Partial
Differential
Equations
Lawrence C. Evans
Graduate Studies
in Mathrmatici
Volume 19
American Mathematical Society
CONTENTS
Preface................................................. xviii
1.	Introduction............................................ 1
1.1.	Partial differential equations..................... 1
1.2.	Examples........................................... 3
1.2.1.	Single partial differential equations......... 3
1.2.2.	Systems of partial differential equations..... 6
1.3.	Strategies for studying PDE........................ 7
1.3.1.	Well-posed problems, classical solutions...... 7
1.3.2.	Weak solutions and regularity................. 8
1.3.3.	Typical difficulties.......................... 9
1.4.	Overview........................................... 9
1.5.	Problems.......................................... 12
PART I: REPRESENTATION FORMULAS
FOR SOLUTIONS
2.	Four Important Linear PDE.............................. 16
2.1.	Transport equation................................ 17
2.1.1.	Initial-value problem........................ 17
2.1.2.	Nonhomogeneous problem....................... 18
2.2.	Laplace’s equation................................ 19
ix
X
CONTENTS
2.2.1.	Fundamental solution.......................... 20
2.2.2.	Mean-value formulas........................... 25
2.2.3.	Properties of harmonic functions.............. 26
2.2.4.	Green’s function.............................. 33
2.2.5.	Energy methods................................ 42
