem 6
in the case of the 2>-spaces. By a partition of unity argument, it suffices to
consider A homogeneous and having constant coefficients on a half-space and
to prove that
c0|M|^2m'p + ci|A|-||^||lp(g> ^ \{AV + XI)u\\Lp(G)*
It suffices by Theorem 1 to take Co = 0, since
||w||^'P S c2\\Avu\\Lv + \\u\\Lp ^ C2{\\(AP + XI)u\\lp + (|A| + 1)H|l*}.
Since we may assume A homogeneous with constant coefficients, it follows
by homothetic transformations that the sought inequality: |A| \\u\\u> ^
C2||(^4p + M)u\\li>, will follow from the uniform inequality
cz\\u\\Lv S \\(AP + (I)u\\^
for all complex numbers £ with Re (|) ^ 0, |f| = 1. This last fact follows
from an integral representation of (Av + £I)~l on the half-space.
One final remark. If the initial function g lies in the common domain of
all the A(t), then w(«, t) will do so for t ^ 0, and we shall have a uniform
bound for its norm in W2m>v(G) (or in C2m>h(G)). By the uniqueness theorem
for solutions of the initial-value problem (2.1) which follows from our
information on the spectrum, we obtain an a priori bound for any solution of
(2.1) with smooth initial function if we know only that for each t > 09u(*,t)
lies in D(AP).
Section 3. Passing in this final section to the case of a second-order
operator A (t) in divergence form
(3.1) %Di(aik(x,t)Dk)9
we consider the regularity of the solution of the parabolic problem (2.1) under
the assumption only of uniform ellipticity and the uniform boundedness of
the coefficients. Di Giorgi [12] obtained the Holder continuity of weak

ESTIMATES FOR ELLIPTIC AND PARABOLIC EQUATIONS 79
solutions of (3.1) in the interior, while Nash [23] obtained similar results for
parabolic equations by a different method. Di Giorgi's methods have been
applied to obtain regularity up to the boundary by Morrey [22] and Stam-
pacchia [31; 32]. Recently J. Moser, in some as yet unpublished3 work, has
obtained a drastic simplification of the Di Giorgi interior regularity proof.
Using a variant of his method, we obtain the following result for parabolic
equations:
Theorem 7. Suppose A(t) of the form (3.1), uniformly elliptic and with
coefficients uniformly bounded for t bounded. Then every weak solution of (2.1)
is Holder continuous for t > 8 > 0.
Applying the quantitative version of Theorem 7, which we do not state
here for the sake of brevity, one may derive existence theorems of the
following type:
Theorem 8. Consider the quasi-linear parabolic equation of the form
(3.2) "^ = " 2 Di(aM*, *> u)D*u)
where a^ is C00 in all its variables, and uniformly elliptic and bounded on a
C™-smooth G for t and u bounded. Consider the initial-boundary value problem
for (3.2) with side conditions,
(3.3) u(',t) satisfies null Dirichlet boundary conditions for t > 0,
(3.4) u(-,t)->g as J -^ 0.
Then there exists one and only one solution u of (3.2, (3.3), and (3.4), which
is C™ for t > 0.
Bibliography
1. S. Agmon, Multiple layer potentials and the Dirichlet problem for higher order
elliptic equations in the plane. I, Comm. Pure Appl. Math. vol. 10 (1957) pp. 179-239.
2. , The coerciveness problem for integro-differential forms, J. Analyse Math.
vol. 6 (1958) pp. 183-223.
3. S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions
of elliptic partial differential equations satisfying general boundary conditions. I, Comm.
Pure Appl. Math. vol. 12 (1959) pp. 623-727.
4. F. E. Browder, On the regularity properties of solutions of elliptic equations, Comm.
Pure Appl. Math. vol. 9 (1956) pp. 351-361.
5. , La theorie spectrale des operateurs aux derivees partielles du type elliptique,
C.R. Acad. Sci. Paris vol. 246 (1958) pp. 520-528.
6. —, Les operateurs elliptiques et les problemes mixtes, C.R. Acad. Sci. Paris vol.
246 (1958) pp. 1363-65.
3 Added in proof. Moser's proof has since appeared in Comm. Pure Appl. Math. vol.
13 (1960) pp. 457-468.

80
FELIX E. BROWDER
7. F. E. Browder, Estimates and existence theorems for elliptic boundary value problems9
Proc. Nat. Acad. Sci. U.S.A. vol. 45 (1959) pp. 365-372.
8. f The spectral theory of strongly elliptic differential operators, Proc. Nat.
Acad. Sci. U.S.A. vol. 45 (1959) pp. 1423-1431.
9. , A-priori estimates for solutions of elliptic boundary value problems. I and II,
Nederl. Akad. Wetensch. Proc. Ser. A vol. 63 = Indag. Math. vol. 22 (I960) pp. 145-159,
160-169; III, to appear.
10. , On the spectral theory of elliptic differential operators. I. Math. Ann.
vol. 142 (1961) pp. 22-130.
U, 9 On the spectral theory of elliptic differential operators. II, to appear.
12. E. Di Giorgi, Sulla differ-enziabilita e Vanaliticita delle estremali degli integrals
multipli regolari, Mem. Accad. Sci. Torino CI. Sci. Fis. Mat. Nat. Ser. 3 vol. 3 (1957)
pp. 25-43.
13. O. V. Guseva, On boundary problems for strongly elliptic systems, Dokl. Akad.
Nauk SSSR vol. 102 (1955) pp. 1069-1072.
14. L. Hormander, On the regularity of the solutions of boundary problems, Acta Math,
vol. 99 (1958) pp. 225-264.
15. T. Kato, Integration of the equation of evolution in a Banach space, J. Math. Soc.
Japan vol. 5 (1953) pp. 208-234.
16. A. I. Koshelev, On a-priori estimates in LP of generalized solutions of elliptic
equations and systems, Uspehi Mat. Nauk. vol. 13 (1958) pp. 29—88.
17. Ya. B. Lopatinski, On a method of reducing boundary problems for a system of
differential equations of elliptic type to regular equations, Ukrain. Mat. Z vol. 5 (1953)
pp. 123-151.
18. J. L. Lions, Lectures on elliptic partial differential equations, Tata Institute
lectures, Bombay, 1957.
19. E. Magenes, Sul problema di Dirichlet per le equazioni lineari ellitiche in due
variabli, Ann. Mat. Pura Appl. vol. 48 (1959) pp. 257-279.
20. E. Magenes and G. Stampacchia, J problemi al contorno per le equazioni differenziali
di tipo ellitico, Ann. Scuola Norm. Sup. Pisa vol. 12 (1958) pp. 247-358.
21. C. Miranda, Teorema del massimo modulo e teorema di esistenza e di unicita per il
problema di Dirichlet relativo alle equazioni ellitiche in due variabli, Ann. Mat. Pura
Appl. vol. 46 (1958) pp. 265-312.
22. C. B. Morrey, Jr., Second order elliptic equations in several variables and Holder
continuity, Math. Z. vol. 72 (1959) pp. 146-164.
23. J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math,
vol. 80 (1958) pp. 931-954.
24. L. Nirenberg, Remarks on strongly elliptic partial differential equations, Comm.
Pure Appl. Math. vol. 8 (1955) pp. 649-675.
25. 9 Estimates and existence of solutions of elliptic equations, Comm. Pure
Appl. Math. vol. 9 (1956) pp. 509-530.
26. J. Peetre, Theoremes de regularity pour quelques classes d'operateurs diffirentiels,
Thesis, November, 1959, University of Lund, Sweden.
27. M. Schechter, Integral inequalities for partial differential operators and functions
satisfying general boundary conditions, Comm. Pure Appl. Math. vol. 12 (1959) pp. 37-66.
28. , General boundary value problems for elliptic partial differential equations,
Comm. Pure Appl. Math. vol. 12 (1959) pp. 457-486.
29. L. N. Slobodetsky, Estimation of solutions of elliptic and parabolic systems, Dokl.
Akad. Nauk SSSR vol. 120 (1958) pp. 468-471.
30. , Inequalities in LP for solutions of elliptic systems, Dokl. Akad. Nauk
SSSR vol. 123 (1958) pp. 616-619.
31. G. Stampacchia, Contributi alia regolarizzazione delle soluzioni dei problemi al

ESTIMATES FOR ELLIPTIC AND PARABOLIC EQUATIONS 81
contorno per equazioni del secondo ordine ellitiche, Ann. Scuola Norm. Sup. Pisa Ser. 3
vol. 12 (1958) pp. 223-244.
32. , Problemi al contorno ellitici con dati discontinui dotati di soluzioni Holder-
iane, Mimeographed Preprint, Genova, February, 1960.
33. H. Tanabe, A class of equations of evolution in a Banach space, Osaka Math. J.
vol. 11 (1959) pp. 121-145.
Yale University,
New Haven, Connecticut

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DIFFERENTIAL EQUATIONS IN HILBERT SPACES
BY
FRANgOIS TREVES
The theory of boundary value problems and initial value (or else Cauchy)
problems for linear partial differential equations in a space-time situation
leads to the study of ordinary differential equations whose coefficients are
linear operators in appropriate Hilbert spaces. We start by showing, on
two significant but particular cases,1 how the transition occurs. We describe
then a method of solution of certain of the problems in the abstract Hilbert
spaces set-up, here again restricting ourselves to a very simplified situation.
1. Mixed problems in cylindrical domains. Let Q, be an arbitrary open set
in the euclidean space Rn (in which the variable will be denoted by x =
(%i,''' ,Xn))- We denote by Hm(£l) the space of functions f(x) whose
derivatives, in the sense of distributions, belong, up to the order m, to L2(Q).
This space is provided with the norm2
i/u = ( 2 f mn^dx)112.
The closure of C${Q) in Hm{Q) will be called JETJftQ).
For every number t ^ 0, we are given a linear partial differential operator
in x:
a= 2 (-i)lfl%(^w+ 2 (-i)lpl-»x(*>*)-
\p\,\q\ = m \p\ = m
We require A to be uniformly elliptic in Q,; no uniformity is required with
respect to t. The coefficients apq(t, x) and ap(ty x) have to be functions of t—
with some kind of regularity—valued in L™(Q,).
We set, for u, v e Hm(£l):
a(t;u,v)= ^ f apq{t,x)Diu(x)D$t>(xjdx+ 2 f ap(t, x)u(x)D>v(x)dx.
\p\,\q\=m J" \p\=m J"
Our restrictions on the coefficients apq(t, x) and av(t, x) imply that there is
function M(t) ^ 0 such that, for every u, v e Hm(Q) :
(1) \a(t\ u, v)\ ^ M(t)\u\m\v\m (for every t).
1 The first of which is well-known.
2 p stands for a multi-index (pu • • •, pn) of non-negative integers ; \p\ = pl + • • • -f-
pn; Dl = (dldxl)*i...(dldxH)'>n.
83

84
FRANgOIS TREVES
The uniform eUipticity of A implies, on the other hand, the existence of a
real function X(t) and of a function a(t) > 0 such that, for every u e Hm(Q):
(2) Rea(t;u,u) + A(OM§ ^ «(*)M5r
A boundary value problem in Q. for the operator A (t being here fixed), in
the weak sense, turns out to be equivalent to the data of a closed linear
subspace V of Hm(Q), H%(Q) c V <= Hm(£l). For instance, given f(x) e
L2(ti), one looks for an element u of V such that, for every v e V,
(3) a(£; -a, v) = f(x)v(x)dx.
Jo
If w is a solution of (3), v ->a(t; u, v) is continuous on V for the norm \v\o
= (Jo |v(z)|2dz)1/2. This fact, together with the explicit form of a(t\ u, v),
leads to the "concrete" interpretation of the problem. Thus one sees that
the choice V = #Jf(£i) generalizes the Dirichlet problem and the choice
V = Hm(Q) generalizes the Neumann problem. Examples of intermediate
cases are easy to construct.
To such boundary value problems in D, correspond mixed problems in the
cylindrical region jRJ x Q, for the "evolution" operator djdt -f A. Given a
function/(£) valued in Ll(Q.), we look for a function u(t) valued in the Hilbert
space V such that:
(i) u(0) equals a given element of V;
(ii) for every v e V and every t > 0,
^(t),Jjo + a(t;u(t),v) = (f(t),v)0.
Of course, some kind of regularity with respect to t has to be imposed on both
f(t) and u(t). The fact that u(t) is valued in V and the equation in (ii)
mean that, for t > 0 and x e ti, du/dt -f Au = / and u satisfies (for every
t > 0), at the boundary of D, the conditions defined by V (all that, in the
weak sense). Moreover, according to (i), the latter is also true at t — 0;
inside D, the function of x u(0) takes pre-assigned values compatible with
the boundary conditions.
From inequality (1) we derive that, for every t, there is a bounded linear
operator A(t):V -> V such that a(t\u, v) = (A(t)u, v)my ( , )m being the
inner product in Hm(£l), and therefore in V. We shall call Ao(t) the her-
mitian part of -4(0- On the other hand, there is a bounded operator
K: V -> V such that (Ku, v)m = (u, v)o. Inequality (2) means that for every
t, Ao(t) -f X(t)K is a positive (bounded) operator on V. And condition (ii)
in our mixed problem reads:
K ^ + A(t)u = Jf(t)
where J is the bounded operator H°(£l) -> V defined by (Jf, v)m = (/, v)0,

DIFFERENTIAL EQUATIONS IN HILBERT SPACES 85
/ g H°(ii), v 6 V. Anyway, we are reduced to an initial value problem for the
ordinary differential operator Kd/dt -f A(t).
One may also generalize classical mixed problems for hyperbolic equations,
by considering the operator Kd^/dt2, -f B(t)djdt -f A(t), in which A(t) is just
defined as above, except that it must be now hermitian, and B(t) is an
operator V -> V such that
\(B(t)uf v)m\ ^ Jf^OMoMo,
in other words "comes" from a bounded operator H°(Q,) -> H°(Q,).
We must underline the fact that this abstract problem covers much more
than what we have said here: for instance, ti, instead of being an open set in
RXi could be some diflFerentiable manifold; or the coefficients apq(t, x) and
ap(t, x) could be N x N matrices functions of t and x. In that case, the
solution/and the data, as functions of (t, x), should be vector-valued, precisely,
should have complex iV-vectors as values. This would cover the case of
certain differential systems. In those various cases, appropriate redefinitions
of the functional spaces involved, like Hm(£l), are of course needed.
2. Initial value problems in a half space. Let us consider a linear partial
differential operator in R\ x R% of the form
A = ^ + 2 A#> x> ^ aF* (m = l)-
For simplicity, we suppose the coefficients of the operators Aj(t, x, Dx) on
B% to be, for every t, (7°° with respect to x and bounded on B% as well as all
their derivatives. Let Sj be the order of Aj(t, x, Dx) and a = sup Sj.
We denote by H% (s being any real number) the completion of C™(RX\) for
the norm
/2
l*|.= (JV + \y\2)s\$(y)\2dy)U2
</>(y) being the Fourier transform of <j>{x). Hs is an Hilbert space, in which
the inner product is given by
(f Ms = J
(1 + \y\*)4{y)${y)dy.
For every t, Aj(t, x, Dx) defines a bounded linear operator H% -> Hsx '*, a
fortiori H%-> H8X~°.
We may then pose problems of the following kind. Given a function
/(£, x) of t valued in HTX, we look for a function u(t, x) of t valued in Hx
satisfying:
(i) Au(t, x) = 0 (in some sense) for t > 0;
(n) u(0, x) = u0, -^ (0, x) = uu • • •, ^rr (0, x) = um-i,

86
FRANCOIS TREVES
Uj being given in Hxi for j = 0, 1,- • •, m — 1, with (usually): s = t0 ^ rx ^
• • • ^ Tm-i. Here again, of course, conditions have to be imposed upon /
and u. And one tries to solve such problems for certain types of operators
A. Classically, for hyperbolic (in that case, a — m) or parabolic (then
a = md for some even integer d > 0). These two types are contained in a
wider one: the class of operators called correct by Gelfand and Silov (and
introduced independently in Treves [3] under the name of solvable).
Existence and uniqueness of the solutions to the above kind of problems can be
proved for correct operators.
We see that we have again to deal with ordinary differential operators, this
time of the form:
(4) L = K-%- + y AM)-;—,
v ' dtm A ;v ' dtm~3
3 = 1
where the coefficients Aj(t) are bounded linear operators from an Hilbert
space V into another one 3. Here, K is the canonical injection of V = Hsx
into 8 = H8x-°.
3. The abstract problem. Let L be a differential operator of the form (4).
One may expect to be able to solve problems of the following sort: given an
Hilbert space H imbedded in H and whose norm is larger than the one induced
by ff, and given a function f(t) valued in H, find u(t) valued in V satisfying:
(i) Lu(t) =f(t) fort > 0;
(ii) u(0), u;(0),- • ., u<m-D(0) are given in V.
Of course u(£) must be sufficiently differentiate and/(£) at least measurable!
But one can also suitably interpret such a problem and extend it to
distributions (valued in Hilbert spaces).
The fact that we cannot take H = S is fairly evident from examples of
hyperbolic mixed problems or correct Cauchy problems. However, for
parabolic mixed problems, i.e., for an operator Kdjdt -j- A(t), where S = V,
one may take H = S — V.
4. Energy inequalities. We introduce now an auxiliary operator
J — Z
whose coefficients are bounded operators V -> H.
In the mixed problem for Kdjdt + A(t), M will be simply the identity map
In the mixed problem for Kd^dt^ 4- B(t)8/dt + A(t), M will be Id/dt.

DIFFERENTIAL EQUATIONS IN HILBERT SPACES 87
In the initial value (or Cauchy) problem for L (see §2), M will be the
derivative of L:
Mv = L(tv) - tLv.
If we set L(t,T) = rm + 2f=i ^j(0Tm~j> we see that, in this case,
M(t,r)=-^L(t,r).
or
The essential tool in our treatment is an inequality of the following kind:3
/•+<» /%-H»
6-*«|4>(*)|5* + e-^HJfcKOl^
/K\ J-00 J-°°
v°/ r+oo
<> Re e-*<«(14(«), Jf<>(0)F*
valid for any continuous differentiate real function p(t) such that p'(t) is
larger, for every t, than a certain continuous function g(t) > 0. On the
other hand, <j> is any (700 function with compact support, valued in a dense
subspace W of F: L<J> and Jf<|> must be valued in F. In the case of a mixed
problem, one can take W = V since the coefficients of L and M are bounded
operators V -> V. In the case of a Cauchy problem with data in the spaces
H8X, if F = H\ one may take W = Hs+° or more generally, W = H8' with s'
large enough. Furthermore, in this case, || || is simply the norm | | v; in the
mixed problems, it is the L2(Q,) norm, i.e., the norm | |o.
5. Outline of the method of solution. We shall try to give here an idea of
how existence and uniqueness of solutions to abstract initial value problems
for the differential operator L are established. We restrict ourselves to the
essential points of the reasoning as it applies to a very simplified case.
First of all, we introduce the space L2(p, V) of measurable functions u(t)
on the real line valued in F, such that Ji^ e~v{t)\u\\rdt < +oo. Let us set
Pm m fim-j
where K* and A* are the adjoints of K and Aj in the usual sense of bounded
linear operators in Hilbert spaces. For u e L2(p, V) and cj> GC00(Tf )4 such
that M4> has a compact support (see later), we may set:
f+oo
Bp(u,+) = I (u(t),L*[e-*MMW)])vdt.
This is a bilinear functional in u and <j>; and obviously, u -> Bp(u,i$>), for <J>
3 | \y is the norm, ( , )y the inner product in V.
4 When TF = H'x'', s' must now be taken larger than we have said above.

88
FRANCOIS TREVES
fixed, is continuous on L2(p, V). Now, if p'(t) ^ g(t) for all f's, we have:
(6) e-«»(\4>(t)\*v + ||M$(t)\\*)dt ^ Re B9{+, +),
J-oo
according to (5). We apply now the following lemma (Lions [1, p. 163]) :
Lemma. Let E be an Hilbert space, $ a linear subspace qfE,\ \a norm on $
larger than the norm of E, b(u, h) a bilinear functional on E x $ with the two
properties :
(i) for every he ©, u-> b(u, h) is continuous on E;
(ii) for every h e ©, c\h\ ^ \b(h, h)\, c > 0.
Then, for any anti-linear functional A on $, continuous for the norm \ \,
there is an element u of E such that b(u, h) — A( h) for every h e $.
We take E = L2(p, V), b( , ) = Bv( , ) and for ©, the set of <J> e C°°(Tf)
n L2(p, V) such that Jf<J> has a compact support. The norm | | of <f> will
be the square root of the left-hand side of (6). Let (( , )) be the inner
product associated to the norm || ||. For every/ e L2(p, V), there is u e L2(p, V)
such that:
(7) Bv(u, <t>) = J ^ *-*w((f(t), M${t)))dt, <t> g ©,
for the right-hand side is an anti-linear functional on $, continuous for | |.
When <|> runs over $, Jf<J> covers the whole of C™( W). This is quite obvious
in the mixed problems, where M = I or Idjdt (and W — V). In Cauchy
problems, we reason by induction on the order m of L. When m — 1, M is
the identity map of V. On the other hand, the derivative of an hyperbolic
(parabolic, correct) operator is hyperbolic (parabolic, correct); and our
assertion, that is to say Jf$ => C™(W), can be derived from the solution of
the Cauchy problem itself. It implies that equation (7) is equivalent to
(8) (Lu, v)v = ((/, v)) for every v e V, in the distributions sense.
But in the parabolic mixed problems and in the Cauchy problems, ((f,v))
— (/> v)v, hence Lu = / in the sense of distributions valued in V. And in
the hyperbolic mixed problems, ((/, v)) = (/, v)0, hence Lu — Kf.
Under favorable circumstances, standard methods prove that, if the
function g(t) > 0 is large enough, the element u of L2(p, V) (with p' ^ g)
verifying (8), will be unique.
Let us assume that/ has its support in the half line t ^ a. Let <E%, a) be
the family of all C1 functions p(t) such that p(a) — 0 and p'(t) ^ g(t).
Observe that/g L2(p, V) for any p(t) e <3>(g, a), provided that g(t) is large
enough. Let then uv be the element of L2(p, V) satisfying (8). There is a
function G(t) ^ g(t) such that, if pi, p^ e Q>(G, a), there is p e Q(g, a) such
that p(t) > pi(t) — c, for some number c (i = 1, 2). Hence, L2(p\, V) and

DIFFERENTIAL EQUATIONS IN HILBERT SPACES 89
L2(p2, V) are both imbedded in L2(p, V). From the uniqueness of up, we
derive uPl = uPr In other words, up does not depend on p(t) e <E>(6?, a);
let us denote it by u. It is easy to prove that there is a constant A such that,
for every p(t) e <3>(g, a),
e-«*\u(t)\%dt :g A e-*<»||/(<)||2*.
J-oo J-oo
This is possible only if u itself has its support in (a, -f oo).
Such an argument can be easily extended to distributions. Let us set
H = V in parabolic mixed problems or in Cauchy problems, H = image of
L2(Q,) under the mapping K when dealing with hyperbolic mixed problems.
If E is any Hilbert space, we denote by S)\ (E) the space of distributions
valued in E, with support in some half-line t ^ a. Given the operator L as
before, our method leads to results of the kind:
Theorem. For every distribution /e &'+(H), there is one and only one
distribution u e &'+(V) satisfying Lu = /. If the support off lies in the half
line t 7> a, so does the support of u.
That such theorems give the existence and the uniqueness of the solutions
of initial data problems (in the sense of distributions) is shown by an
argument of Schwartz [2]. Other standard techniques derive the solution of a
wide class of initial data problems, in the sense of functions, from the results
about distributions.
There is a very abundant literature on the subject of equations of the type
d/dt + A(t). Of foremost importance are the works of Visik, Ladizenskaya,
Lions, Kato. We must also mention the works of F. E. Browder. Various
articles of Lions deal with second order equations, say of the type d2\dt2
-f B(t)dldt -f A(t). Also noteworthy results have been gathered in the case
of noncylindrical mixed problems, and more generally when the space V
varies with the time.
References
1. J. L. Lions, Sur les problemes mixtes, Ann. Inst. Fourier vol. 7 (1957) p. 143.
2. L. Schwartz, Les equations devolution liees au produit de convolution, Ann. Inst.
Fourier vol. 2 (1950) p. 19.
3. F. Treves, Problemes de Cauchy et problemes mixtes en theorie des distributions, J.
Analyse Math. vol. 7 (1959) p. 104.
University of California,
Berkeley, California

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a maximum property of cauchy's problem in
three-dimensional space-time1
BY
H. F. WEINBERGER
1. Introduction. It was shown by Germain and Bader [8] that the
Tricomi problem for Tricomi's equation
,, . d2u d2u
has the following property. If AP and BP are characteristics and AB is a
segment of the X-axis, and if u vanishes on AP, then its maximum in the
triangle APB is attained on AB.
This property was generalized by Agmon, Nirenberg, and Protter [1] to a
class of hyperbolic second order equations in two variables, with variable
coefficients.
The author [17] showed that Cauchy's problem for a class of second order
operators of the form
„ ftX r d I du\ d L du\ du , du
(1.2 Lu ee — \a — \ - 7T~ 6— - c- d —
y } dx\ dx) dy \ dy/ dx dy
also has a maximum property. Namely, if
du
dy
du
(1-3) 7T ^ 0
on the initial line y = 0 and if
(1.4) Lu ^ 0 for y > 0,
then u attains its maximum at y — 0. The class of operators having this
property is characterized by inequalities between the coefficients and their
first derivatives.
M. H. Protter [12] showed that Cauchy's problem for any operator of the
form (1.2) has a maximum property in an extended sense. That is, the
maximum of u divided by a suitable function of the form e^(l — j8e~a^)
over a sufficiently small strip 0 ^ y ^ yo is attained on y = 0.
1 This research was supported in part by the United States Air Force through the
Air Force Office of Scientific Research of the Air Research and Development Command
under Contract No. 49 (638)-228.
91

92
H. F. WEINBERGER
These maximum properties yield some interesting theorems on ordinary
differential equations. In particular, they generalize results of S. Bochner
[2] on the positivity of certain eigenfunction expansions.
The maximum properties also lead to bounds in the maximum norm of
the solution of Cauchy's problem in terms of the Cauchy data. These are
useful for approximating the solution of Cauchy's problem. Analogous
results in higher dimensions would be even more useful.
A. Weinstein [18; 19] found from an explicit solution that Cauchy's
problem for the wave equation
OS) £ - A. = ».
where A is the ra-dimensional Laplace operator, has the following maximum
property. If
(1.6) Au ^ 0, A2u ^ 0, • • •, Apm ^ 0, px ^ |(m - 1),
and
(1.7) 1^0, 4^0, .... ii.^0, P3 £*<»»- 3),
on the initial plane t = 0, then u takes its maximum value at t = 0.
The following closely connected theorem is obtained by applying the
Volterra-Tedone solution [3, pp. 399, 402] of the inhomogeneous wave
equation to the function dm-*u\dtm-*.
Theorem 1. Let u satisfy the differential inequality
/d\™-*[d2u A 1
<L8> (*) h*-H*0>
where A is the m-dimensional Laplace operator. Let
du dm~2u dm~xu
(1.9) «(0, *) = - (0, *) = •• -1—j(0,x)-0, ._(0,*)S0,
X ~ (X1,- • ', Xm).
Then
(1.10) u(t,x) ^ 0 for* > 0.
If we apply this theorem to u minus a partial sum of its Taylor series in t,
we find the following corollary.
Corollary 1. Let

CAUCHY'S PROBLEM IN THREE-DIMENSIONAL SPACE-TIME 93
and
dm-lu
Then
m^? tk ?fiu
(1-13) u(t,x) H J UW{0,X)' O^tZT.
The ^-derivatives of u on the initial plane t = 0 can be found from the
Cauchy data. In particular, Corollary 1 includes the result of Weinstein
[18; 19].
Theorem 1 encourages us to seek a class of second order hyperbolic
operators with variable coefficients having a similar maximum property.
In this paper we find such a class of operators in three-dimensional space-
time. Our class includes the wave operator.
The maximum property of (1.2) was proved by means of the integral
equation obtained by replacing the unknown Riemann function in Riemann's
method [3, p. 311] by a constant. Our conditions on the coefficients imply
that the kernel of this integral equation is nonnegative, which, in turn,
yields the maximum property.
The use of integral equations to study the solution of Cauchy's problem is
not new. It was introduced by Hadamard [9] and has been used by several
authors [5; 6; 7; 11; 13; 14; 15] to obtain existence and uniqueness theorems
and to investigate Huygens' principle.
The derivation of the integral equation involves the introduction of normal
coordinates. While this is adequate for theoretical purposes, the explicit
introduction of such coordinates frequently poses insurmountable difficulties.
For the purpose of establishing a maximum property, we need to prove that
a certain kernel, depending upon the normal coordinates, is nonnegative.
Since the maximum property itself is invariant under coordinate
transformations, we seek a criterion which is invariant under such transformations.
That is, our criterion must depend upon the invariants associated with the
differential operator. For the sake of simplicity we restrict the present
investigation to a class of operators having only a single invariant.
2. The integral equation. We consider an operator of the form
G U
(2.1) Lu = — - Mu
in the three-space with coordinates (t, x1, x2). Here
(2.2) ^^^li^S)-
The matrix g{3 is symmetric and uniformly positive definite. Its elements

94
H. F. WEINBERGER
are four times continuously difFerentiable functions of x1 and x2. They are
independent of t. The function g is defined by
(2.3) g = [det0«]-i.
The inverse gtj of the matrix gW defines a Riemannian metric. Its only
invariant is the Gaussian curvature K (see, for example, [16]).
We choose an arbitrary point P with ^-coordinate T as the origin of a
system of geodesic normal [16] (polar) coordinates (r, 6) in the plane t = T.
That is, r is the geodesic distance of any point Q in this plane from P. 0 is
the angle between the geodesic PQ and a fixed direction at P. In some
neighborhood of P we have a one-to-one transformation (x1, x2) <-> (r, 0).
We extend this transformation to a transformation (t, x1, x2) <-* (t, r> 0) for
alH ^ 0 in a neighborhood of the line r = 0. The differential operator M
becomes
<"> ^--[!(^K(v-'£)]
where y is a nonnegative function of r and 6 such that the element of length
ds = {dr2 -f y2dB2}112. We derive the integral equation from the Green's
identity (cf. [4; 10])
///[->-sH»*-«--rr[^-raH
r = €
(2.5) -jE jo [vMu--- + uLv\vdrd6
* = o
r + t=T
Here d/dr is the total derivative along the characteristic conoid r -f t — T.
It is easily verified that the function (cf. [7])
T - t + UT - t)2 - r2W2
(2.6) v = (27r)-iri/y-1/2 log U - ^—
r
satisfies the conditions
v = Lv = 0 on r + < = I7,
(2.7) , ^
v ' limy— = -(277)-1,
(2.8) ^ = -(27r)-iri/2y-1/2[(^ - t)2 - r2]~i/2,

CAUCHY'S PROBLEM IN THREE-DIMENSIONAL SPACE-TIME 95
and
d_
dt
-2y-3^ + 5y-4
(LV) = (SK)-W2y-V2[iT _ t)2 _ r2]-l/2|_2y-l g + y'^Y
We let u satisfy the initial conditions
(2.10) u(0,xi,x*) = —(0,x\x*) == 0
and use the v defined by (2.6). Letting e -> 0 in (2.5), we obtain the integral
equation
(2.11) u{P) = [[[\ujt{Lv) + (27r)-iri/V1/2{(^ - 02 - r*}-U*Lu\ydrdOdt
cP
where Cp is the part of the retrograde characteristic cone through P where
t ^ 0.
This equation holds for each point P in the half space t > 0 provided its
t -coordinate T is so small that each point in the plane t = T at distance
r ^ T from P is connected to P by a single geodesic of length r. If the
curvature K is bounded above by a constant fc2, this will be satisfied if
IcT < tt.
3. The maximum property. Suppose that
B
(3.1) l(Lv)^0
and that
(3.2) Lu ^ 0.
Then we can show that
(3.3) u ^ 0
in the following manner. Suppose first that the inequalities (3.1) and (3.2)
are strict for t > 0. Then by (2.10) and (3.2) u is negative in some
neighborhood 0 ^ t ^ To. If P is the point with the smallest value of T for which
u vanishes, the integral equation (2.11) gives a contradiction. Thus u cannot
become positive. By continuity, this result remains valid even when the
inequalities (3.1) and (3.2) are not strict. Thus, (3.1) implies a maximum
property for L.
It follows from (2.9) that the inequality (3.1) is independent of t and
depends only upon y and its first and second derivatives. If y were known

96
H. F. WEINBERGER
explicitly, we could compute the right-hand side of (2.10), and verify (3.1).
However the function y cannot in general be found without finding the
geodesies from P. This involves solving a one-parameter family of systems
of nonUnear second order ordinary differential equations, and is frequently
impossible.
The function y is connected with the curvature K by the equation
(3.4) g + Ky = G
and satisfies the initial conditions
y(0, 6) = 0,
(3.5)
i^-1-
The curvature K is defined in terms of the metric gy as a function of x1 and
x2. When the geodesies are not known, K cannot be expressed as a function
of r and 0, and hence (3.4) and (3.5) cannot be used to find y explicitly.
However, we can use bounds on K and its derivatives to bound y and its
derivatives by means of (3.4) and (3.5)
We note that by (2.9) the kernel (djdt)Lv is equal to a positive factor times
the function
(3.6) Kr, 0) = - 2y-i g + y-2(g)2 - 2y-a g + 5y-4(|)2 - r~>.
Taking the limit as r -> 0 using (3.4) and (3.5), we find
(3.7) *(O,0) = |z(J»).
Thus, a necessary condition for \fs to be nonnegative is that the curvature K
be nonnegative. We suppose that K is nonnegative and bounded, so that
(3.8) 0 ^ K(xi, x2) ^ k2.
It follows immediately from (3.4) that
(3.9) k cot kr gy"1^ r"1
or
for
(3.10) kr <, 7T.
Therefore
kr1 sin kr g y g r,
(3.11)
COS AT ^ -r ^ 1.

CAUCHY'S PROBLEM IN THREE-DIMENSIONAL SPACE-TIME 97
Consequently the function tfi defined by (3.6) satisfies
(3.12) ip ^ y~3<f>
where
<"»> ♦("•)--v5 + ,(|),-4 + 3r-.(i)'-,
Differentiating and using (3.4) and (3.5) we show that <f> satisfies the
differential equation
+ Y [drdO y 8r BO) J
and the initial conditions
(3.15) <f>(0,6) = ^ (0, 6) = 0.
It is easily seen that <f> and hence tjt is nonnegative if the right-hand side of
(2.15) is nonnegative. Thus we have proved the following maximum
property.
Theorem 2. Let the Gaussian curvature K of the metric satisfy the
inequalities
0 ^ K g k2,
(3.16)
for all
(3.17) a ^ k cot kT
where
(3.18) 0 ^ T <> irk-1
and for (x1, x2) in a domain Do of the initial plane t = 0. Let u(t, x) satisfy the
initial conditions
Oil
(3.19) u{0, x\ x2) = — (0, x\ x2) = 0
and the differential inequality
(3.20) Lu ^ 0 for 0 ^ t ^ T.
Then
(3.21) u ^ 0
in the part of the domain of dependence of D0 where 0 S t ^ T.

98
H. F. WEINBERGER
Applying this theorem to the difference of u and the first two terms of its
Taylor series in t, we find the following corollary.
Corollary 2. Let K satisfy the hypotheses of Theorem 2. Let
(3.22) L[u(t, x\ x*)] + M
Then
u(0
> s1, *2) + ' "77 (0, x\ x2) \ £ 0 for 0 g t ^ T.
ot J
3u
(3.23) w(*, x\ x2) ^ u(0, x*, x*) + t— (0, x\ x*) for 0 ^ t ^ T.
Ob
We note that if K is uniformly positive and has bounded second derivatives,
the criterion (3.16), (3.17) is satisfied for sufficiently small T. On the other
hand, the second inequality in (3.16) is certainly satisfied if
(3.24) KM{K)>=K*-\lg«™™.
The criterion (3.16), (3.17) is sufficient, but not necessary for the maximum
property. For example, if K is any nonnegative constant we find by direct
computation from (2.9) that
(3.25) | (Lv) > 0,
so that the maximum property holds for T ^ ttK-1'2.
Bibliography
1. S. Agmon, L. Nirenberg, and M. H. Protter, A maximum principle for a class of
hyperbolic equations and applications to equations of mixed elliptic-hyperbolic type, Comm.
Pure Appl. Math. vol. 6 (1953) pp. 455-470.
2. S. Bochner, Sturm-Liouville and heat equations whose eigenfunctions are ultra-
spherical polynomials or associated Bessel functions, Proceedings of the Conference on
Differential Equations, University of Maryland, 1955, pp. 23-48.
3. R. Courant and D. Hilbert, Methoden der Mathematischen Physik, vol. II, New
York, Interscience, 1937.
4. J. B. Diaz and M. H. Martin, Riemunn's method and the problem of Cauchy. II,
The wave equation in n dimensions, Proc. Amer. Math. Soc. vol. 3 (1952) pp. 47^-483.
5. A. Douglis, The problem of Cauchy for linear hyperbolic equations of second order,
Comm. Pure Appl. Math. vol. 7 (1954) pp. 271-295.
6. , A criterion for the validity of Huygens' principle, Comm. Pure Appl. Math.
vol. 9 (1956) pp. 391-402. Also Transactions of the Symposium on Partial Differential
Equations, Berkeley, 1955, pp. 93-104.
7. , On linear hyperbolic equations of second order, Technical Note BN-139,
Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, 1958.
8. P. Germain and R. Bader, Sur le probleme de Tricomi, Rend. Circ. Mat. Palermo
vol. 2 (1953) p. 53.
9. J. Hadamard, Lectures on Cauchy"s problem, New Haven, Yale University Press,
1923.

CAUCHY'S PROBLEM IN THREE-DIMENSIONAL SPACE-TIME 99
10. M. H. Martin, Riemann's method and the problem of Cauchy, Bull. Amer. Math. Soc.
vol. 57 (1951) pp. 238-249.
11. M. Mathisson, Eine neue Losungsmethode fur Differentialgleichungen von normalem
hyperbolischen Typus, Math. Ann. vol. 107 (1932) pp. 400-419.
12. M. H. Protter, A maximum principle for hyperbolic equations in a neighborhood of
an initial line, Trans. Amer. Math. Soc. vol. 87 (1958) pp. 119-129.
13. M. Riesz, U integrate de Riemann- Liouville et le probleme de Cauchy, Acta Math,
vol. 81 (1949) pp. 1-223.
14. S. Sobolev, Sur une generalisation de la formule de Kirchoff, Dokl. Akad. Nauk
SSSR N.S. vol. 6 (1933) pp. 258-262.
15. S. Sobolev, Methode nouvelle a resoudre le probleme de Cauchy pour les equations
hyperboliques normales, Mat. Sbornik N.S. 1, vol. 43 (1936) pp. 39-71.
16. J. L. Synge and A. Schild, Tensor calculus, University of Toronto Press, 1952.
17. H. F. Weinberger, A maximum, property of Cauchy's problem, Ann. of Math. vol.
64 (1956) pp. 505-513.
18. A. Weinstein, On a Cauchy problem with subharmonic initial values, Ann. Mat.
Pura Appl. vol. 4 (1957) pp. 325-340.
19. , Hyperbolic and parabolic equations with subharmonic data, Symposium
on the Numerical Treatment of Partial Differential Equations with Real Characteristics,
Prov. Intern. Computation Centre, Rome, 1959, pp. 74-86.
Institute for Fluid Dynamics and Applied Mathematics,
University of Maryland,
College Park, Maryland

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COMMENTS ON ELLIPTIC PARTIAL DIFFERENTIAL
EQUATIONS1
BY
LOUIS NIRENBERG
In this talk we shall describe some of the developments in partial
differential equations since the symposium held here in 1955, with particular emphasis
on elliptic differential equations. Thus we will not discuss the very general
and interesting investigations that have been carried out for equations of
arbitrary type with constant coefficients—chiefly by L. Ehrenpreis, L.
Hormander and B. M. Malgrange. We shall also omit the important work
of E. de Georgi and J. Nash, as well as its extensions, since these are
described in the talk by C. B. Morrey in these proceedings; in general we try to
minimize the overlap with other papers in the symposium.
We start with some remarks on existence and uniqueness in the Cauchy
initial value problem. In 1950 Garding [11] characterized all "hyperbolic"
differential operators with constant coefficients, i.e., those for which the
Cauchy initial value problem for a function is well posed. A. Lax [14]
described all hyperbolic operators with variable coefficients for a function of
two variables. Yamaguti and Kasahara [22] have characterized all'' strongly
hyperbolic" first order systems with constant coefficients for a vector u
du ^ . 8u _ .
here Aj and B are constant matrices; they call the system "strongly
hyperbolic" if there is existence and uniqueness in the initial value problem (with
the initial surface xn — constant) for the systems with the matrices Aj fixed
for arbitrary (constant) matrices B. Thus the problem of characterizing
"hyperbolic" equations may be close to solution.
It has been known for some years that uniqueness in the initial value
problem for a noncharacteristic initial surface does not hold in general, even
for equations with (7°° coefficients. Recently (unpublished) both A. Plis and
P. Cohen have even exhibited elliptic equations with regular coefficients for
1 This paper represents results obtained at the Institute of Mathematical Sciences,
New York University, under the sponsorship of the Office of Naval Research, Contract
Nonr-285(46). Reproduction in whole or in part permitted for any purpose of the
United States Government.
101

102
LOUIS NIRENBERG
which there is no uniqueness in the initial value problem. These equations
have multiple (complex) characteristics. In fact Calderon [7] has proved
uniqueness for any equation with simple characteristics.
To describe this condition we fix some notation. Consider equations for
functions u defined in a bounded domain 2 in n-space (later in discussing
elliptic operators we shall assume, for convenience, that the boundary 3) of
3f is of class C00), points in the space are denoted by x = (xi, • • •, xn) and
D = (Dlt...,Dn), A = ^"
denotes the differentiation vector; derivatives of order m of a function
u(x) are denoted by Dmu. A linear differential operator L is then a
polynomial in D with (complex) coefficients which are functions of x> L = L(x; D).
The leading part of Ly the part of highest order, is denoted by L\ The
operator L is elliptic if, for every point x, L'(x9 £) ^ 0 for every real vector
i # o.
Consider then the initial value problem in a half neighborhood of the origin
with xn > 0 for an operator L of order m
(1) Lu = 0, with DJnu = 0, j < m on xn = 0.
Calderon [7] proved that zero is the only solution in case the coefficients of
U are real provided the equation has only simple characteristics, i.e.,
provided that for any real scalars fi,- • •, f»-i, not all zero, the complex
roots r of the polynomial 1/(0; £i,- • •, fn-i, t) are all simple. (In [7] the
case of dimension n — 3 is excluded; however, Calderon has since succeeded
in proving the result also for this case.) In case L is elliptic Hormander [12]
improved Calderon's result by eliminating the hypothesis that L' have real
coefficients and by requiring slightly less smoothness of the coefficients of L'.
Calderon's work in [7] is based on the method of Calderon and Zygmund
[8] of representing differential operators in terms of singular integral
operators. A linear differential operator L of order m is written in the form
L = KA™,
where A is the square root of the Laplace operator, i.e., Au(x) is the inverse
Fourier transform of the function |£\u(£); here u(£) is the Fourier transform
of u. K is a singular integral operator transforming square integrable (or
Lp) functions into square integrable (Lp) functions. The singular integral
operators of the kind considered form an algebra.
This method of singular integral operators seems to be a very powerful
tool and will prove very useful in other differential problems. It enables
one to apply Fourier integral techniques, which normally are used fcfr
equations with constant coefficients, also to equations with variable coefficients.
It has already been applied to give a rather direct derivation of (known)

ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS 103
energy inequalities for hyperbolic differential equations, see Mizohata
[18; 19] where other references may be found. In addition I. Katz
(unpublished) has used the method to obtain inequalities for a certain class of
differential operators L, enabling him to assert that for C00 functions / there
exist solutions of the equation Lu = /.
In this connection we mention here that Lewy [16] exhibited an operator
L (in fact of first order and with linear coefficients) with the property that in
general for a given C00 function / the equation Lu = f has no solution, even
locally. And now Hormander [13] has proved the following striking result.
Let L be a differential operator of order m with C00 coefficients in S and let
L' be the part of L of order m. Let L' be obtained from L' by complex
conjugation of the coefficients of L'; the commutator L'L' — L'L' has order
less than 2m. Let C<2,m-\(x\ D) be the part of the commutator of order
2m — 1. A necessary condition for the equation Lu = f to have a
(distribution) solution for every C00 function / with compact support in S is that any
real vector £ satisfying L\x\ £) = 0 at some point x in S satisfies also
C2m-i(#; £) = 0. This shows that in general equations with complex
coefficients do not possess solutions.
To conclude this discussion of uniqueness in the initial value problem we
mention that P. Cohen has recently constructed a first order equation with
complex coefficients for which uniqueness does not hold.
Before taking up elliptic equations we mention that Friedrichs [10] has
initiated work on boundary value problems for symmetric first order systems
of differential equations, independant of the "type" of the equations. This
yields existence theorems for certain equations of mixed type (elliptic and
hyperbolic in different regions). See also Lax and Phillips [15]. Extensions
of this important development to higher order equations have yet to be made.
We turn now to elliptic partial differential equations. Consider in a
bounded domain S with Cico boundary S an elliptic operator L of even
order 2m. We first describe various estimates that have been established
for general boundary value problems. For this purpose we introduce several
norms. For k a nonnegative integer, 0<a<l,l<p<oowe set
\u\k = l.u.b. \D>u(x)\,
, , \D*u(x) - Dku{y)\
here l.u.b. is taken with respect to all derivatives of u of the orders shown.
We also set
M*.p= 2 \&u\*dx\ ,
IJs $tk I
where summation is over all derivatives D% of order not greater than k.
We shall use (u, v) to denote Lz scalar product of functions u, v in S.

104
LOUIS NIRENBERG
Consider the boundary value problem for the elliptic operator L
(2) Lu = / in 3f, BjU = 0 on Of, j = 1, • • •, m.
Here Bj is a differential operator of order rrij; in the case of the Dirichlet
problem BjU is equal to the (j — 1) order normal derivative of u. Under
mild smoothness conditions on the coefficient of the operators Bj and L,
necessary and sufficient conditions (algebraic in nature) on the operators
have been given for the following estimates to hold for solutions of (2) (we
assume here for simplicity that each nij < 2m):
(a) Lp estimates:
MUm.p ^ Ci||Llft||0,p + C2H|o,p;
(b) pointwise estimates of Schauder type:
|^|2m+a ^ C3\Lu\a + C4|^|o;
the constants ci, • • •, C4 are independent of u. In case the solution is unique
these inequalities hold with C2 = C4 = 0.
These results are contained in Agmon, Douglis, Nirenberg [4] where
analogous estimates are proved, without the restriction rrtj < 2m; see also
Browder [5; 6]. The proofs of the estimates are based on a careful analysis
of equations (and boundary operators) with constant coefficients and only
highest order terms in a half space bounded by a hyperplane. These
equations are solved explicitly with the aid of "Poisson kernels".
From now on when discussing boundary value problems (2) we shall
assume that the system satisfies the algebraic conditions required for these
estimates. We note first that the estimates yield a proof of the finite
dimensionality of solutions of (2) for/= 0.
The main reason for studying inequalities with such norms rather than
merely the L2 norms, which are indeed the easiest and most convenient for
most linear problems, is in connection with nonlinear problems. The
inequalities yield regularity theorems for solutions of nonlinear elliptic
equations both in the domain and at the boundary—under suitable (possibly
nonlinear) boundary conditions. In addition, with the aid of (b) it is possible
to prove perturbation theorems for nonlinear problems. Suppose, namely,
that u is a solution of a nonlinear elliptic boundary value problem; one seeks
a solution v close to u for a system resulting from a slight perturbation of the
original. If the first variation, at u, of the original system is a system of
the type (2) which is uniquely solvable for all sufficiently smooth / then
the perturbed system admits a unique solution close to u (see [4]).
We consider now the existence theory for boundary value problems (2);
we shall say that there is existence if (2) is solvable for all "smooth"
functions/. We also describe some extensions of the estimates (a), (b).
(i) The Dirichlet problem ; BjU = normal derivative of u of order (j — 1).
By application of the Riesz theory for compact operators one finds with the

ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS 105
aid of the estimates (a) or (b) that if there is uniqueness in (2) then, for any
operator M of order less than 2m, existence and uniqueness hold for the
operator L -f \M for all but at most a denumerable discrete set of complex
values of A. Furthermore we have the well known alternative: uniqueness
in (2) implies existence. It is not known whether for any given L there
always exists a complex number A such that uniqueness holds for the operator
L + A.
We mention here another open question which is purely algebraic. Let
P{£w ' *> £n), Q(£i,- ' *, £n) be two homogeneous polynomials of degree 2m,
with complex coefficients, in n > 2 variables, £ = (£i, • • •, £n), such that
P(£), Q(£) are nonzero for any real £ ^ 0, i.e., P and Q are "elliptic". Can
P and Q be connected by a one parameter family of such "elliptic"
polynomials of order 2m?
(ii) General boundary operators Bj. The system of boundary operators is
called normal if their orders rrij are different and if the boundary is nowhere
characteristic with respect to any of them. For normal operators the adjoint
problem in terms of the Lz scalar product may also be described by means of
a system like (2) with normal boundary operators. Furthermore the adjoint
problem continues to satisfy the conditions for the estimates (a) (b) to hold.
In general (2) has a solution for given/ if and only if/ is orthogonal (L2) to
the null space of the adjoint problem; see Schechter [20], Browder [5],
Agmon (to appear). If the system of boundary operators is non-normal
the adjoint problem may no longer be equivalent to a system of the form (2).
Nevertheless it is possible to describe the adjoint problem, and Schechter
[21] has proved that uniqueness in that problem implies existence for (2).
(iii) Distribution solutions: Both Agmon (in [2] and unpublished work)
and Schechter have treated these with the aid of inequalities extending those
in (a). Schechter's work is described in his paper in these proceedings,
Agmon has obtained existence theorems by first proving the regularity of
generalized solutions near the boundary. We mention one of his extensions
of the Lp results in (a) for the Dirichlet problem. Let L* be the formal
adjoint of the elliptic operator L and let wbea function in S that belongs
locally to Lq for some q > 1 and satisfies
\(U, L*V)\ g Cu\\v\\2m-ktp'
for every function v with zero Dirichlet boundary data. Here l/p -f l/p'
= 1, k is a fixed positive integer, and Cu is a constant independent of v.
Then u has derivatives up to order k in Lv and
(3) INI*.* = ciGu + c2|H|o,i>
with constants Ci, C2 independent of u. The proof makes use of the " Poisson
kernels" of [4].
(iv) Further estimates. The estimates (a) for p — 2 are very special cases
of the general coerciveness problem of Aronszajn. We remark only that

106
LOUIS NIRENBERG
this problem has been solved in general by Agmon [1] and Hormander
(unpublished). See [1] for complete references. Agmon (unpublished) has also
solved the following Lp coerciveness problem (generalizing the results of (a)):
Let Ai,---,Ak be partial differential operators of order m. Consider
functions u satisfying a finite number of differential boundary conditions
BjU = 0 on <2>, j = 1,- • •, N, where the orders of the Bj are less than ra.
Give necessary and sufficient conditions on the operators so that for some
positive constants C, c the inequality
2 Mj^IIo.p + C|M|o,j> ^ cHlm.p
3
holds for all functions satisfying the boundary conditions BjU — 0.
The Schauder type estimates (b) have also been extended. In an
interesting paper [9] (and in further unpublished work), P. Fife has considered
problems (2) in which the given data (for example, the function /) satisfy
Holder conditions in some but not all variables, e.g., in n — 1 independent
directions, and has derived estimates analogous to the Schauder ones.
(v) Agmon's maximum principle. Agmon [3] has recently proved a
striking generalization of the maximum principle for elliptic equations of
arbitrary order, extending work of Miranda [17]. Let wbea solution of the
Dirichlet problem for the elliptic operator L of order 2ra
Lu = 0 in 3), D^lu = fa on @f, j = 1,- • •, ra.
Here Dn denotes differentiation in the normal direction. The maximum
principle in its general form is expressed in terms of norms \u\-j for j = 0.
These are defined as follows. Let u be expressed as a finite sum of derivatives
of order ^ j of functions /* and set
M-* = inf2lM°>
k
where inf is taken over all such representations. With similar norms
defined for the functions fa on the boundary the maximum principle asserts:
for 0 ^ I S m - 1
m
(4) \u\i g ci 2 |<H'-*-i + c2H|o,i,
1 = 1
where c\, C2 are constants independent of u; in case of uniqueness we may set
c2 = 0.
The proof makes use of explicit approximate solutions in a neighborhood
of any boundary point, these are obtained with the aid of the Poisson kernels
of [4]. It also uses the Lv estimates (3). We shall illustrate how (3) is used
by presenting the example given in [3]. Let the domain be a circle in the
plane and let L = A2 -f Zi, where A is the Laplace operator and L\ is an
operator of order less than four; we consider the case 1=1.

ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS 107
Let uo be the solution of A2^o = 0 with the same, given, Dirichlet data as
u. From the explicit formulas (via the Poisson kernels) expressing uo in
terms of its Dirichlet data one finds easily that
\u0\i S constant (|<£0|i + \<f>i\o)-
The function u\ = u — uo has zero Dirichlet data. For any v with zero
Dirichlet data
(uu L*v) = -(Luo9v) = —{LiUo,v)
so that, after partial integration, we find
\{ui, L*v)\ g constant* ||^o||i,p'||v||2,p'
for arbitrary finite p > 1. We may then apply Agmon's result of (iii); by
(3) we have
IMh.p g constant (||^o||i,p + ||^i||o,i)-
For p > 2 however there is the well-known general inequality
\u\\i S constant ||wi||2,j>.
Combining these inequalities we obtain easily the desired result (4).
Bibliography
I. S. Agmon, The coerciveness problem for integro-differential forms, J. Analyse Math,
vol. 6 (1958) pp. 183-223.
2. , The Lp approach to the Dirichlet problem, Ann. Scuola Norm. Sup. Pisa
Ser. 3 vol. 13 (1959) pp. 49-92.
3. , Maximum theorems for solutions of higher order elliptic equations, Bull.
Amer. Math. Soc. vol. 66 (1960) pp. 77-80.
4. S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions
of elliptic partial differential equations satisfying general boundary conditions. I, Comm.
Pure Appl. Math. vol. 12 (1959) pp. 623-727.
5. F. Browder, Estimates and existence theorems for elliptic boundary value problems,
Proc. Nat. Acad. Sci. U.S.A. vol. 45 (1959) pp. 365-372.
6. , A priori estimates for solutions of elliptic boundary value problems, Nederl.
Akad. Wetensch. Indag. Math. vol. 22 (1960) pp. 145-159 and pp. 160-169.
7. A. P. Calderon, Uniqueness in the Cauchy problem for partial differential equations,
Amer. J. Math. vol. 80 (1958) pp. 10-36.
8. A. P. Calderon and A. Zygmund, Singular integral operators and differential
equations, Amer. J. Math. vol. 79 (1957) pp. 901-921.
9. P. Fife, A remark on potential theory and Schauder estimates, Stanford University
Technical Report No. 91, August 1960.
10. K. O. Friedrichs, Symmetric positive linear differential equations, Comm. Pure
Appl. Math. vol. 11 (1958) pp. 333-418.
II. L. Garding, Linear hyperbolic partial differential equations with constant coefficients,
Acta Math. vol. 85 (1950) pp. 1-62.
12. L. Hormander, On the- uniqueness of the Cauchy problem, II, Math. Scand. vol. 7
(1959) pp. 177-190.
13. , Differential operators of principal type, Math. Ann. vol. 140 (1960) pp.
124-146.
Differential equations without solutions, ibid. pp. 169-173.

108
LOUIS NIRENBERG
14. A. Lax, On Cauchy's problem for differential equations with multiple characteristics9
Comm. Pure Appl. Math. vol. 9 (1956) pp. 135-169.
15. P. D. Lax and R. S. Phillips, Local boundary conditions for dissipative symmetric
linear differential operators, Comm. Pure Appl. Math. vol. 13 (1960) pp. 427-455.
16. H. Lewy, An example of a smooth linear partial differential equation without
solution, Ann. of Math. vol. 66 (1957) pp. 155-158.
17. C. Miranda, Teorema del massimo modulo e teorema di esistenza e di unicitd per il
problema di Dirichlet relativo alle equazioni ellittiche in due variabili, Ann. Mat. Pura
Appl. vol. 46 (1958) pp. 265-311.
18. S. Mizohata, Systemes hyperboliques, J. Math. Soc. Japan vol. 11 (1959) pp. 205-
233.
19. , Une note sur le traitement par les operateurs dHntegrale singuliere du
problems de Cauchy, J. Math. Soc. Japan vol. 11 (1959) pp. 234-240.
20. M. Schechter, General boundary value problems for elliptic partial differential
equations, Comm. Pure Appl. .Math. vol. 12 (1959) pp. 457-486.
21. , Various types of boundary conditions for elliptic equations, Comm. Pure
Appl. Math. vol. 13 (1960) pp. 407-425.
22. M. Yamaguti, K. Kasahara, Sur le systeme hyperbolique a coefficients constants,
Proc. Japan Acad. vol. 35 (1959) pp. 547-550.
Institute op Mathematical Sciences,
New York University,
New York, New York

SOME UNUSUAL BOUNDARY VALUE PROBLEMS1
BY
MARTIN SCHECHTER
1. Introduction. We shall outline some recent findings which developed
from the study of general boundary value problems for higher order elliptic
equations. They concern an elliptic operator A = 2uis2r alx(x)D^ in a
bounded domain G of En, where /x = (/xi,- • •, /xn) is a multi-index of non-
negative integers, |/x| = 2* /"-*> an(i -D" = d^/dx^- - -dx%*. The boundary
conditions are given in terms of differential operators of the form Bj =
zLM^m, bj(JL(x)D^, 1 S j S t, with orders m^ < 2r. Given a function /, we
discuss solutions u of the problem
(1) Au =finG\ Bju = 0 on 0G, 1 ^ j ^ £,
where ^G is the boundary of G. For the "usual" problem it is assumed
that t = r, that A is properly2 elliptic in G, that the set {JBi}J=1 is normal2
and covers2 ^4, and that fe L2(G). Quite general results concerning this
problem have been obtained (cf. [8; 4]). In this note we shall describe what
happens when some of the above hypotheses are removed. In §2 we
consider the case when / is a distribution. We show how a "solution" of
problem (1) can be defined and how the existence and regularity theory of
[7; 8] can be carried over. In §3 we remove the hypothesis that the set
{-Bj}jLi is normal. Certain important theorems remain valid in this case.
In §4 we state some results which hold when t ^ r. Indeed, important
existence theorems can be proved even when a different number of boundary
conditions are prescribed on different sections of the boundary.
For convenience we shall assume throughout that dG is of class (700 and
that the coefficients of A and the Bj are in (7°°(G). The letters s, p, q will
denote nonnegative integers.
2. Generalized solutions. Let V be the set of C^G) functions u which
satisfy BjU = 0 on dG, 1 <; j «£ r, and let V be the set of those v e C°°((?)
satisfying (Au, v) — (u, A*v) for all ueV, where (u, v) = ja^vdx and
^* = Zui^2r (— iy^D^a^x)'), the formal adjoint of A. When the set
{Bj}rj:=1 is normal, it was shown in [3] that v e V if, and only if, B]v = 0 on
1 The work presented here was obtained at the Institute of Mathematical Sciences,
New York University, under the sponsorship of the U.S. Atomic Energy Commission
Contract AT(30-1)-H80.
2 For definitions cf. [7].
109

110 MARTIN SCHECHTER
8G, 1 S j ^ r, where {B'j}rj=J is some normal set depending on {B-\ri=l and A.
For u e C°°((?) we define
H^ t>eC°°(G) i'^ll5
\u\-s = l.u.b.
ueF' ||v !| s
Denote the closures of C°°(C?) with respect to those norms by HS(G), H~S(G),
and 3~~S(G), respectively. One easily defines (u, v) for u e H~~s(G)i v e HS(G)
and for ue H~s(G), veV, satisfying \(u, v)\ ^ IM|-*IMI* an(i |(^> v)\
^ |^|__s||?;||s, respectively. For/e H~2r~s(G) we shall say that u e H~S(G) is
a solution of problem (1) if there is a sequence {ut} of functions in V such
that \\uk — u\\-8~>0 and \Aujc —/|_2r-s->0 as &-> oo. The set of all
ueV (resp. v e V) which satisfy Au = 0 (resp. A* v= 0) will be designated
by N (resp. N').
Theorem 1. For f e S~2r~s(G)y there is a solution ueH~s(G) of problem
(1) if, and only if, (/, v) = 0 for all v e N'.
Theorem 2. Assume that f e H~2r-S(G), ueH-*-*(Q) and that (u, A*v)
— (/> v) for aM v G V- Then actually u e H~s(G) and is a solution of (I). If
fEH*(G)y then ueH^+p(G) and \\u\\2r+p S K{\\f\\p + H-*-*), where K
does not depend on f or u.
Theorem 3. If ueH~~^{G) and \(u, A*v)\ S co\\v\\-8 for all ve V\ then
actually ueH2r+8(G) and \\u\\2r+s S K(cq -f ||^||-5), where K does not depend
on u.
We let (u, v) — $dG uvdv be the L2(dG) inner product and define
<M>_S = l.u.b. Ig^l-
veC^G Flk+1
Theorem 4. There are constants k and ki such that
\\u\\-8 ^ hl\Au\-2r-8 + 2 <-B^>"m/-5 + H|-2r-s)
for all u e C°°(£) and
\\u\\-s ^ kil\Au\-2rs + 2 <jB^>-»/-«)
for all u e C°°(G) satisfying (u, N') = 0.
In proving Theorems 1-4 we make use of the following:
Lemma 1. If M is a finite dimensional subspace of HS(G), then for every
u e H~S{G), u = uf + u\ where u' e M and (u", M) = 0.

SOME UNUSUAL BOUNDARY VALUE PROBLEMS 111
Lemma 2. If ue H~S(G) and (u, A*v) = 0 for all v e V, then ue N.
Lemma 3. For every bounded linear functional F(v) in H~~S(G) there is a
function h e H*(G) such that F(v) = (v, h) for all v e H~S(G).
Lemma 3 is similar to a representation theorem due to Lax [5]. We do
not have space to prove Theorem 4; Lemmas 1-4 are elementary and their
proofs will also be omitted.
Proof of Theorem 1. The necessity is obvious. To prove sufficiency,
assume that (/, N') = 0. Since fe H-2r~s(G), there is a sequence {/*} <=
C™(G) such that \fk - f\ _2r_s -> 0. Each fk = fk + /;, where /; e N' and
(fi, N') = 0 (projection theorem). Thus if v e N', (fk, t;) = (/* -/,*)-* 0.
Since N' is finite dimensional, it follows trivially that the fk converge in
L2(G) to zero. Hence \f — /|_2r-s~>0. Since (f^N') = 0, there is a
ukeV such that (uk, N) = 0 and Auk - fk ([8]). That there is a u e H~S(G)
such that ||ttfc — u\\-8 ~> 0 follows from the second estimate in Theorem 4.
Proof of Theorem 2. Clearly (/, N') = 0. Hence, by Theorem 1,
there is a w0 6 H~S(G) which is a solution of (1). Thus (w — uq, A*v) = 0 for
all w g F'. By Lemma 2, u — uq e N and hence u e H~S(G) and is a solution
of (1). If fe Hv(G), then u0 e H2r+v(G) [8] and the above reasoning gives
u g H2r+v{G). The estimate comes from a known one for u0 [2; 4; 6].
Proof of Theorem 3. Clearly F(v) = (^4*?;, w) is a bounded linear
functional over the subset V of H~~S(G). Thus by the projection theorem
and Lemma 3, there is an fe HS(G) such that (A*v, u) — (v,f) for v e V.
Moreover \\f\\s g cq. An application of Theorem 3 completes the proof.
3. Non-normal boundary conditions. We now assume that / e (7°°(G) but
remove the stipulation that the set {B3)]z=l is normal. Functions in V need
not now satisfy differential boundary conditions and the methods of [8; 4] do
not apply. However the following can be proved.
Theorem 5. // (/, N) = 0, there is a function v eV such that A*v = /
in G.
Theorem 6. // AT/ = 0, then for every f e C°°(6y) there is a ue V such that
Au = / in G.
Theorem 7. There is a constant K such that \v\%r S K(\\A*v\\q -h ||v||o)
for all v e V. Thus A* is coercive over V.
Theorem 8. Assume that u e L2(G) and that (u, A*v) = (Aw, v) for some
w e V and all v e V. Then ue V and Au = Aw,
Lack of space does not permit us to include the proofs of Theorems 5-7.
We prove Theorem 8 by observing that (u — w, A*v) = 0 for all v e V.
Since N is a finite dimensional subspace of C°°((?) [4; 8], u — w = w' -f w",
where w" e N and (wf, N) = 0 (projection theorem). Thus (w\A*v) = 0

112 MARTIN SCHECHTER
for v e V. But by Theorem 5 there is a v e V such that A*v = w'. Hence
w' = 0 and u = w + w" e V.
4. Underdetermined and overdetermined boundary conditions. We now
restore the assumption that {Bj}rj=1 is normal. By adding appropriate
operators we may assume that {Bj}rj==1 is part of a normal set {Bj}j1Ll
containing 2r operators (Dirichlet system). For every such system we can find
another set {BfijL i such that
(2) (Au, v) - (u, A*v) = £ <B,v, B'2r_j+lv>
for all u, v g C°°(<?). Let diC? be a portion of dC? with smooth n — 2
dimensional boundary and set #2$ = dG — d\G. Let 5 and t satisfy either
0^s^t^rorr<:S<:tS2r. We have
Theorem 9. There is a solution u e C°°(6?) of
(3) 4t* = / in G,
(4) S^ = 0 on diG, 1 ^ j S s,
(5) 5^ = 0 on d2G, 1 ^j S t,
if, and only if, (f, v) = 0 for all v e (7°°(G) satisfying
(6) ^*v = 0 wi G,
(7) £>; = 0 on
(8) B-v = 0 on
(Equation (4) (resp. (8)) is £o be suppressed when s
Corollary 1. A necessary and sufficient condition that there exist a
u e (7°°(6?) satisfying Au = f in G and having zero Cauchy data on dG is that
(/, v) = Ofor all v e C°°(6?) such that A*v = 0 in G.
Corollary 2. A necessary and sufficient condition that there exist a
u g C°°(6?) satisfying Au = f in G is that (/, v) = 0 for all v e C°°(6?) having
zero Cauchy data on dG and satisfying A*v = 0 in G.
Corollary 3. Assume that the coefficients of A are analytic in G and
that dG is an analytic surface. Then for every f e C00(G) there is a solution
ueC™(G)of Au =/.
The proof of Theorem 9 is too long to be reproduced here. Corollaries
1-3 are immediate consequences.
5. Concluding remarks. We have shown that some of the assumptions
made in the usual problem can be dropped without affecting certain im-
hQ,
82G,
)hen s -
1 ^ j ^ 2r - s,
1 ^ j S 2r - t.
= 0 (resp. t = 2r).)

SOME UNUSUAL BOUNDARY VALUE PROBLEMS 113
portant theorems. Which theorems remain unaflFected depends, of course,
upon which particular assumptions are removed. It should be noted,
however, that we have assumed throughout that A is properly elliptic and is
covered by {Bj}rj=sl. One cannot drop these hypotheses without altering
completely the nature of the results obtained.
A complete exposition of the results announced here will appear elsewhere
(cf.[9;10]).
Bibliography
1. Shmuel Agmon, The coerciveness problem for integro-differential forms, J. Analyse
Math. vol. 6 (1958) pp. 183-223.
2. Shmuel Agmon, Avron Douglis and Louis Nirenberg, Estimates near the boundary
of solutions of elliptic partial differential equations satisfying general boundary conditions,
I, Comm. Pure Appl. Math. vol. 12 (1959) pp. 623-727.
3. Nachman Aronszajn and A: N. Milgram, Differential operators on Riemannian
manifolds, Rend. Circ. Mat. Palermo Ser. 2 vol. 2 (1953) pp. 1-61.
4. F. E. Browder, Estimates and existence theorems for elliptic boundary value problems,
Proc. Nat. Acad. Sci. U.S.A. vol. 45 (1959) pp. 365-372.
5. P. D. Lax, On Cauchy's problem for hyperbolic equations and the differentiability of
solutions of elliptic equations, Comm. Pure Appl. Math. vol. 8 (1955) pp. 615-632.
6. Martin Schechter, Integral inequalities for partial differential operators and functions
satisfying general boundary conditions, Comm. Pure Appl. Math. vol. 12 (1959) pp. 37-66.
7. , General boundary value problems for elliptic partial differential equations,
Comm. Pure Appl. Math. vol. 12 (1959) pp. 457-486.
8. , Remarks on elliptic boundary value problems, Comm. Pure Appl. Math.
vol. 12 (1959) pp. 561-578.
9. , Various types of boundary conditions for elliptic equations, Comm. Pure
Appl. Math. vol. 13 (1960) pp. 407-425.
10. , Negative norms and boundary problems, Ann. of Math. vol. 72 (1960) pp.
581-593.
New York University,
New York, New York

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A NEW PROOF AND GENERALIZATIONS OF THE CAUCHY-
KOWALEWSKI THEOREM TO NONANALYTIC AND
TO NON-NORMAL SYSTEMS
BY
AVNER FRIEDMAN i
1. Normal systems. A function g(x, t, v) is said to belong to class (Si, 82,
S3) in a region D if
\D\DlDktg\ ^ HoH^+^i + Sgj + S3fc)! (H0, H constants)
for all (x, t,v)eD and for all the derivatives. Here, x — (x±, • • •, xn),
v = (vi9- • •, vp), (Sm)! = T(8m -f 1). Note that if 8n = 1 for some n, then
g is analytic in the corresponding variable. The most general Cauchy-
Kowalewski (or normal) system is
(1) 1%** = fi(x,*, *i, • • •, uA, • • •, Dl«iDrt«iu0, • • •),
(2) Dtufa 0) = 0 (0Si^ft-l,lgigA)
where
(3) pi ^ qtaj + riaj, pa > riaj for all i, a,j.
Here, for simplicity, we take homogeneous initial values.
Theorem 1. Let the functions ft(x, t, v) (v = (v±9- • *, vt%) for some Ti)
belong to class (8, 1, 1) in a region x e B, \t\ ^ <0> 2 vf = vo> for some t0 > 0,
Vo > 0, where 1^8^ minf,aj (pi — riaj)lqiaj. Then there exists one and only
one solution Ui(x, t) (1 S i ^ A) of (1), (2) which is analytic in t near t = 0.
Furthermore, the Ui(x,t) belong to class (8, 1) for x e B, \t\ ^ t± for some
h > 0.
The choice 8 = 1 gives the Cauchy-Kowalewski theorem. We note that
our restriction on S is sharp. Thus, for instance, for equations Dfu = D%u
-f f(x, t) (p ^ q), the theorem is false if 8 > pjq [6; 5; 4]. Special cases of
Theorem 1 for linear equations were considered in [4; 2] and by others
mentioned there, and one special case of nonlinear equations in [3]. Our
method, however, is new even in these special cases. It is based on direct
estimates of derivatives, using the following lemma.
1 Prepared under Contract Nonr 710(6)(NR 044 004) between the Office of Naval
Research and the University of Minnesota.
115

116
AVNER FRIEDMAN
Lemma 1. Let W(x, t) be a function satisfying for all m ^ 0
iz*zwo, o)i * {Mo0M77i8;+* -4)! if l = * =n'
where, by definition, a\ = Oi/a ^ 0. TJien there exists a constant K depending
only on 8 such that ifj^2 then, for any m ^ 0,
l-DyDfWrMH'l.-v-o
/iOfoV-1
[ 0 if
MoMn+u-^hm + jfc - 4)! if 1 ^ jfc ^ n,
jfc = 0.
The proof is elementary. The lemma enables us to estimate derivatives of
analytic functions of W provided we have appropriate bounds on the
derivatives of W.
We prove Theorem 1 by estimating, using (1), (2), the sequence of functions
D%D$u(x, 0). There is no need to transfer the differential system into a
first order system. Furthermore, the method can be modified to yield
existence theorems for systems with several ^-variables (Goursat problem).
Duff [1] has considered certain linear differential analytic systems with
boundary values given on two different intersecting surfaces. He applied the
method of majorants to prove existence of analytic solutions. Our method
can be extended to general nonlinear systems which contain Duff's system
as a very special case. Thus, a special typical case of our result is: Consider
Dtv = f(x, a, t, v, w, Dxv, Dxw, D„v),
Daw = g(x, a, t, v, w, Dxv, Dxw),
v = h(x, ar, w) on t = 0,
w = k(x, t, v) on a = 0.
If f, 9> h, k are analytic in appropriate domains and if the equations v = h,
w = k are satisfied at one point A = (x°, 0, 0, v°, w°) and 1 + hwkv ^ 0 at A,
then there exists a unique analytic solution u, w in a neighborhood of (x°, 0, 0).
Proofs and further details will be given somewhere else. (Added in proof:
See Trans Amer. Math. Soc. vol. 98 (1961) pp. 1-20.)
2. Non-normal systems. If (3) does not hold but nevertheless
(3') pa > riaj for all i, a,j
then the system (1), (2) is called non-normal. There are very simple linear
non-normal systems without an analytic solution, such as [7]
(4) Dtu = D%u + (1 + x + x2 + xs)u +1 (q ^ 2), u(x, 0) = 0.

THE CAUCHY-KOWALEWSKI THEOREM 117
On the other hand, for equations of the form
(5) Dfu = ^ BiWDiu,
(6) Dfu= ZUt)D>
with nonhomogeneous initial values, Asadulin (see [3]) and Friedlender and
Salehov [4] respectively proved the existence of analytic solutions provided
the initial values are of class 8 and, in (5), provided the Bjc(x) are polynomials
in a; of degree n < p. Here 8 ^ p\q for (6) and 8 ^ (p — n)/(q — n) for (5).
We shall prove a theorem which contains the above results.
Theorem 2. Consider the system
(7) Dfm, = 22 attx> t)DlijoD^ua + f.(x, t) (1 £ * £ A)
and the initial conditions (2) and let p{ > rija. Assume that the aja are
polynomials in x with coefficients analytic in t, \t\ ;S to, the degree being n)a. Let 8
satisfy
(8) 0< 8 £ 1, S(qijo - n)a) £ ft - rija - n)a
and let the fi(x, t) belong to class (8, 1) for x e B, \t\ ^ to. Then there exists
a unique solution of (7), (2) analytic in t near t = 0; it is of class (8, 1) in
(x, t), x e B, \t\ ^ hfor some h > 0.
The case of nonhomogeneous initial values of class 8 follows immediately
from Theorem 2 by change of variables. The example in (4), with q = 2,
shows that the restriction (8) on 8 is sharp. Thus, if we delete the term
x3u in (4), the equation will have an analytic solution. Since the proof of
Theorem 2 is not long, we describe it here.
For simplicity we take only the case of one equation, namely,
L
(9) Dfu = 2 *;(*, t)Dyirtm + f(x, t).
3 = 1
We first note that by differentiating (9) and using (2) we can define
successively a sequence DfD$u(x, 0). We shall prove by induction on n that for
every m ^ 0
(10) \DfD?+pu(x, 0)| ^ HQHm+an+aP(8m + n + p)!
where Ho, H, a are constants. Then, if we define u(x, 0 = 2 vk{%)tk where
k\vk(x) = D\u{x, 0), u is analytic in t and, as is easily seen, it satisfies (9).
Now, by (2), (10) is true for n < 0. Hence we may take n ^ 0 in the
inductive passage from w — 1 to w. If w = 0 some of the calculations
below have to be modified; hence, for simplicity, we take n > 0. Ho is

118 AVNER FRIEDMAN
determined by the case n = 0. H and a will be determined in passing from
n — 1 to n, n > 0.
We shall use the assumptions of the theorem
«») i^^')i-Mroir,Lfo4r>°;;;iis""
(*2) |DiD#(;M)| ^ A0Ai+k(8i + jfc)!.
We shall write D£ = Z>* wherever r = s. This is justified since the estimates
below are symmetric in the xt.
Applying D™D? to (9) and using (11), (12) and the inductive assumptions,
we get
\D?D?+*u(x, 0)| g K f f ("U^—'(n - „)!
j=lv=0 \v/
2 I ) H0H»+<*i+av+ari (8/x + Sg^ + v + rj) + ^o^m+w(Sw + n)\,
X
where K is used to denote any constant independent of m, n, H, a. Taking
H > 2A, and using the elementary inequalities
(*W - v)\(v + y)! ^ (n + y)\ (if y £ 0),
l.u.b. (8/x +y) ( J £ K(8m - 8n, + y + w,)! (if 0 < S ^ 1, y ^ 0)
we obtain
iD^Df+M^, 0)| £ KH0Hm+an ^ Hqt + ttri(8m - Srij + 8^ + ti + r,- + ^)!
i = i
+ ^0-4m+w(Sm + 7i)!.
Now if #/ -f ar/ < ap,
8m — hrij -f 8g;- -f n -f r^ -f rij ^ 8m -f n -f p,
then by taking H sufficiently large (depending only on H0, H, Ao, A, K, a)
the proof of (10) is completed. The first inequality is used to define a.
The second inequality is a consequence of (8).
References
1. G. F. D. Duff, Mixed problems for linear systems of the first order equations, Canad.
J. Math. vol. 10 (1958) pp. 127-160.
2. V. R. Friedlender, On the Cauchy-Kowalewski problem for some partial differential
equations, Uspehi Mat. Nauk vol. 12 no. 3 (1957) pp. 385-388.
3# 1 On the analyticity of solutions of the Cauchy problem for some nonlinear
partial differential equations, Mat. Sb. (N.S.) vol. 47 (1959) pp. 17-44.

THE CAUCHY-KOWALEWSKI THEOREM 119
4. V. R. Friedlender and G. S. Salehov, On the question of the inverse of the Cauchy-
Kowalewski problem, Uspehi Mat. Nauk vol. 7 no. 5 (1952) pp. 169-192.
5. E. Holmgren, Sur Vequation de la propagation de la chaleur, Ark. Math. Astr.
Physik vol. 4 (1908) pp. 1-4.
6. Le Roux, Sur les integrales analytiques de Vequation d2u/dy2 = du/dx, Bull Sci.
Math. France vol. 19 (1895) pp. 127-129.
7. M. Riquier, Sur Vapplication de la mithode desfonctions majorantes a certains systemes
differentiels, C. R. Acad. Sci. Paris vol. 75 (1897) pp. 1018-1019.
University of Minnesota,
Minneapolis, Minnesota

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REGULARITY OF CONTINUATIONS OF SOLUTIONS
BY
FRITZ JOHN
Let there be given a solution u of a partial differential equation in a
region jB determined by data/ on a manifold <f>. What regularity properties
of u can be inferred from regularity of/? In the case of a well posed problem
it can be expected that regularity of sufficiently high order in the data will
produce solutions that are as regular as desired (provided the regions
involved are also sufficiently regular). A certain fixed number of derivatives
may be lost (as in the case of the Cauchy problem for hyperbolic equations
in several dimensions), but / in Coo should produce only 11 in Goo.
Of more interest is the situation where the determination of u from / does
not represent a well posed problem. In that case u will in general not exist
for prescribed/. But if u exists, will it have to be regular for very regular/?
We will of course have to assume that u is determined uniquely by/, i.e.,
that B lies in the domain of determinacy of the data. Our question also
becomes meaningless in eases where high regularity of the solution is already
a consequence of the differential equation alone, as is the case for analytic
equations of elliptic type.
Instructive examples for irregular behavior of continuations of solutions
are furnished by the common equation of waves
(1) uxx + uyy = utu
The data/ shall be the values of u itself in a solid cylinder <j> with generators
parallel to the 2-axis. By the uniqueness theorem of Holmgren u is
determined everywhere by /. We consider first special solutions of the form
(2) u= \ £-* exp [i(ax + 0y 4- yt)]d£,
where a, j8, y are functions of £ satisfying the identity
a2 + £2 = y2.
Formally u will be a solution of (1). We take here a, /?, y of the form
« = €, P = -W)> v = € - Hi)
where g(g) is an arbitrary function and h(£) is given by
m ~ £[i + (i-g2lt2)1/2]'
121

122
FRITZ JOHN
Then
(3) u = f° £-* exp [i(x + t){ + yg{£)] exp [-ifc(f)]#.
Choose first k real and
gtf) = log f, £o = 1.
We have then
/*00
M = £*-* exp [i(x + Ofl exP [-#*(£)]<*£
where
»<« - ^
One verifies easily (by moving the path of integration into the complex)
that u is analytic for real x, y, t outside the plane x + t = 0. In points of
the plane x -f t = 0 with y < k — s — 1 the function w is of class C8. For
y increasing towards the value k — s — 1 some of the derivatives of u of order
s become infinite. However all tangential derivatives of u along the plane
exist for y < k — 1.
This particular solution u thus is analytic outside the characteristic plane
x -f- t = 0. The plane x -f £ = 0 itself is broken up into strips by bi-
characteristic lines x + t =■ 0, y — k — s — 1 where s is an integer. The
order of differentiability of u varies by 1 from strip to strip, till finally u
itself ceases to exist for y ^ k — 1. The solution u is determined uniquely
everywhere by its values in the half-space
y < k - 101,
where it is of class Cioo- The regularity of the solution decreases with
increasing distance from that half-space, u being of class C99 for y < k — 100,
of class C98 for y < k — 99, etc. Finally u ceases to exist for certain points
with y — k — 1.
We choose next in formula (3) for k an integer and
g(£) = log log f, £0 = e.
Again u will be analytic for x -f t / 0. In the points ofx + t = 0 the
solution u will be of class CV_i for y < — 1, but only of class Cjc-2 for y ^ — 1.
Here the number of permitted differentiations differs by 1 in the two half-
planes x + y = 0, y < —1 and x -f t = 0, y ^ — 1.
A different type of discontinuous behavior appears when we consider
solutions of (1) in polar coordinates r, 6 of the form
00
u = ^ *nn~kJn{nr) exp [m(0 + £)]

REGULARITY OF CONTINUATIONS OF SOLUTIONS 123
where en is bounded, k is a real positive number, and Jn denotes the Bessel
function of order n. It is known that J7t(nr) behaves like tt-1/2 for r > 1,
like n~1/d for r = 1, and decreases exponentially with n for r < 1. It
follows that u is of class Os everywhere for s < k — 2/3. For r < 1 we have
that u is analytic. [Following Hans Lewy this can be seen most easily by
observing that u only depends on r and 6 -f t = T. The differential
equation
Urr + r~lUr -f (r~2 — \)Utt = 0
satisfied by ^ is elliptic for r < 1 and hyperbolic for r > 1.] If we choose in
particular
J1 for w = 2m, m = integer,
[0 for all other n
and make use of the precise asymptotic expansions for Jn{nr) for large n we
find that u is precisely of class C* on r = 1 for 5 < fc + 1/3 and in r > 1 for
s < k -f 1/2. If we take for & a value with
5 8
2<k<Z
we have in w a solution of the wave equation which is analytic inside the
cylinder r = 1, is of class 6Y2 on the boundary of the cylinder, and of class C3
outside. Here the "surface of discontinuity" r = 1 is not a characteristic
surface of the differential equation.
The solution u just constructed can be considered as the unique solution
of a (not well posed) Cauchy problem, with Cauchy data prescribed on the
strip
1 1
y = 0, —- < x < -? —00 < t < -foo.
The data are then analytic. By Cauchy-Kowalewski the solution has to be
analytic near the initial manifold. Our example shows that farther away u
can still exist without being analytic.
We see then that regularity of a solution of the wave equation inside the
cylinder of radius 1 about the £-axis implies nothing about regularity outside.
The same holds of course for any circular cylinder with generators parallel
to the £-axis. More generally, if ^ is a bounded set and P a point that can be
separated from <f> by a plane parallel to the £-axis, then regularity of u on <f>
implies nothing about regularity at P. This follows from the fact that there
will then exist a circular cylinder with generators parallel to the £-axis which
contains <j> but not P, and a solution which is analytic in </> but only of class
C2 at P. By affine transformation we obtain the same result when there
exists any time-like plane separating P from <f>.
In contrast to these negative statements we have the following result:
Let ubea solution of a differential equation L(u) — 0, where Lisa differential

124
FRITZ JOHN
operator of order m with constant coefficients. Here u shall be of class Cm in a
closed bounded region R, where the boundary of R consists of a surface S and a
portion of a hyper-plane. The surface S shall be noncharacteristic and of class
Coo. If then the Cauchy data of u on S are in Coo, it follows that u is in 0 oo
throughout the interior of R.
The proof of this theorem can only be sketched here.1 The assumptions
permit for every N to construct a function v of class Coo in the closure of R
which has the same Cauchy data as u on S and for which L(v) vanishes of
order N -f 1 onS. Let w = u — v and / = L(w) = — L(v). Then w has
vanishing Cauchy data on S and is of class Cm in the closure of jB, while / is
of class Coo and vanishes of order N -f 1 on S. Let the plane bounding R be
given by x\ = 1 and let R be contained in the slab
0 < x\ < 1.
We continue w as identically 0 in the points outside R with x\ = 0. Then
w is of class Gm in the half space
R'\ xx < 1,
while/ = L(w) is of class Cn in R''. Let
||/||r= sup |I>/|
|a| ^ r\ xx < 1
and let \w\$ be denned similarly. By an inequality of L. Nirenberg2 there
exist for every 0 between 0 and 1 constants C, a with 0 < a ^ 1 such that
(4) \w(x)\ ^ C(\\f\\r)*(\\w\\a)i-* for Xl < 6.
Here r and s only depend on the coefficients of L, the order m, and the number
of dimensions. The existence of higher derivatives of w (and hence of u) is
established by deriving from (4) a priori estimates for the derivatives of w in
terms of \u\8 and ||/||r' where r' may have to be large. This is sufficient for
our purposes since the number of derivatives of / known to exist can be
taken arbitrarily high by choosing N sufficiently large, whereas we do not
want to increase the regularity assumptions on w itself in R'.
The principle of deriving from (4) estimates of the desired type for
derivatives of w is the following. Let £ be a vector with £i > 0. Then for any
positive h the function w(x — h£) is again defined in R' and L(w(x — h£))
= f(x — h£). Let dh denote the difference operator defined by
. . x w(x) - w(x - h£)
dhw(x) - -^ ^ —•
h
1 For the complete proof see the author's paper Continuous dependence on data for
solutions of partial differential equations with a prescribed bound, Comra. Pure Appl.
Math. vol. 13 (1960) pp. 551-585. A different proof has been given independently by
B. Malgrange in a forthcoming paper in the Bull. Math. Soc. Sci. Math. Phys. R. P.
Roumaine.
2 Uniqueness of Cauchy problems for differential equations with constant leading
coefficients, Comm. Pure Appl. Math. vol. 10 (1957) pp. 89-105.

REGULARITY OF CONTINUATIONS OF SOLUTIONS 125
Let now & be a number between 1/2 and 1. Then by (4)
(5) \dhw - dhkw\ S G(\\dhf - dhkf\\r)°(\\dhw - dhkw\\sy-° for xi < 6.
Now
dhf _ dhkf = _ __
can be estimated in terms of second derivatives of/, and hence
\\dhf - dhkf\\r S -4*||/||r+2.
On the other hand
\\dnw - dhkw\\s < Bh-tWwWs.
We then form for any j the telescopic series
(dhw - dhkw) + (dhkW - dhtfw) -f • • • + {dhww).
We find from (5)
\dhw - dhkm\ S CA*Bi-*h**-Hl + fc2-"1 + • • • + ^(2a-1);')(||/||r+2)a(||^||8)1-a.
If here a > 1/2 we obtain an inequality of the form
\dhw - dh<w\ ^ C"Aa--1(||/||r+2)-(||w||.)1-«
valid for x\ < 0 and 0 < h' < h. It follows then that the directional
derivative
i. w(x) — w(x - M) ^- >
lim dhW = lim 7 = y ^iWx.(x)
exists, and that it can be estimated in terms of ||/||r+2 and \w\s- If « ^ 1/^
we have to apply more complicated diflFerence formulae. Thus for a > 1/3
one can consider instead of (5) estimates for the operator
kdh - (1 + k)dhk + dhk*
applied to w. This operator applied to / can be estimated in terms of third
derivatives of/. Replacing h by hlc, hlc2, hksy- ■ •, and summing again gives
rise to a telescopic series. One obtains then estimates for 2< £iwx{ in terms
of \\f\\ r+3 and ||w\\s. Analogous expressions can be found for any positive a,
yielding an a priori estimate for the first derivatives of w. Repeating the
argument one can then derive estimates for the derivatives of second order,
etc.
With the method described there is still some difficulty in proving the
existence of the higher derivatives for which a priori estimates are obtained.
Following a suggestion of L. Nirenberg this difficulty can be avoided
completely by first working with "mollofied" functions w, which are in Goo*
This has the further advantage of permitting one to replace s by 0 in formula
(4). One obtains then for the original function u the result that the value of

126
FRITZ JOHN
any derivative of u of order j at an interior point P of R can be estimated in
terms of the maximum of \u\ in R and of a finite number of derivatives of the
Cauchy data of u on S (the number depending on j and the location of P).
The same arguments can be used to show more generally that a function
w satisfying a Holder condition of the type (4) is in Coo, if/ = L(w) is in Coo.
Essential for the proof is only the assumption that I is a linear operator
which is invariant under translations, and that the region R' where w is
denned is such that R — f is contained in R for a set of vectors £ forming a
convex cone.
New York University,
New York, New York

SOME LOCAL PROPERTIES OF ELLIPTIC EQUATIONS
BY
DAVID GILBARG
1. We review here a number of results on the behavior of solutions of
second order elliptic equations at isolated singular points, boundary points,
and at infinity. The common feature of these results for linear equations
will be the qualitative dependence of the solutions on the regularity properties
of the coefficients when little or no regularity is assumed of the coefficients
at the points in question. In the final sections we consider some applications
to nonlinear equations.
It will be of interest to compare the behavior of solutions of two classes of
elliptic equations:
and
du
aik(x) —
dxi
0,
(2) Lusai*{x)i^-k + bt{x)ter0'
where x = (x±, • • •, xn) and the summation convention is adopted.
Self-adjoint equations of divergence form (1) arise naturally in the
discussion of nonlinear equations derived from variational problems, while
equations (2) of non-divergence form arise in the theory of general quasi-
linear equations. The a priori Holder estimates of De Giorgi [5], Nash [23],
and Moser [21] for solutions of (1) and their extensions by Morrey [18] have
led to a fairly complete theory of nonlinear equations of variational type (see
Morrey [19]), whereas the analogous a priori estimates for (2) required in the
theory of general quasilinear equations are still lacking. As we shall see,
where local properties are concerned, the corresponding linear equations (1)
and (2) often behave differently under weak regularity assumptions on the
coefficients, and it is possible that this behavior reflects a basic difference in
the a priori estimates that hold for the two "classes of equations.
Obviously if the coefficients in (1) are sufficiently smooth, this equation
can be considered a special case of (2) and the same results apply to both.
Hence important differences between the two classes of equations can be
expected to appear only when the coefficients are rough and, indeed, for
most of the problems considered here, essentially the same results will hold
127

128
DAVID GILBARG
for both (1) and (2) provided the coefficients are Holder continuous or, more
generally, Dini continuous,1 at the points in question.
2. Probably the oldest and best known result independent of regularity
properties of the coefficients is the maximum principle, and we first consider
some results related to it. As proved by E. Hopf [11], in a completely
elementary way, the (strong) maximum principle asserts that any G2 function
u(x) satisfying the inequality
(3) Lu ^ 0(^0),
where L is a locally uniformly elliptic operator, cannot achieve its maximum
(minimum) in the interior of its domain of definition unless it is a constant.
We recall that L (or D) is uniformly elliptic in a region R when there are
positive constants Ai, A2 such that: (i) aik(x)^^k ;> A^ff for all real gl9 • • •, £n,
and for all x e R\ (ii) |a<*|, |6<| S A2. L is locally uniformly elliptic when
it is uniformly elliptic in compact subdomains. An essential point here is that
no regularity is assumed of the coefficients.
In the following the Hopf maximum principle will be applied to solutions
of equations of divergence form (I) under the assumption that the coefficients
are in class C1 and the solutions in O2 in the interior of the domains under
consideration. However, it should be remarked that the strong maximum
principle holds for uniformly elliptic equations (1) even when the coefficients
are only measurable and the solution u is a weak solution in the sense that u
is continuous, has (first derivatives in ^2, and jaacUx.(pxkdx = 0 for all C00
functions 9 with compact support in the domain of u. Most of the following
results concerning (1) can be stated for weak solutions.
3. The extended maximum principle. Suppose now that u is a non-
constant solution of (1) or (2) defined in the punctured sphere S:0 < \x\
= r ^ ro. We set
m(p) = min u(x), M(p) = max u(x)
\x\=p \x\=p
and shall say that u{x) satisfies the extended maximum principle in 8 if
(4) m(p) < u(x) < M(p)
for all x such that 0 < |o:| < p; or, in other words if m(p) and M(p) are
monotonically decreasing and increasing respectively.
The extended maximum principle is proved for equations of form (2) by
comparison arguments based on Hopf's maximum principle (see below).
When the equation is uniformly elliptic the extended maximum principle is
known to hold [10] if the coefficients a^ are Dini continuous at r = 0 and
u = o(r2~n), n > 2 ; or if the coefficients aik are continuous at r = 0 (with no
1 f(x) is Dini continuous at *q if \f{x)—f(xo)\ ^(p(\x — xq\), where I -— dr< 00.

SOME LOCAL PROPERTIES OF ELLIPTIC EQUATIONS 129
prescribed modulus), and u — 0(r2_n+d) for some 8 > 0 and n > 2. In the
latter case counterexamples show that the hypothesis cannot be relaxed to
allow u — o(r2~n). For given 8, the result remains true if u = 0(r2~n+s) and
the coefficients are discontinuous at r — 0 provided their oscillation is
sufficiently small (depending on 8). However, if the equation is uniformly
elliptic and no other limitation is placed on the behavior of the coefficients at
r = 0, counterexamples show that the extended maximum principle need
not be true even for bounded solutions.
The counterexamples mentioned above are provided by the following
equation, which proves instructive in other connections as well (see [10]).
,X\ A d2U A A * , / N XiX*
(5) AikJ^T0' A*-** + M-;r■
This equation has radially symmetric solutions u — u(r) satisfying the
ordinary differential equation
u" 1-ri
u' r(l + g)
and given by the formula
The connection between properties of the coefficients Am in (5) and the
corresponding solutions u(r) is of course determined by g(r). Thus if
(7) ^) = (*-Jg>;°f2-2, r>0;rt0) = »-2,
equation (5) is uniformly elliptic for sufficiently small r and has the bounded
solution
u(r) = a -f 6/log r, a,b = const,
which does not satisfy the extended maximum principle.
The contrast between the cases n — 2 and n > 2 is of interest. Thus far
we have considered only the latter. When n = 2, if the coefficients att are
Dini continuous at x = 0, and u = o(log r), the extended maximum principle
holds. However, if the coefficients are continuous and no further restriction
is placed on the modulus of continuity, the result is untrue in general even if
the solution is bounded (unlike the situation for n > 2). This is seen, for
example, by setting n = 2, g(r) — — 2/(2 -f log r) in (5); the corresponding
solution (6) is u = a -f 6/log r, which does not obey the extended maximum
principle.
The preceding results have been extended by Meyers and Serrin [17] to
include certain cases of non-uniform ellipticity at the singular point. For
example, in (2) with bi bounded, let aik(x) —^ Qj«v 3*S X -> 0, and suppose the

130
DAVID GILBARG
matrix (a%k) has k positive eigenvalues. Then if u = 0(r2~~*+6), 8 > 0, the
extended maximum principle holds. Thus, when n > 3, this result may
apply even if the ellipticity breaks down at x = 0.
The proof of the extended maximum principle is essentially the same in all
cases for equations (2) of non-divergence form. Namely, let u be the given
solution, which we may suppose is non-negative on \x\ = ro, and let y(r) be
the growth allowed the solution by hypothesis: max|£|=r \u(x)\ g y(r). The
proof hinges on determining a comparison function T(x) with the properties:
(i) r ^ 0 in 0 < r ^ r0; (iij L(T) ^ 0; (iii) yjT -> 0 as x -> 0. The
possibility of finding such a function depends in an essential way on the properties
of the coefficients of L near x = 0. Thus, if the a^ are Holder continuous
with exponent a at x = 0, arid aik(0) = 8^, T may be chosen in the form
r2~n(l + kra) for a suitable constant k. The proof continues by the following
standard argument. For given e > 0 consider w€(x) = u(x) -f tT(x). We
have that w€(x) ^ 0 on |a;| = ro, w€(x) ^ 0 if \x\ is sufficiently small (since
u(x)jT(x) -> 0 as x —> 0), and L(tv€(x)) S 0. Hence, by the maximum
principle, w€(x) ^ 0 in 0 < \x\ < ro. Now fixing x and letting e -> 0, we
find that u ^ 0 in this region, and hence either u > 0 in 0 < \x\ < ro or
u = 0. This gives one half of the required result (4), namely, u(x) > m(ro)
if \x\ < ro. Similarly, by choosing e < 0, one obtains u(x) < M(ro) if
\x\ < r0.
As is well known, the preceding argument can be used to obtain a theorem
on removable singularities whenever it is possible to find a solution of the
Dirichlet problem for a sphere about the singular point, taking as boundary
values on the sphere the values of the given solution. Applying the extended
maximum principle to the difference of these two solutions, one sees that the
two are identical. For a proof of theorems on removable singularities
without the maximum principle, see Bers [3].
The extended maximum principle is true under much more general
circumstances for equations of divergence structure (1) than for equations of
form (2). Namely, let the coefficients a^ in (1) be bounded and symmetric,
and suppose the equation is elliptic, but not necessarily uniformly elliptic, in
the punctured sphere 0 < \x\ = r S R] that is, aik(x)^^k ^ A(a;)2£f, where
X(x) > 0. The possibility that X(x) -> 0 as x -> 0 is permitted, and no
assumptions are made concerning the continuity of the coefficients at x = 0.
Suppose that u = o(r2~n) if n > 2, or u = o(log r) if n = 2, as r -» 0; then u
obeys the extended maximum principle.
A proof has been given by H. Royden. An earlier proof, by the method of
differential inequalities, appears in [10] under the more restrictive hypothesis :
u - O(|log rl1'8) if n - 2, and u = 0(r2-»+5) if n > 2, for 8 > 0. Royden's
argument proceeds as follows when u is a non-negative solution vanishing on
\x\ = R (this contains the essence of the proof for the general case). In the
spherical shell Ar: 0 < r :§ \x\ S R, let v(x) be the solution of (1) satisfying
the boundary conditions.

SOME LOCAL PROPERTIES OF ELLIPTIC EQUATIONS 131
v = 0 on \x\ = R,
v = M(r) = max u on \x\ = r.
By the maximum principle v ^ u in Ar and
auc(v - u)xvjcds ^ 0
l
)\x\-R
where v denotes the inward drawn unit normal. Let
(n2-n _ P2-n\
n > 2,
be the harmonic function coinciding with v on the boundary of Ar. Then we
have, from (1) and the Dirichlet principle,
M(r) aikuxyi4s ^ M(r) aikvxvjcds = vaikvxyi4s
= a>ikVx.vXkdx ^ auchx.hXkdx ^ const JMdz.
This gives, for a suitable constant 1£,
KMHr) C
M(r)C(u) ^ —2=n ' w^ere ^(^) = aikUxyjcds.
Since C(^) > 0 if w ^ 0, it follows that either u = 0 or Jf (r) ^ const r2_n,
which is the required result.
4. In the following theorem, also intimately connected with the maximum
principle, the situation contrasts with the preceding in that equation (2) now
yields stronger results.
Let u be a positive solution of (2) in an open region N. It is assumed that
(2) is uniformly elliptic but that the coefficients are otherwise arbitrary.
Let P be a boundary point of N such that some sphere lying entirely in N has
P on its boundary, and suppose that u(P) = 0. Then the inward normal
derivative at P is strictly positive. (If the derivative does not exist, the
same result can be stated for the lower derivate.)
This theorem is a simple consequence of the maximum principle (Hopf
[12])-
For equations (1) of divergence form—at least in two independent variables
—the same result holds if the coefficients are Dini continuous at P. However,
it is false in general if the coefficients are continuous without additional
restriction on the modulus of continuity [9].
5. Regularity properties at isolated singular points. Consider uniformly
elliptic equations of non-divergence form (2) for which n = 2. We have

132
DAVID GILBARG
seen in this case that the extended maximum principle need not hold for
bounded solutions at an isolated singular point—even if the coefficients are
continuous there. However, the following limit theorem is true. If the
solution is bounded on one side, it will have a limit (possibly infinite) at the
singular point, provided only that the equation is uniformly elliptic (no
continuity assumptions on the coefficients). This result is proved in [10] by
a general Harnack inequality; the argument is outlined in § 6. Concerning
the modulus of continuity at the limit, we observe that if the coefficients are
assumed continuous, the solution need not approach its limit Dini
continuously. This is shown by example (5) when g(r) = — 2/(2 -f log r) and
u = a + ft/log r.
The situation for n > 2 is not yet completely clear, and important
questions remain open. Thus, still in the uniformly elliptic case for (2), if no
regularity is assumed of the coefficients, it is not known whether a bounded
solution must have a limit at an isolated singular point. Examples show
that the limits, when they exist, need not be taken on Dini continuously.
This is seen by inserting (7) in (5) and (6), the corresponding solution being
again a -f 6/log r. If the coefficients are sufficiently regular, say Holder
continuous in the neighborhood of the singularity, the classical theorem on
removable singularities is valid: a solution that is o(r2-n) as r -> 0 can be
defined at x — 0 so as to be in class C2 in the neighborhood.
Between these extremes is a variety of possibilities. Thus if the
coefficients are continuous, or have sufficiently small oscillation at r = 0, a
solution that is 0(r2~n+8), 8 > 0, as r -> 0 will be in class C1 and have strong
second derivatives. The proof of this result (conjectured in [10]) is the same
as that outlined in § 3 of the above stated theorem on removable singularities.
It differs only in using the C1 solutions of the Dirichlet problem constructed
by Morrey [20] and the maximum principle for these solutions. The precise
quantitative dependence of the regularity of the solutions on regularity of
the coefficients is still unexplored. A detailed discussion of the local behavior
of solutions when the coefficients are Holder continuous, and also for higher
order elliptic equations, has been given by Bers [3].
For equations of divergence form, somewhat more is known concerning
regularity at isolated singular points. If the equation (1) is uniformly
elliptic and the solution is o(r2~n) as r -> 0, n > 2, then (under no additional
hypothesis on the coefficients) the solution is Holder continuous at the
singularity. This can be inferred from the De Giorgi-Nash a priori Holder
estimate of solutions of (1) by approximating the given equation with smooth
equations, while keeping the same ellipticity constants, proceeding to the
limit with a suitable convergent subsequence of solutions of these equations,
and observing from the extended maximum principle that the Holder
continuous limit function is identical with the given solution. The same
result can also be obtained directly by an application of Moser's proof of the
De Giorgi-Nash estimate. Namely, the solution is bounded as a consequence

SOME LOCAL PROPERTIES OF ELLIPTIC EQUATIONS 133
of the extended maximum principle, hence is easily seen to have square
integrable first derivatives in a neighborhood of the singularity, and is thus
a weak solution to which Moser's theorem applies; the Holder continuity of
the solution follows.
If the coefficients are Holder continuous in a neighborhood, the solution
has Holder continuous first derivatives up to and including the singularity.
For, by the extended maximum principle, the given solution must be identical
with the smooth solution that coincides with it on a sphere about the singular
point, and the latter solution has Holder continuous derivatives (see, for
example, Lichtenstein [15], or Hopf [13]). If the equation is not uniformly
elliptic at the singular point, examples show that a bounded solution need
not have a limit [6].
6. Asymptotic behavior. The problem of asymptotic behavior of
solutions has been only slightly explored and to date most results in this direction
appear to have been motivated by physical applications. The subject merits
further investigation.
We make the obvious remark that the theory of behavior of solutions at
infinity is not a simple extension of that at finite points. Even in the case
of elliptic equations whose coefficients are well-behaved at infinity, say
having a power series expansion in negative powers of r, the formal solution
expansions differ in an essential way from those for harmonic functions.
Thus, while any bounded harmonic function defined in the exterior of a
sphere has an expansion of the form a0 + ^Lm^n-2 ct>m(0)r~m, where 6 = xj\x\
denotes the angle variables, the general expansion of a bounded solution of
(2) (with bi — 0) takes the form
rn
(8) a0 + r*~n ^ 2 aim(0)(log r)« *-».
m^O / = 0
The terms in powers of log r are typical—but not universal—in expansions
about infinity for equations with variable coefficients. As is well known,
such terms do not appear in the expansions about finite points. The
existence of such expansions were already observed in connection with the theory
of subsonic flows (see [7] and the accompanying references).
A general theory of the asymptotic behavior of solutions has been obtained
by Meyers [16] for linear elliptic equations whose coefficients have a partial
expansion in negative powers of r and satisfy a Holder condition in the
neighborhood of infinity. He proves in particular that the general asymptotic
expansions of bounded solutions are of the form (8).
Consider next the limit behavior of solutions at infinity when the
coefficients have weaker regularity properties. For equations (2) of non-
divergence form with bi = 0, if the equation is uniformly elliptic and no
regularity is assumed of the coefficients, a bounded solution defined for
r > ro always has a limit when n = 2, but it is not yet known whether this is

134
DAVID GILBARG
true when n > 2. In the latter case, the limit, if it does exist, need not be
approached Dini continuously, as example (5), (7) shows. If the coefficients
are continuous at infinity, or have sufficiently small oscillation at infinity,
the solution has a limit, which is approached at the rate 0(r2_w+6), where 8
depends on the oscillation and can be taken arbitrarily small if the
coefficients have a limit. If the coefficients approach their limits Dini
continuously then 8 = 0. These statements are proved (in [10]) by a comparison
argument almost identical with the proof of the extended maximum principle
at finite points.
An extension of these results to include certain types of non-uniform
ellipticity has been given by Meyers and Serrin [17], and can be stated in one
form as follows. Suppose the coefficients in (2) (bi = 0) approach limits,
aik(x) -> a^t, as x -> oo, and let the matrix (a%k) have k positive eigenvalues.
Then if u has the limit uo> it follows that u — uo = 0(r2~k+8) for any 8 > 0.
Thus, the equation need not be elliptic at infinity provided k ^ 3. In the
general non-uniformly elliptic case for (2) (bi = 0), when the coefficients do
not have limits at infinity, a bounded solution need not have a limit [14].
These limit theorems plus the maximum principle yield obvious extensions
of Liouville's theorem that bounded entire solutions are constants. Whether
such a Liouville theorem is true for uniformly elliptic equations (2) (bi = 0)
without additional assumptions in the coefficients is not known for n > 2.
In two dimensions (but not for n > 2 [14]) the result is true even for non-
uniformly elliptic equations, as a consequence of Bernstein's geometric
theorem [1]. However, in the uniformly elliptic case a stronger result can be
stated; namely, that a solution defined over the entire plane and bounded on
one side is necessarily a constant [10]. This is false in general for non-
uniformly elliptic equations.
Consider now the asymptotic behavior of equations of divergence form (1),
which we assume to be uniformly elliptic in an exterior domain r > r$.
Without further assumptions on the coefficients we have the following limit
theorem. A solution that is bounded on one side for r > tq has a limit
(possibly infinite) at infinity.
This result is based on the following Harnack inequality. Let u be a
positive solution of the uniformly elliptic equation (1) in a sphere S of radius
r, and let S' be the concentric subsphere of radius r/2. Then there is a
constant K depending only on the ellipticity modulus of (1) and the
dimension n (but not on r), such that the inequality
(9) u(x) S Ku(y)
holds for all x, y eS'.
For n = 2 this Harnack inequality is proved by a simple estimation of the
Dirichlet integral of log u (Bers and Nirenberg [4]). For general n ^ 2 the
result was asserted by Nash without details of proof [23, p. 953] and was later
proved by different methods by Moser [22]. Analogous Harnack inequalities

SOME LOCAL PROPERTIES OF ELLIPTIC EQUATIONS 135
for equations of non-divergence form (2) have been proved by Serrin [25] and
were used to obtain some of the earlier stated limit theorems for these
equations. However, when n > 2 his results presuppose a modulus of continuity
for the coefficients.
The above limit theorem follows easily from the Harnack inequality. For
let u be the given solution, and suppose for convenience that u ^ 0. If u
does not have the limit + oo, it has a finite lower limit, and without loss of
generality we may assume lim inf u(x) = 0 as x -> oo. Hence for any e > 0,
there is a sequence of points x% —> oo on which 0 ^ u(xt) < e. We may
infer from the Harnack inequality (9), by using a chain of spheres, that
0 ^ u(x) S Au(xt) < Ae on the spheres \x\ = |x<|, where A is an
appropriate constant independent of the sequence x\. By the maximum principle,
the same inequality holds in the annular regions between the spheres,
therefore for all sufficiently large \x\, and hence lim sup^oo u(x) ^ Ae. Since e
was arbitrary, it follows that lim sup^-^oo u(x) — 0, and thus lim^oo u(x) = 0.
The limit statement can be made more quantitative in some cases. When
n = 2, it follows readily from the Holder estimates of Morrey that if the
solution is bounded it approaches its limit Holder continuously. This limit
behavior contrasts with that for solutions of (2). When n > 2, if the limit
is uo and the coefficients of (1) are continuous at infinity then, for arbitrary
S > 0 (cf. [8, p. 295]),
u — uo — 0(r2"w+6) as r-» oo.
For given 8, the same statement is true if the oscillation of the coefficients at
infinity is sufficiently small.
Added in proof. Since this paper was written Moser has announced the
following optimal result. Let (I), n > 2, be uniformly elliptic in r > ro, and
let u be a solution in this region, bounded in magnitude by M and having the
limit Uq at infinity. Then
\u - uo\ ^ CMr2-n,
where C depends only on the ellipticity modulus and n.
Finally, we remark that the above limit theorem and the maximum
principle together immediately imply the following Liouville theorem: Let
equation (1) be uniformly elliptic; then a solution defined throughout space
and bounded on one side is a constant. If n = 2, and the solution grows no
faster than o(log r), the Liouville theorem remains true even if the equation
is not uniformly elliptic [10] (it is assumed that the coefficients are symmetric
and bounded). Whether an analogous result is true for n > 2 is an open
problem.
7. Nonlinear equations—removable singularities. It has been observed by
Bers [2] and Finn [6] that certain nonlinear equations, unlike linear ones, do

136
DAVID GILBARG
not admit solutions with isolated singularities. Their results, proved for
equations in two variables, can now be generalized to n variables, largely as
a result of the De Giorgi-Nash a priori Holder estimate for solutions of (1).
We start with the following result of Finn [6]. Let Ajc(u) = At(ui, • • •, un),
k = 1,. . ., n, be C2 functions of u and suppose that the domain of values u
for which -4*,iM&£* > ®
is convex, where A^y\ = dAjcj diif, and £i,* • *, £n
are real, 2 £? ¥" 0. Let cp(x) be a C2 solution of the equation
,,0, ^(,1>...,„)^,„M^- = 0, „-£
in the punctured sphere S:0 < \x\ = r £ r0. Suppose this equation is
elliptic with respect to q> and that 22= i M*| = o(rl~n) as r -> 0. Then the
first derivatives cpx., i — 1, • • •, n, satisfy the extended maximum principle in
S.
This result implies directly that the first derivatives of <p are bounded and
that (10) is uniformly elliptic in S. We can conclude from this that the
singularity at x = 0 is removable. Namely, consider the linear equation,
satisfied by any first derivative v = dcpjdxj, obtained by differentiating (10)
with respect to Xj (j = 1,- • •, n) and then setting a^x) = Ajcti(u(x)). This
equation is
By the preceding, the coefficients are bounded and the equation is uniformly
elliptic; thus v (which is bounded) satisfies the conditions of the limit theorem
at the end of § 5 and is therefore Holder continuous in a neighborhood of
the origin. Since this is true of all the first derivatives it follows that the
coefficients aa in (11) are likewise Holder continuous in a neighborhood of the
origin. The solution v must therefore have a Holder continuous derivative
in this neighborhood. We conclude that cp e C2 in a region containing x = 0
and the singularity is removable.
The preceding theorem involved hypotheses on the growth of the functions
Ajc(ui,- • •, un) at the singularity, and these are fulfilled in practice either
because of the structure of the equation, or by virtue of known growth
properties of the gradient of the solution. An example of the former is the
minimal surface equation,
(12) s lo + z<rJ " "•
for which the functions Ak = <px /(l + 2 <pl)112 are automatically bounded.
The latter situation is exemplified in the theory of subsonic flows where the
gradient of the solution—the flow velocity—is known to be bounded.
When the theorem on removable singularities is stated in terms of the

SOME LOCAL PROPERTIES OF ELLIPTIC EQUATIONS 137
admissible growth of the solution itself, a different argument is necessary.
Suppose the following conditions are satisfied:
(i) The differential operator (dAk/dxic)(<pXv- • •, <pXn) is uniformly elliptic
with respect to arbitrary cp; namely, there is a constant A > 0 such that
(13) A-i2££4.*&&^2ff
for arbitrary cp.
(ii) |^4«f*| ^ const, i,k = 1,- • •, n. (By virtue of (i) this assumption is
superfluous if Ai^ = Ak,i, as in variational problems.)
(iii) cp is a C2 solution of (10) in 0 < \x\ = r ^ r0 with the property that
cp = o(r2_w) as r -> 0.
Under these hypotheses cp has a removable singularity at x = 0.
The proof requires the existence of the solution of the Dirichlet problem
for (10) under the assumption (i). This is discussed below. Suppose then
that cp is a C2 solution of (10) in the sphere 0 S \x\ S r0, coinciding with cp
on |a;| = ro. The function t/r = cp — cp, by the theorem of the mean, satisfies
a uniformly elliptic linear equation (in 0 < \x\ ^ ro),
where the coefficients are given by
aik~dXi = Jo JtAk^Xl ^ W*!'" ',(PX» + ^n)^'
By (ii), the coefficients aik are seen to be bounded. Since \fs = o(r2~n) as
r -> 0, the extended maximum principle shows that 0 = 0, and hence cp = <p.
The singularity of cp is therefore removable.
The crucial step in the above arguments is of course the existence of the
solution of the Dirichlet problem for (10). We outline the existence proof
here and present the complete details in a later work. The method is similar
to one used by Finn and the author [8] in proving the existence of subsonic
flows.
Let D be a bounded region in En with boundary D and let / be a function
defined on D (D and/ are here assumed sufficiently smooth). Let the
functions Ai(u\r • •, un), ut = cpx.9 i = 1,- • •, w, satisfy the uniform ellipticity
condition (13). We establish the existence of a C2 solution cp of (10)
coinciding with/ on D.
Let 2 be the Banach space of vectors u(x) = (u\(x), • • •, un(x)) defined in D
with the norm
.... ii.. , t \u(x) — u(y)\
\\u\\ = lub \u\ + lub^-V^ r^-
11 " d ' ' d \x - y\a
where 0 < a < 1. The particular choice of a will be determined later.

138
DAVID GILBARG
Let an arbitrary element u of 2 be inserted in the coefficients Akj(u) of
(10), and consider the linear uniformly elliptic equation thus obtained,
namely
<14> ^^ = 0>
where a^x) = Ak,i(u(x)). Suppose 0(x) is the solution of this equation in
D having the boundary values O = / on D. Since the coefficients a^ are
Holder continuous with exponent a in D, such a solution exists and U =
grad Oe2. This procedure defines a transformation u -> Tu = U of 2 into
itself. The above boundary value problem for (10) is solved if the equation
u = Tu is shown to have a solution.
Consider the family of equations
(15) u - kTu - 0, 0 S k S 1.
By virtue of the Leray-Schauder fixed point theorem—in this case a
simplified version due to Schaefer [24] suffices—this family of equations has a
solution u{x] k) for each k, 0 ^ k S 1, provided that (a) T is completely
continuous in 2—which follows easily from the Schauder theory of linear
elliptic equations; and (b) the solutions of (15) are uniformly bounded in
2 : \\u(x; k)\\ < C, where C is independent of k.
To prove the latter a priori estimate we observe to begin with that the
first derivatives of the solution of (14) are bounded on Z) by a constant
depending only on A, /, and D,
(16) lub |gradO| ^ M - if (A,/, D).
b
(This is proved by a simple comparison argument based on the Hopf
maximum principle.) Hence solutions of (15) also enjoy this inequality.
Furthermore, \u\ ^ M throughout D. For each first derivative v = dyjdxj,
j = 1, • • •, n, satisfies equation (11), and since the solutions of (11) obey the
maximum principle, we infer that \u\ — |grad 0| ^ M in D.
The results of De Giorgi, Nash, and Moser show that solutions of (11) in
any compact subregion D' <= D satisfy
(i7) '"''"iWI4"H
f ~~ y\a ~~ da d
where C = C(A, n), a = a(A, n), 0 < a < 1, and d is the minimum distance
between D' and i). This value of a is the one appearing in the definition of
2. By the preceding we have that the solutions of (15) obey the inequality
\u(x) - u(y)\ ^ const
\x - y\a ~ da
where the constant is independent of k (0 ^ k S 1).

SOME LOCAL PROPERTIES OF ELLIPTIC EQUATIONS 139
It remains to show that a similar inequality holds in a neighborhood of D.
Combined with (18) and the previous bound \u\ ^ M, this will establish the
required uniform bound for \\u\\. In a neighborhood of each boundary point
P a suitable point transformation x <—> £ can be found which takes the surface
element of D at P into a plane element £n = 0 and the equation (10) into
one of the form
—- = 0, where Bk = Bk((ji, • • •, £„> <p«,» • • •, <pen)>
By subtracting an appropriate function from cp we may suppose also that
cp = 0 on D.
The equation analogous to (11) satisfied by the first derivative v = dcpldtjj,
j = 1,. .., n, is now of the form
Wk
0.
This equation and the solution v can be defined by reflection across the plane
£n = 0, the coefficients being discontinuous in general on the plane.
Interior Holder estimates obtained by Morrey [18] for equations of this form
now establish the desired Holder property for solutions of (15) in the
neighborhood of the boundary. We forgo the details here and present a
complete account elsewhere.
Various generalizations are possible in both the existence theorem and its
application to removable singularities. These allow more general ellipticity
assumptions than in (13) and permit the independent variables to appear in
the functions At. These extensions use the results of Morrey [18; 19].
8. Nonlinear equations—asymptotic behavior. Let cp be a uniformly
elliptic solution of (10) with bounded gradient in the exterior of a sphere.
As we have seen, each derivative v = <px (j = 1,- • •, n) satisfies a linear
uniformly elliptic equation (11) with bounded coefficients. It follows from
the results stated in § 6 that u = grad cp has a limit uq at infinity. This
implies in turn that the coefficients in (11) are continuous at infinity, and
hence \u — uq\ = 0(r2~n+6)y as r->co, for any 8 > 0. Equation (10),
viewed as a linear equation, has Holder continuous coefficients at infinity,
and cp — uox is a bounded solution. It follows (by § 6) that for some
constant (po
cp — <pQ = uqx -j- 0(r2~n) as r-~> oo.
This bootstrap argument, if pursued further, yields an asymptotic expansion,
in general of the form (8). The existence of later terms in the expansion
depends of course on suitable differentiability of the functions Ai(u) in (10).
Further details concerning these expansions will appear in a later work.
An immediate application is to the theory of subsonic flows in which the

140
DAVID GILBARG
equation for the velocity potential in a special case of (10). By the above,
if a strictly subsonic flow (i.e., a uniformly elliptic solution) is defined in the
exterior of a sphere, the velocity vector has a limit uo at infinity. It can be
shown, furthermore, that the potential has an asymptotic expansion of the
special form
V = «o*+ I ^ (0 = zl\z\).
Such an expansion holds also for solutions of the minimal surface equation
(12) having a bounded gradient in the exterior of a sphere.
In the case of plane flows, or minimal surfaces over the plane, analogous
results are true under the weaker hypothesis that the solution is pointwise
rather than uniformly elliptic (cf. [2; 7]).
References
1. S. Bernstein, Vber ein geometrisches Theorem und seine Anwendung auf die
partiellen Differentialgleichungen vom elliptischen Typus, Math. Z. vol. 26 (1927) pp.
551-558.
2. L. Bers, Isolated singularities of minimal surfaces, Ann. of Math. vol. 53 (1951)
pp. 364-386.
3. , Local behavior of solutions of general linear elliptic equations, Comm. Pure
Appl. Math. vol. 8 (1955) pp. 473-496.
4. L. Bers and L. Nirenberg, On linear and nonlinear boundary value problems in the
plane, Atti Convegno Internazionale sulle Equazioni Derivate Parziali, Trieste, 1954,
pp. 141-167.
5. E. De Giorgi, Sulla differ enziabilita e Vanaliticita delle estremali degli integrali
muUipli regolari, Mem. Accad. Sci. Torino Ser. 3a vol. 3 (1957) pp. 25-43.
6. R. Finn, Isolated singularities of solutions of nonlinear partial differential equations,
Trans. Araer. Math. Soc. vol. 75 (1953) pp. 385-404.
7. R. Finn and D. Gilbarg, Asymptotic behavior and uniqueness of plane subsonic
flows, Comm. Pure Appl. Math. vol. 10 (1957) pp. 23-63.
8. , Three-dimensional subsonic flows and asymptotic estimates for elliptic
partial differential equations, Acta Math. vol. 98 (1957) pp. 265-296.
9. D. Gilbarg, Some hydrodynamic applications of function theoretic properties of
elliptic equations, Math. Z. vol. 72 (1959) pp. 165-174.
10. D. Gilbarg and J. Serrin, On isolated singularities of solutions of second order
elliptic differential equations, J. Analyse Math. vol. 4 (1955-1956) pp. 309-340.
11. E. Hopf, Elementare Betrachtungen uber die Losungen partieller
Differentialgleichungen zweiter Ordnung vom elliptischen Typus, S.-B. Preuss. Akad. Wiss. vol. 19
(1927) pp. 147-152.
12. , A remark on linear elliptic differential equations of the second order, Proc.
Amer. Math. Soc. vol. 3 (1952) pp. 791-793.
13. , Zum analytischen Charakter der Losungen reguldrer zweidimensionaler
Variationsprobleme, Math. Z. vol. 30 (1929) pp. 404-413.
14. , Bemerkungen zu einem Satze von S. Bernstein aus der Theorie der
elliptischen Differentialgleichungen, Math. Z. vol. 29 (1928) pp. 744-745.
15. L. Lichtenstein, Zur Theorie der konformen Abbildung. Konforme Abbildung
nicht analytischer singularitatenfreier Fldchenstuicke auf ebene Gebiete, Bull. Internat.
Acad. Sci. Cracovie A, 1916, pp. 192—217.

SOME LOCAL PROPERTIES OF ELLIPTIC EQUATIONS 141
16. N. Meyers, Expansion of solutions of second order linear elliptic equations about
infinity, Notices Amer. Math. Soc. vol. 5 (1958) p. 335.
17. N. Meyers and J. Serrin, The exterior Dirichlet problem for second order elliptic
partial differential equations, J. Math. Mech. vol. 9 (1960) pp. 513-538.
18. C. B. Morrey, Second order elliptic equations in several variables and Holder
continuity, Math. Z. vol. 72 (1959) pp. 140-164.
19. , Existence and differentiability theorems for variational problems for
multiple integrals, Partial Differential Equations and Continuum Mechanics, University
of Wisconsin Press, 1961, pp. 241-270.
20. , Second order elliptic systems of differential equations, Ann. of Math. Studies
No. 33, Princeton University Press, 1954, pp. 101-159.
21. J. Moser, A new proof of de Giorgi*s theorem concerning the regularity problem for
elliptic differential equations, Comm. Pure Appl. Math. vol. 13 (1960) pp. 457-468.
22. , On Harnack's theorem, Notices Amer. Math. Soc. vol. 7 (1960) p. 981.
23. J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math,
vol. 80 (1958) pp. 931-953.
24. H. Schaefer, Vber die Methode der a priori Schranken, Math. Ann. vol. 129 (1955)
pp. 415-416.
25. J. Serrin, On the Harnack inequality for linear elliptic equations, J. Analyse Math,
vol. 4 (1955-1956) pp. 292-308; also Notices Amer. Math. Soc. vol. 5 (1958) p. 52.
Stanford University,
Stanford, California

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ESTIMATES AT INFINITY FOR STEADY STATE SOLUTIONS
OF THE NAVIER-STOKES EQUATIONS
BY
ROBERT FINN
This paper outlines the second part of our study on the behavior of
time-independent solutions of the Navier-Stokes equations
/xAw - pw»Vu> - Vp = 0,
(1)
V-iv = 0
which are defined in a neighborhood of infinity in two- or three-dimensional
Euclidean space. The notation is the usual one of vector analysis. In (1),
u> = u>(x) can be considered as the velocity field of an incompressible fluid
flow. The constant /x is the viscosity coefficient of the fluid, and fcAw
represents the shearing forces in the motion. The density of the fluid is p
(assumed constant), and piv*Vw corresponds to the inertial reaction of the
fluid elements. The term Vp denotes the forces due to changes in the
pressure p. Thus, the first equation in (1) expresses the equilibrium of these
three forces at all points of the flow field.
In what follows, we shall assume that /x = p = 1. This normalization
can always be achieved by multiplication of u> and of p by constant factors,
and we shall assume this done. The relations (1) then take the form
Aw — W'Vw — Vp = 0,
(la)
V-u> = 0.
A vector field w(x) will be called a solution of (la) in a region <?, provided
there exists a scalar p(x) such that (la) is satisfied by the pair (u>, p). We
shall assume throughout that $ contains a neighborhood of infinity.
In marked contrast to the solutions of the equations of potential fluid
flow (see, e.g. [2; 3]), the solutions of (la) which tend to a limit u>0 at infinity
need not be representable by asymptotic expansions. In particular, there
is in general no positive y such that \w — u>o| = 0{r~y) as r-~>co. An
example is provided by the two-dimensional solution1 defined by the relations
u> = (wi, w2)9 x = (xu x2),
1 In three dimensions, the situation is less clear; see the remarks in §7.
143

144
ROBERT FINN
2
^2 = -(1 - a)r"° T - (1 + a) -g
Xl /i . \*2
(1 + a) —
r r2
for any real a. For any a in the range 0<a<l,w->0 and ra| u>| —>■ 1 — a
as # -> oo. Letting a^Owe see that no estimate of the proposed form can
exist.
Nevertheless, under certain natural assumptions it is possible to derive
information of an a priori character on the behavior of the solutions.
1. We have proved in [1] that if w(x) is a three-dimensional solution of (la)
in $ and if the Dirichlet Integral2 of w(x) is finite, i.e., if /«? |Vu>|2dF < oo,
then there exists a vector u>0 such that w(x) -> u>0 as x -> oo. As a corollary
to this result, we have shown that the (generalized) solution of the exterior
boundary value problem given by Leray in [4] necessarily assumes the
prescribed data at infinity. We have, however, been unable to determine
whether every such solution admits an asymptotic development in terms of
given functions of r.
2. In [5] we discard the assumption that the Dirichlet Integral is finite;
instead, two alternative types of hypothesis are introduced.
Hypothesis A: There exists a vector w0 such that w(x) —>- w0 as x —>- oo.
From Hypothesis A we derive the consequence that all derivatives of w(x)
tend to zero, \Vw\ -> 0 as x -> oo. The proof depends on a representation for
the solutions of (la) by means of a fundamental solution tensor x(#> y) = (xu)
associated with the linearized equations 3
Au> — u>o-Vu> — Vp = 0,
(2)
V.u> - 0.
Such a tensor has been determined explicitly by Oseen [6, p. 34]. Setting
u(x) = w(x) — w0i we obtain the representation, analogous to a classical
formula of potential theory,
(3) u(x) = j> {u.Tx -x-Tu + (x-u)(">o-n)}dSy + f X-("-Vu)iF,
for any solution of (la) defined in a region ^ bounded by a closed surface 2.
2 This integral can be interpreted physically as half the sum of the rate at which energy
is converted into heat by shearing motion in the flow, and the total vorticity in the
flow.
3 These are the equations satisfied by infinitesimal perturbations of the solution
tv = wq of (la).

SOLUTIONS OF THE NAVIER-STOKES EQUATIONS 145
Here
is the stress tensor associated with the motion, and
The term X'^u is to be understood in the sense (x*Tu)i = xai^11)^71^
where n = (wi, n^ w3) is the exterior directed unit normal, and summation
is extended over repeated indices.
Let w(x) -> w0 at infinity, and let m(r) = max|*|^r |w>(#) — u>o\. We
apply (3) to a unit sphere centered at x, and compute formally a difference
quotient Shu(x) by taking differences of x(#, y) under the sign. The crucial
point is, it is then possible to deform (3) so that no derivatives of u(x)
appear on the right hand side, and so that no differences of x(*> y) (which
are not integrable in the limit) appear in the volume integral. The proof of
this fact is not intrinsically difficult, but it is technical and uses an existence
theorem. We are led to an inequality of the form
(4) yh(r) ^ Cm(r - \)yh(r - 1) + Cm(r - 1)
where we have set yh(r) = max(X|^r |8'*u(^)|. (Since m(r) -> 0, so does
yh{r) for fixed h; hence the maximum is achieved at a finite point.) In (4),
C does not depend on ft.
Since m(r) -> 0, there exists an A > 0 (independent of ft) such that
Cm(r — 1) < 7] < 1 for r ^ A. Iteration of (4) shows that
(5) yh(r) ^ Cm(r - lW^-^A) + Gm(r - 1) ——
1 - 7]
where we have chosen N as the largest integer such that r — N ^ A. From
(5), we conclude the existence of a number B > A such that yh(B) ^ yh(A)\2
uniformly in an interval 0 < |h| < 8 (unless yh(A) -> 0 as h -> 0, in which
case the result is trivially correct). It follows that the maximum of |Shtt(#)|
in |*| ^ A will be achieved at a value of \x\ in the interval A ^ \x\ ^ B
for all sufficiently small h. But as h --> 0 these difference quotients tend
uniformly in this annular region to the corresponding derivatives of u(x),
and hence remain uniformly bounded. We then conclude from (5) the
existence of a constant C such that yh{r) ^ Cm(r — 1), and the desired
result follows by letting h -> 0.
Similar estimates on the pressure p, and an integral equation satisfied
by the solution, can also be derived. Details are included in [5].
3. If a suitable rate of decay of the solution to its limit is assumed, much
more far-reaching results can be obtained. We consider here only three-
dimensional solutions of (la).

146
ROBERT FINN
Hypothesis B. There exist positive constants C and e, and a vector u>0 ^ 0,
such that \w(x) — u>o| < Cr~1/2~e as x -> oo.
Any solution of (la) in £ which satisfies Hypothesis B behaves asymptotically,
up to terms of smaller order, as a^yj^x, 0) + 6V(l/r), where a is a constant
vector, b is a constant, and x is the fundamental solution tensor for the system (2).
From this, we can demonstrate the existence of a paraboloidal "wake"
region in the direction of the vector u>0, in which \w(x) — u>0| < Cr'1.
Outside this region, the solution tends to its limit progressively more rapidly
until, exterior to a semi-infinite cone whose axis extends to infinity along the
direction of u>0, we have \w(x) — u>0| < Cr~2. For any closed surface S
whose exterior is S, w(x) admits the representation (obtainable formally
from (3) by an integration by parts)
(6) w{x) = w0 + <j> {u-Tx - X'Tu + (X-^X^o-n) + (X'U)(u*n)}dSy
- j^u.{u.Vx)dVy
where u(x) = w(x) — u>o. The above expression, fl*x(#>'0) + 6V(l/r), is
obtained by examining the integral over S for large values of r. The essential
difficulty lies in the estimation of the volume integral. We show in [5] that
this integral, considered as an operator applied to any vector field u(x)
which satisfies Hypothesis B, yields a new field u(x) which satisfies an
improved estimate in directions outside the wake region. But the same
operator applied to u gives rise to a vector field which tends to zero in all
directions more rapidly than was known for u(x). Iteration of these
estimates a finite number of times then shows that the solution w(x) in (6) tends
to u>o in any direction at least as rapidly as does the surface integral. The
order of decay of this integral is, however, easily estimated. Inserting the
estimate in the volume integral shows that this last term behaves, for any
8 > 0, as 0(r~3/2+8) in the wake, and then decays progressively more
rapidly, until outside the wake the order is 0(r~5/2+6). These results, and
corresponding estimates for the derivatives of w(x) and for the pressure, are
stated precisely in [5]. They can be interpreted as yielding an estimate for
the contribution of the nonlinear terms in (6) to the values of the solution
near infinity. The demonstrations are elementary, being based for the most
part on the type of reasoning encountered in classical potential theory, but
they are much too complicated in detail to reproduce here.
4. We consider now an application of the developments in §3. Suppose
that $ is the exterior of a smooth closed contour 2 and that u> = 0 on 2.
The solution can then be interpreted physically as representing a steady flow
of a viscous fluid past a rigid obstacle to which the fluid adheres. Using
methods which are essentially classical, the force exerted on £ by the fluid
can be computed by means of integrations taken over a sphere S« of large

SOLUTIONS OF THE NAVIER-STOKES EQUATIONS 147
radius which encloses S in its interior. If the calculations are carried out,
it is seen that if w(x) satisfies Hypothesis B, then the nonlinear terms in (6)
yield no contribution to the force. The force on S is completely determined by
an appropriate solution of the linear equations (2), and in this case w(x) — tt>o
behaves at infinity, up to terms of smaller order, as 3F -x(x> 0)» where 9* is the
force exerted on 2 in the motion.
5. As a second application, we prove that if w(x) is a solution of (la) in &
if w(x) satisfies Hypothesis B, and if w(x) = w* = const, on S, then the scalar
product of (u>o — w*) with the force exerted on 2 by the fluid is equal to the rate
at which energy is converted into heat in the motion. This theorem is obtained
by a formal integration by parts, the only technical difficulty being to
estimate an integral over the spherical outer surface E#. The estimate is easily
effected with the aid of the results of §3. A particular consequence of this
result is that in a nontrivial flow past S with adherence on S, the component of
the force exerted on S in the direction u>o cannot vanish A Another consequence
is that if tv(x) = const, on S and if \w(x).— u>o| = o(r_1), then* u>(x) = u>0 in
S. In fact, it follows easily from the estimates of §3 that if \w(x) — u>o|
= o^r"1) then the force on S vanishes, hence energy is conserved in the flow,
and we conclude easily that the solution represents a rigid body motion.
Since w{x) -> u>0, the result follows.
6. The developments in §3 permit us also to prove a theorem of Liouville
type. If w(x) is a solution of (la) throughout three-dimensional Euclidean
space, and if w(x) satisfies Hypothesis B, then w(x) = u>o. In fact, we then
have for a sphere Vr bounded by S/j,
f |Vw|W = <j) U'-Tp - 2UH"-n) - -zuHwo-n) - p(f*.n)
dS
with u(x) = w(x) — Wq. The results of §3 show that the surface integral
vanishes in the limit, hence u>(x) = const. = u>0, q.e.d.
7. Because of the implications of Hypothesis B, it would be desirable to
prove the existence of a solution of the exterior boundary value problem in a
class of functions which satisfy this condition. The best theorem presently
available asserts the existence of a solution which is continuous at infinity
and has finite Dirichlet Integral [4; 1]. Using the results of [5], this problem
can be reduced to the solution of an integral equation in which the integral
operator transforms the desired class into itself. The step which remains
4 We remark the contrast between this result and the known result that in any
potential flow past S, the force on S necessarily vanishes [2; 3].
5 A theorem of this type appears first in Berker [7], under rather elaborate
assumptions on the velocity field. Granting the existence of solutions which satisfy
Hypothesis B, the result we give is best possible in the sense that the actual behavior of a
nontrivial solution must be \w(x) — w0\ = O^-1) and not ofr-1).

148
ROBERT FINN
outstanding is to obtain an a priori bound on the constant (7, depending only
on prescribed data.
Another problem of interest is to determine whether there are solutions
which satisfy Hypothesis A and do not satisfy Hypothesis B. The example
of §0 shows that in two dimensions, singular behavior of this type must be
expected. In three dimensions, the situation is less clear, but there is
some evidence to suggest that the solutions are less singular at infinity.
In fact, [8], if w(x) -~> w$ as x -~> oo in three dimensions, and if w(x) has finite
Dirichlet Integral, then in almost all radial directions from the origin, \w(x)
— wq\ < C\x\~~ll2. This question is of importance in connection with
iterative methods of constructing solutions of (1); see, e.g. [9].
References
1. R. Finn, On steady-state solutions of the Navier -Stokes partial differential equations,
Arch. Rat. Mech. Anal. vol. 3 (1959) pp. 381-396.
2. R. Finn and D. Gilbarg, Asymptotic behavior and uniqueness of plane subsonic flows,
Comm. Pure Appl. Math. vol. 10 (1957) pp. 23-63.
3. , Three-dimensional subsonic flows, and asymptotic estimates for elliptic
partial differential equations, Acta Math. vol. 98 (1957) pp. 266-296.
4. J. Leray, iStude de diverses equations integrates non lineaires et de quelques problemes
que pose Vhydrodynamique, J. Math. Pures Appl. vol. 12 (1933) pp. 1-82.
5. R. Finn, Estimates at infinity for solutions of the equations of viscous incompressible
fluid flow, Technical Report No. 84, Stanford University, 1959.
6. C. W. Oseen, Neuere Methoden und Ergebnisse in der Hydrodynamik, Leipzig,
Akademische Verlagsgesellschaft m.b.H., 1927.
7. R. Berker, Sur les forces exercees par unfluide visqueux sur un obstacle, Rend. Circ.
Mat. Palermo vol. 2 (1952) pp. 260-280.
8. R. Finn, On the steady-state solutions of the Navier-Stokes equations, III, to appear
in Acta Math.
9. I. Imai, On the asymptotic behavior of viscous fluid flow at a great distance from a
cylindrical body, with special reference to Flion's paradox, Proc. Roy. Soc. London
Ser. A. vol. 208 (1951) pp. 487-516.
Stanford University,
Stanford, California

INTERIOR ESTIMATES FOR SOLUTIONS OF ELLIPTIC
MONGE-AMPERE EQUATIONS
BY
ERHARD HEINZ
Introduction. A partial differential equation of the form
(1) F(x,y,z,p,q,r,s,t) = 0
(p = Zx, q = Zy, r = ZXx, 8 = %xy> * — zyy)y
is called elliptic for a given real solution z = z(x, y), if
(2) D = FrFt -\F*> 0.
In this paper we shall be concerned with Monge-Ampere equations of the
form
(3) F = Ar + 2Bs + Ct + (rt - s2) - E = 0
(A = A(x, y,z,p,q),-', E = jB(x, y, 2, p, g)).
For any solution z(x, y) of (3) we have
(4) D = J?1^ - i Jf = (^ + t)(C + r) - (B - s)2 = AG - B2 + E.
Ellipticity of (3) therefore means that
(5) D = AC - B* + E > 0.
An important step in proving existence theorems for equation (3) consists
in deriving interior estimates for the second derivatives of the solutions. In
this paper we shall establish such estimates for a large class of Monge-
Ampere equations and thus generalize our previous results.1 As in Lewy's
fundamental papers our main result (Theorem 2) strongly depends on the
intimate connection between equation (3) and certain nonlinear elliptic
systems of second order. These will be considered first.
1. Preliminary results on elliptic systems of second order.
Theorem 1. Hypotheses.
(I) The functions £ = £(u, v) and 17 = r)(u, v) are real-valued and of class C2
1 See [2], also for further references on this subject.
149

150 ERHARD HEINZ
for u2 + v2 < 1 and map the unit disk u2 + v2 ^ 1 homeomorphically onto
the unit disk £2 + 772 ^ 1 swcA £to£
(1.1) £(0,0) = 7,(0,0) = 0
and
(i.2) £„77V - ivrju ^0 (142 + „2 < i).
(II) TAe functions £(u, v) and rj(u, v) satisfy for u2 + v2 < 1 the partial
differential equations
+ A3(£ *?)fo5 + *?5) + KiL v)iLvv - ZvVu)
and
Ar, = «,(£, ,)(£ + ^) + fat, r,)(iuVu + fo.)
+ h(L V)(vl + Vv) + k(L V)(LVv ~ Liu)-
Here the coefficients hi(£, 77), • • •, A.j(£, i) are of class C1 for £2 + tj2 ^ 1 and
satisfy for g2 + -q2 ^ 1 the inequalities
(1.5) | (\h.{g,r,)\ + \k(i,V)\) < Mo < oo
and
(!••) 2(1^1 +
4 /I
< M i < oo
^77
where the functions cov(£, 77) (r = 1,- • •, 4) are defined by the equations
(1.7) «n(£ l) = *i(f, >j),
(1.8) «>s{$,-n) = W£,v)-h(£,ri),
(1.9) o,8(£, i?) = ML l) - Ht, ij),
and
(1.10) «m(£, 17) = Aa(£, I)-
(Ill) JFe have the inequality
(l.H) IT (£5 + vl+& + rfadudv g iV < 00.
U* + V2 < 1
Conclusion. Let 0 ^ p < 1 and 0 < v < 1. TAen there are three fixed
positive numbers \\ = Ai(ilfo> Jfi, N, p), A2 = A2(jlfo, Mi, N, p, v), arid A3 =
\s(Mo, Mi, N, p) such that the following estimates hold:
(1.12) |£u|,-'-,M S Ai (1*2 + „2 <g ^2),

SOLUTIONS OF ELLIPTIC MONGE-AMPfiRE EQUATIONS 151
(1 13) \^(u2,v2) - {u(ui,vi)\ ^ X2.[(u2 - u±)2 + (v2 - vi)2?/2,
\y]v{u2,V2) - -qv(UuVi)\ ^ A2-[(^2 ~ UiY + (v2 - ^l)2]^2
(u2 + v\ ^ p2, u\ + v\ <t p2),
and
(1.14) \£uyv - £vnu\ ^ A3 (u2 + v2 ^ p2).
This theorem is a generalization of Theorems 10 and 11 of [1]. For a proof
see [3, §2].
2. Elliptic Monge-Ampere equations. Our next object consists in
exhibiting the connection between the Monge-Ampere equation (3) and
elliptic systems of the form (1.3) and (1.4). This is done in the following two
lemmas. For a proof we refer to [2, pp. 19-24].
Lemma 1. Let z — z(x, y) be a real-valued function of a class C2 in a
domain D, of the xy-plane, which satisfies for (x, y) e ii the Monge-Ampere
equation (3), where the coefficients A = A(x, y,z, p, q),- • • ,E = E(x, y, z, p, q),
considered as functions of the variables (x, y, z, p, q), are of class C2 in the
vicinity of each point of the hyper surf ace
(2.1) T - {(x, y, z(x, y), p(x, y), q{x, y)); {x, y) e £}}.
Furthermore let
(2.2) D = AC-B2 + E>0
for (x, y, z, p, q) e T, and let the disk (x — x0)2 + (y — yo)2 ^ p2 (p > 0) be
contained in ti.
Then there exists a pair of real-valued functions x = x(u, v) and y = y(u, v)
with the following properties:
(I) x = x(u, v) and y = y(u, v) are of class C2 for u2 + v2 < 1 and map the
unit disk u2 + v2 ^ 1 homeomorphically onto the disk (x — xo)2 -f (y — yo)2
^ p2 such that
(2.3) x(0, 0) = xo, y(0, 0) = y0,
and
(2.4) xuyv - xvyu #0 (u2 + v2 < 1).
(II) The functions x = x(u, v) and y — y(u, v) satisfy for u2 + v2 < 1 the
partial differential equations
(2 5) A* = k^X* V^X" + °^ + k^X' y^X^u + XvV^
+ hfa y)(yl + yl) + ^fo y)(x^v - *«#.,)>
and
(2 6) ^y = ^x* y^x* tx^ + ^Xi y)(x«y«„+ x«yJ
+ h(*> y)(yl + yl) + h(x> y)(xu2/v - xvyu),

152 ERHARD HEINZ
where the coefficients ki(x, y), ■ ■ ■, k^x, y) are defined, by the following
expressions:
ki(x, y) = Bq- ^ (Dx + Dzp - DPC + DqB),
h{x, y) = -A, - Bp - 2q(Dv + Dzq + DPB - DqA),
(2.7)
and
h(x, y) = Ap,
k*(x> V) = 7pu72 \AX + B„ + A& + B4 - APC
+ (Aq + BP)B - BgA - ^ Dp\,
h{x, y) = Gg,
h(x,y)= -Bg-Gp- ^(Dx + Dzp - DPC + DqB),
h(x, y)~ BP- ^ (D„ + D4 + DPB - DqA),
(2.8)
U(x, y) = jp—2 (Cy + Bx + Czq + Bzp - BPC
+ (Bq + CP)B -CgA - ±Dq\.
(Ill) For u2 + vz < 1 we have the representations
A + t = x\ + x\ B - s = xuyu + xvyv
{D)V2 xuyv - xvy„' (D)1'2 xuyv - xvyu'
(2.9)
C + r = yl + y\
(D)1'2 xuyv - xvyu
Lemma 2. Let the hypotheses of Lemma 1 be satisfied. Furthermore, let
(2.10) \A\,.--,\E\^«,
and
(2.11) D = AG - S2 + E ^ a-i
for (x, y, z, py q) e T, where a is a fixed positive constant, and
(2.12) \p(x,y)\,\q(x,y)\SlS< cc

SOLUTIONS OF ELLIPTIC MONGE-AMPERE EQUATIONS 153
for (x, y) e O. Then for the mapping functions x = x(u, v) and y = y(u, v),
defined in Lemma 1, we have the estimate
(2.13) IT (xl + yl + xl + yl)dudv <, 2naWp2 + ina^p.
U* + V*<1
We are now in a position to generalize Theorems 2 and 3 of [2] and to establish
the principal result of this paper.
Theorem 2. Hypotheses.
(I) The function z = z(x, y) satisfies all the hypotheses of Lemma 1.
Furthermore we have
(2.14) \p(x,y)\,\q(x,y)\ g /30 < oo
for (x, y) e £2.
(II) The coefficients A = A(x, y, z, p, q),- • •, E = E(x, y, z, p, q), occurring
in the Monge- Ampere equation (3), satisfy for (x, y, z, p,q) e T the inequalities
(2.15) \A\,--.,\E\ g a0,
(2.16) D = AG - B* + E ^ of1,
and
(2.17) \At\,...,\E9\£au
where ao and ai are two fixed positive constants.
(III) The functions
(2.18) <f,1(x,y) = Ap,
(2.19) <£2(z, y) = AQ + 2BP,
(2.20) fa(x, y) = Cp + 2Bg,
and
(2.2i) m*, y) = cQ
satisfy for (x, y) eQ. the inequality
(2.22) 2 (IXsI + \f\)^ yo<°°-
Conclusion. Let p > OandO < v < 1. Furthermore, let Qp be the subset of
all those points (x, y) e O, whose distance from the boundary of Q. is greater than

154
ERHARD HEINZ
p. Then there exist two fixed positive numbers 0o = 0o(<*o, <*i, j8o, yo , p) wnd>
0X = 0!(ao, ai, j3o, yo, p, v) such that the following estimates hold:
(2.23)
and
(2.24)
M.M.M ^®o
((», i/) g tip),
|r(*2,2/2) - r(xl9 yi)| £ 0i[(z2 - *i)2 + (2/2 - yi)2]"'2,
\t(x2,2/2) - *(*i,2/i)| ^ 0i[(*2 - *i)2 + (2/2 - 2/i)2]"/2
((m, i/i) g tip, (s2,2/2) e tip).
Proof. In virtue of Theorem 1 of [2] the inequality (2.24) is a consequence
of (2.23). Hence we may restrict ourselves to the proof of (2.23). Let
(#o, 2/o) be an arbitrary point in tip and let x = x(u, v) and y = y(u, v) be the
mapping functions defined in Lemma 1. Then from Lemmas 1 and 2 it
follows that the functions
(2.25)
and
(2.26)
£{u, v) =
r}(u, v) =
x(u, v) -
p
y(u, v) -
- x0
- 2/o
satisfy the conditions of Theorem 1. Using the formulas (2.7) and (2.8) we
obtain
(2.27)
(2.28)
(2.29)
(2.30)
(2.31)
and
(2.32)
hence
(2.33)
and
(2.34)
Mf> 1) = pM^o + pL 2/0 + m) (v = 1>-
M£> 1) = pk(x0 + pf, 2/0 + m) {v = i,-
a>i(f, -q) = Si(f, 7?) = fxf>4{x0 + pf, 2/0 + pq),
Wf > V) = *i(f> *?) - Mf, V) = pH(xo + /of, 2/0 + *"?),
<*>3(f> 1?) = h2(£, rj) - A3(f, rj) = -p<f>z(x0 + pf, y0 + /w?),
a>4(f, 7?) = Mf > *?) = p^i(^o + pf, yo + p^),
4
2 (iMf.*?)! + |M£*?)|) < -Mo(«o, «i, j80, p) = Jf0 < 00
4),
4),
2(
;1F
+
dcov
drj
\ < Mi(y0i p) = M1 < 00

SOLUTIONS OF ELLIPTIC MONGE-AMPERE EQUATIONS 155
for f2 + r)2 <; 1. Furthermore, on account of Lemma 2, we have the
inequality
(2.35) IT (£ + r,l + ev + rjDdudv ^ 2na^ + 4*rc#%p"1 = tf < 00.
tt* + I?" < 1
If we now apply Theorem 1 and use (2.9), we obtain the estimates
(2.36) \r(x0, y0\, \s(x0, y0)\, \t(x0i y0)\ ^ 0O
where
A (M M N 0\2
(2.37) 0O = aQ + 2(al + «o)m' £{x'0t Mu Nto} = ©o(«o> «i>/Wo> P) < <*>•
Since (#o, 2/o) is an arbitrary point in tip the theorem is established.
Bibliography
I.E. Heinz, On certain nonlinear elliptic differential eqtiations and univalent mappings,
J. Analyse Math. vol. V (1950-1957) pp. 197-272.
2. 9 On elliptic Monge-Ampere equations and WeyVs embedding problem, J.
Analyse Math. vol. VII (1959) pp. 1-52.
3. , Neue a-priori-Abschdtzungen fur den Ortsvektor einer Fldche positiver
Gauss'scher KriXmmung durch ihr Linienelement, Math. Z., to appear.
Stanford University,
Stanford, California

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ZERO ORDER A PRIORI ESTIMATES FOR SOLUTIONS
OF ELLIPTIC DIFFERENTIAL EQUATIONS
BY
H. 0. CORDES
This is a small contribution toward the following still unsolved problem.
Let
be a second order elliptic partial differential operator with coefficients being
defined and uniformly Hoelder-continuous in a bounded region R. No
terms of lower than second order are taken into this consideration.
Let the boundary T of jB consist of a finite number of simple noninter-
secting closed hyper-surfaces having all derivatives up to the order 2
continuous.
Assume that there exist two positive constants m and M such that
(2) m % g g | aik(x)tt* * M % &
i = l itk = l i = l
holds for all x e R and all real numbers £i, • • •, f n.
Let u = u(x) be a twice continuously differentiate solution of the
differential equation
(3) Lu=f, xeR
taking on the boundary values u = 0 at the boundary T of R.
(4) u(x) = 0, x g T.
Assume that f(x) is uniformly bounded over R. Accordingly the norms
(5) ll/Ho = sup \f(x)\,
xeR
as well as
(6) ||ft||0 and ||^||i = sup < |^(a?)|,
xeT I
are well defined numbers. Further, we define the Hoelder-constants
157
du
ydxl
? * * * >
du
dxn

158
H. O. CORDES
(7) Ha(u\ R) = sup
xl ^x1; xl,xleR
and
\u(xi) - u(x2)\
[x1 — x2\a
(8) H'a(u; R) = Max sup
r2 a
Question. Does there exist a constant c which can be determined already
from knowing the region R and the constants m, M, ||/||o, \\u\\o> and a such
that
(9) Ha(u;R) S c
holds for all solutions u(x) of the differential equation (3) having all derivatives
up to the order two continuous in jB and satisfying the boundary condition
(*)?
Or even more, does there exist a constant c which can be determined from
knowing the constants m, M, ||/||o> Mo, a and the region jB only such that
(10) H'a(u\R) £ c
holds for all solutions u(x) of the differential equation (3) having all
derivatives up to the order two continuous in R and satisfying the boundary
conditions (4)?
In particular it is of importance that the constant c can be chosen
independently of the continuity properties of the coefficients Oik(x).
An estimate of the type (9) or (10) with a constant c having the properties
described will be called a "zero order estimate".
Such estimates get their fundamental importance from the fact that they
imply existence of the solution of Dirichlet's problem connected to quasi-
linear second order elliptic equations. If (9) can be proven then Schauder's
fixed-point-theorem yields existence of a solution of the general equation
-A d2u
(11) A (Hk(x, u) = 0, xeR,
ijfc^i dXidXic
taking on given boundary values at the boundary T of jB provided that this
equation is elliptic and that certain smoothness conditions are imposed
upon the region R and the coefficients a<* as well as upon the boundary data.
Likewise the stronger a priori estimate (10) together with a similar a priori
estimate for the first derivatives of u implies a corresponding existence
theorem for the more general equation
(12 > aik\x,u, —,• . .,—1 = 0,
rather than only for equation (11).

ZERO ORDER A PRIORI ESTIMATES
159
Nevertheless the proof of (10) or even only (9) so far has remained an open
question, at least in this desired generality.
In a previous paper [1] the author has given a proof of estimate (9) and
(10) in the case where the uniform ellipticity condition (2) is replaced by a
somewhat stronger condition which he called ^-condition and ^-condition
respectively.
The best way perhaps to describe the character of these conditions is the
following.
Suppose A — ((due)) is a real symmetric n x n-matrix; then A has n real
eigenvalues Ai,--,An, which are not necessarily distinct. The condition
of uniform ellipticity
(13) m I g H I «*&* ^Mt$
can be expressed by stating that
(14) m ^ Xi ^ M, *= 1, ••-,*.
This amounts to the following statement: A second order linear differential
operator of the form (1) is uniformly elliptic in the region R if and only if
the w-component vector
(15) X(x) = (Ai(s),..-,A.(*))
has its endpoint in a hypercube m < Xi ^ M,i = 1, • • •, n, being a subset of
the domain Xi > 0, i — 1,- • •, n, where A*, i — 1,- • •, n, denote the
eigenvalues of the matrix A(x) = ((aa(x))). Note that for symmetry reasons
the ordering of the Xi(x) is not of importance in selecting the components of
the vector X(x).
We shall say that the matrix A = ((due)) satisfies a iL€-condition if the
corresponding vector A = (Ai, • • •, Xn) lies in the interior of a certain circular
hyper-cone K€ with axis falling into the line Ai = • • = An. For any
e > 0 this cone also is a subset of the domain Ai > 0, • • •, Xn > 0; for
e = 0 the cone will be the (uniquely determined) circular hyper-cone with
axis Ai = A2 = • • • = An and being tangent to each of the planes A{ = 0 in
the manifold A< = 0, Ai = • • * = A*_i = A*+i = • • • = An. The precise
definition of the cone K€ will follow in this section.
The ^-condition can be described in a very similar manner as a condition
of the type
(16) XeK'€
where K'e again denotes a circular hyper-cone with axis Xt = A2 = • • • = A„.
However this cone K'€ is somewhat smaller than the cone K€. More precisely,
if r€ and r'e for an instant denote the radii of the spherical cross sections of K€

160
H. O. CORDES
and K'e respectively with any plane perpendicular to the common axis of Ke
and K'e then we have
For n = 2 this quotient is equal to 1, for n = 3 it takes the value a/8/11
» 0.85, and for n->oo it tends toward 1/a/2 a 0.707. Accordingly we
have the K€- and X^-condition coinciding for n = 2. It proves that for
w = 2 the ^-condition essentially amounts to the condition of uniform
ellipticity. For increasing n the conditions get more and more restrictive.
Definition. A matrix A = ((a^)) is said to satisfy a ^-condition if the
eigenvalues Xt,i = 1, • • •, n of A satisfy the inequality
(18) (n - 1) ^ (Ai - A*)2 £ (1 - e)( 2 A4)2.
The matrix A is said to satisfy a ^-condition if the inequality
rather than only (18) is satisfied.
In two earlier papers [1; 2] the author was proving the following two
statements.
Theorem 1. The zero order a priori estimate (9) is satisfied if in addition
to the conditions on u,\, L and R mentioned above for every x e R there exists a
nonsingular constant n x n-matrix T£ such that Ai = T'±AT± satisfies a
K€-condition for a neighborhood \x — x\ ^ S relative to R. Hence 8 > 0 is
independent of x e R. T'± denotes the transposed of the matrix T±.
Theorem 2. The zero order a priori estimate (10) is true if in addition to
the general conditions imposed initially upon L,u,^, and Rfor every x e R there
exists a nonsingular constant n x n-matrix T± such that A± == T'±A T± satisfies a
Recondition for all x of a neighborhood \x — x\ ^ 8 of x relative to R where
8 > 0 is independent of x e R.
The proofs of Theorems 1 and 2 as given in [1] are very complicated.
Accordingly the author has attempted to simplify these proofs. A simplified
proof for Theorem 1 is given in [2]. In the following we shall deal with a
further simplification which again shall be discussed for Theorem 1 only.
It shall be indicated, however, how the same idea can be employed for a
simplification of the proof of Theorem 2. For both cases we shall make one
further restriction by assuming the region to be convex.
The author believes that the simplifications are considerable; in particular,
the very unpleasant boundary considerations are eliminated.
We shall restrict ourselves in the following to the case of dimension n not

ZERO ORDER A PRIORI ESTIMATES
161
less than 3 this being justified by the fact that zero order estimates are
fairly well investigated in the case of n = 2 (see [4 ; 5; 6]). The case n = 2
can be treated by a variant of the method discussed here.
We find it convenient to define a somewhat generalized concept of a
ife-condition for formal reasons only.
Definition. We shall say that the symmetric n x w-matrix A = ((a^))
satisfies a Kr or ^-condition with respect to the symmetric positive definite
n x 7i-matrix C = ((c^)) if the eigenvalues Ai, • • •, Xn of the problem
(20) Axfs = Xxfs
satisfy conditions (18) or (19) respectively.
By comparing the definitions we immediately notice that A satisfies a
K€- (K'€~) condition in the sense of the first definition if and only if it satisfies
a Ke- (K'€-) condition with respect to the n x w-unit-matrix In = ((8<*)). If
<E> is any nonsingular n x n-matrix then the statement that A be satisfying
a K€- (K'€-) condition with respect to C is equivalent to the statement that
<!>fA<S> satisfies a K€- (K'€~) condition with respect to <E>'C<E>, where <E>' denotes
the transposed of the matrix <E>.
Accordingly T AT satisfies a Ke- (K'€~) condition if and only if A satisfies a
K€- (K'€-) condition with respect to the matrix C = T"-1?1"1. This amounts
to the fact that Theorem 1 and Theorem 2 can be expressed in the following
form.
Theorem 3. Let n ^ 3 and let R be a convex region of (xi,- • •, xnyspace
with twice continuously differentiable boundary Y. Let
(21) C(x) = {{Cik{x)))itk*,l9...9n
be a fixed twice continuously differentiable symmetric positive n x n-matrix
function defined for all x e R. Assume that e, p, P are given fixed positive
numbers. Let A(x) — ((atk(x))) be any symmetric n x n-matrix with bounded
measurable coefficients satisfying
n
(22) p £ ^ au(x) = P' xeR>
i = l
and such that A(x) satisfies a K^condition with respect to the above matrix
C(x) for every x gT. Let the differential operator L be defined by
(23) L=Xai*(x)^
Assertion. There exist possible constants
(24) a = a(R;C(x);c,P9P), i = 1,2,
such that
(25) Hi/2(u; R) g ci sup (Lu)2\x - ±\*-*dz + c2 uUx.
zeR JR JR

162
H. O. CORDES
The constants c\ and c% can be determined from knowing the region R, the
matrix C(x) and the constants e, p, P only.
Theorem 4. Let the assumptions of Theorem 3 be satisfied, but in addition
let the matrix A(x) satisfy a Recondition rather than only a Recondition.
Assertion. There exists an a > 0 and positive constants
(26) a = a(R; C(x); e, p, P, a), i = 3, 4,
which can be determined from knowing R, C(x), e, p, P and a only such that
H'a(u; R) < c3sup {Lu)2\x - x\2~n-2adx
-f c4 sup (u(x) — u(x))2dx.
xeR JR
In the following the symbols c, ci, C2, • • •, c', c", • • • always denote positive
constants which can be determined from knowing the region jR, the matrix
C(x) and the constants listed in Theorem 3 or Theorem 4 respectively only.
Writing the same symbol c in different equations will not necessarily imply
that the corresponding constants coincide.
We give an outline of the proof of Theorem 3 as follows. We start from
the following differential identity which can be proved by simple partial
integration only.
If ((btk(x))) is a twice continuously differentiate symmetric n x n-matrix
function denned in R, then
i
i,j,kj = l
(28)
2 bikbji(u\ijU\ki - u\ikU\ji)dx
= 2 (*<*fy* ~ bijbki)\jiU\iU\kdx
JR i,j,k,l = l
t* n
+ 2 bacbjiU\i(njU\ki - nku\ji)da
/• n
+ 2 (°ikbji)\kU\j(u\ini - niU\i)da.
Jr i,jtk,i = i
Here n^i — 1,- • •, n, denote the components of the exterior normal on T,
and do- denotes the surface element on Y. The symbols u\{, u\ik, etc., denote
the derivatives dujdxi, d2u\dxidxk respectively.
We use this identity for the matrix btk = Cik(x)\x — x\^d~n^2. Then the
left-hand side takes the form
(29)
2 CikCjlU\ijU\kl\x - x\*-ndx - I I 2 ciku\ik) \x - x\3~ndx.

ZERO ORDER A PRIORI ESTIMATES 163
The isolated singularity at x = x will not disturb the applicability of identity
(28).
Next it can be shown that the boundary integrals of the right-hand side
of (28) are non-positive, due to the convexity of the region R. In addition
it proves that the first integral of the right-hand side of (28) can be estimated
by an expression of the form
c 2 iu\i)2\x - %\2~ndx.
JR i=i
Thirdly it can be proved that
n n
(30) 2 cikc1lu\ilU>\kl ^ Co ^ (U\ik)2
i,j,A:,Z = l t,*=l
where c0 stands for any lower bound of the eigenvalues of C(x), x e R.
Accordingly we obtain the estimate
coe' 2 (^K*)2!^ ~~ %\3~ndx
JR i,ffi
(31) =2 [Ci*Ckl ~ (L ~~ c')CikCji\u\ijU\ki\x - x\*~ndx
+ c 2 (un)2\x - x\2~ndx.
Finally the ^-condition will establish the inequality
n
(32) 2 \pifiki - (1 ~ e')cikCji]u\ijU\ki ^ c'(Lu)2.
Therefore we arrive at
2 (^U*)2IX — &\3~~ndx
JR i,t^i
£ c \ (Lu)2\x - ±\*-*dx + c' J y (u\t)*\x - x|2-»dx.
JR JR iti
Now it can be proved that
f V (^U*)2|* - x\3~ndx
jr ijfeii
can serve to estimate the integral
| (u(x) - u(x))*\x - x\-l~"dx
(33)

164 H. 0. CORDES
and that the last term of (33) can be absorbed in this estimate. Hence we
arrive at
(34)
(u(x) — u(x))2\x — x\-l~ndx
^ c \ (Lu)2\x - x\*~ndx + c' u2dx.
But by a lemma proved by the author in an earlier paper [1] we have
(35) H1/2(u; R) ^ c sup (u(x) - u(x))2\x - x\-l~ndx.
Accordingly the statement of Theorem 3 results.
The proof of Theorem 4 runs quite similarly. Again identity (28) is used,
this time for hue = Cit\x — £|(2-w-2a)/2 Again the boundary terms have the
right sign to be neglected. However, it is not any more possible to estimate
the other integral of the right-hand side of (28) in the same manner than in
Theorem 3. Instead a somewhat tricky but sharp estimate carried out
essentially already in [1] will furnish
/• n
2 fyubn - bi]bki)\jiU\iU\kdx
J& i,j,k,l=l
<36> * ((»?W-1) + °(a)) in ( Jml CH2|X - ^-^
/* n
+ C y (U\i{x) - U\i(x))2\x - xli-n-^dx.
J* »ti
Accordingly we get the estimate
r n
c0€f ^ (M\ifc)2\x - x\2~n~2adx
JR i,k = l
<3,) s /. J., [('+ (»*V.-d + °w)w' -(i - <*«*.
-u\ijU\ki\x — x\*-n-2adx
f ^r
+ c > (u\i(x) - u\i(x))2\x — x\l-n~2adx
JR i = i
instead of an analog of (31). Therefore we now need the estimate
.2 K1 + (n + n(n _ i) + °(«))CWC** - (1 - *')CikCji\u\ijUm
(38) ^ c'(Lu)\

ZERO ORDER A PRIORI ESTIMATES 165
However, this estimate can be proved only if the ^-condition is imposed
rather than only the i£e-condition.
In this way we arrive at
i
2 (u\ik)2\x - x\2~~n~2adx
(39) ^ c \ (Lu)*\x - x\2~n~2adx
/» n
+ c I 2 (u\*(x) ~~ u\{(x))2\x — x\1-n~2adx.
JR i = i
All other estimates turn out to be analogous to those carried out in the proof
of Theorem 3. Some proofs are even simplified. Accordingly we first get
/• n
^ (M\i(x) - u\i(x))2\x - x\~~n~2adx
(40) jHi = 1
^ c (Lu)2\x - x\2-n-*«dx + c' (^(x) - ^(x))2da;.
JR JR
This and the lemma used before again yield
H'a(u; R) ^ csup I (Lu)2\x ~ x\2-n~-2adx
(41) " J"
+ c' sup (u(x) - u(x))Hxy
±€R JR
which proves Theorem 4.
The author is preparing a paper containing the detailed proofs of the first
theorem listed [3]. Also it is to be mentioned that in a very similar way one
also can obtain an estimate of the form
(42) sup \uH(x)\ S c||/||o + c'Ho.
xeR
Bibliography
1. H. O. Cordes, Ueber die erste Bandwertaufgabe bei quasilinearen Differential-
gleichungen zweiter Ordnung in mehr als zwei Variablen, Math. Ann. vol. 131 (1956) pp.
278-312.
2. y Vereinfachter Beweis der Existenz einer Apriori-Hoelderkonstanten, Math.
Ann. vol. 138 (1959) pp. 155-178.
3, 9 Zero order estimates for second order elliptic differential equations, to
appear.
4. J. Leray and I. Schauder, Topologie et equations fonctionellesy Ann. Sci. ficole
Norm. Sup. vol. 51 (1934) pp. 45-78.

166
H. 0. CORDES
5. C. B. Morrey, On the solutions of quasilinear elliptic partial differential equations,
Trans. Amer. Math. Soc. vol. 43 (1938) pp. 126-166.
6. L. Nirenberg, On nonlinear elliptic partial differential equations and Hoelder-
continuity, Comm. Pure Appl. Math. vol. 6 (1953) pp. 103-156.
University of California,
Berkeley, California

INDEX
A priori estimates, 73-74, 138
for elliptic and parabolic boundary value
problems, 73
for mixed initial-boundary value
problems for general parabolic equations,
73
A priori Holder estimate, 127, 132
Adjoint
boundary problem, 76
pairing, 26
Admissible (norm), 26
Associated
norms,^23, 26-27
space V, 23
spaces, 23, 26-27
Asymptotic
behavior, 133, 139
expansions, 133, 139
Bernstein's inequality, 58
Bessel potentials, 23, 29, 33
class Pa of of order a in JR", 29
Boundary conditions, 111
overdetermined, 112
underdetermined, 112
Boundary value problems, 109
adjoint of, 76
estimates for, 103
exterior, 144, 147
Leray's, 144
mixed initial-, 78
regular, 74
(See also Elliptic)
Calculus of variations, existence theorems
in, 20
Canonical
pairing [B*, B, <&*, 6>], 28
self-pairing, 28
Cauchy
initial value problem, 101
-Kowalewski system, 115
problem, 51
Characteristics, 102
Class
Pa of Bessel potentials of order a in R",
29
Comparison
arguments, 128
function, 130
Compatible (norms), 25
Compensating function, 31
Compensation method, 24, 31
Conditionally coercive (quadratic forms),
24
Conjugate norm, 26
Correct operators, 86
Curvature, 96
Vq>, 5
Derivative of an operator, 87
Dini continuous, 128
Dirichlet
problem, 84, 104, 137
system, 112
Dirichlet Integral, 144, 147
of order j3 in D, 30
Distribution, 109
Domain
G of class C1, 4
of polyhedral type and class Cv*>\ 30
Elliptic
equations, 109
operator, 102, 109
systems of second order, 149
uniformly, 128
Elliptic boundary value problems
existence of solutions to in
operator-theoretic terms, 76
regularity of solutions of, 73
Ellipticity, non-uniform, 134
Energy inequalities, 86
Equation
of the type d/dt + A(t), 89
of waves, 121
second order of the type d^/dt2 +
B(t)d/dt + A(t), 89

168
INDEX
Equivalent norms (|| ||x ~ || ||2), 29
Estimates
for boundary value problems, 103
of Di Giorgi-Nash type, 73
zero order, 158
(See also A priori)
Existence
of solutions to elliptic boundary value
problems in operator-theoretic terms,
76
theorems in the calculus of variations,
20
Extended maximum principle, 128
Extension theorem, 24, 30
Exterior boundary value problem, 144,
147
Extremal, 8
Force, 146
Frechet derivative, 57
Fundamental solution tensor, 144, 146
General restriction theorem, 31
Generalized solutions, 109
Goursat problem, 51, 71, 116
Harnack inequality, 132, 134
Hyperbolic, 101
Initial value problems, 85
Integral
equation, 93
singular operators, 102
Interpolated norms, 25
Interpolation of norms, 23
Isolated singular points, 131
Isomorphism {S, T) of the pairing
[F, W, <v, w >] into (or onto)
[V, W, <v'9 v/y], 28
K€- [K'€-] condition, 159
with respect to the symmetric positive
definite n x n matrix C = ((crt)), 161
Kowalewski-Cauchy system, 115
Leray
exterior boundary value problem of, 144
-Schauder fixed point theorem, 138
Limit theorem, 132
Liouville, 147
theorem, 134-135
Majorant method, 51
Majorants, 62
Maximum
principle, 21, 106, 128
property, 91, 95
Minimal surface equation, 136, 140
Mixed
initial-boundary value problem, 78
problems in cylindrical domains, 83
Monge-Ampere equations, 149
Navier-Stokes equations, 143
Neighborhood of infinity, 143
Neumann problem, 84
Newton's method, 54
Nonlinear equations—removable
singularities, 135
Non-normal systems, 116
Non-uniform ellipticity, 134
Norms
associated, 23, 26
conjugate, 26
equivalent, 29
interpolated, 23, 25
standard ||w||a,Z)> 30
Normal (boundary conditions), 109
Operator
correct, 86
derivative of, 87
elliptic, 109
singular integral, 102
Pairing, 23, 26
adjoint, 26
of vector spaces, 23
proper, 26
(See also Canonical and Isomorphism)
Perturbation methods, 54
Properly elliptic, 109
Quadratic interpolation, 24—26
theorem, 26
Quasi-linear parabolic equation, 79

INDEX
169
Regular boundary value problem, 74
Regularity
of continuations (of solutions), 121
of solutions of elliptic boundary \
problems, 73
Removable singularities, 130
Restriction theorem, converse to, 31
Singular integral operators, 102
Space
associated, 23, 26-27
associated F, 23
C\(G)9 1
Cj>h(G), 74
@l(E), 89
H\ 85
HUD), 2
HMG), 4
AS, 37
Lipc (0), 4
Pa(^#n), 30
P&(^n)> 30
JT'»*(G), 74
Standard norm ||w||a,z), 30
Stokes, Navier- equations, 143
Stress tensor, 145
Stronger (norm) (>), 28
Strongly hyperbolic, 101
Subsonic flows, 139
Surface area, 19
Telescopic series, 125
Variational integral
definition of, 17-18
lower semicontinuity of, 18
Variational problems, 17
Velocity field, 143
Wake, 146
Wave equation, 92, 121
Weak solution, 128
Weaker (norm) (-<), 28
Well posed problem, 121
Zero order estimate, 158

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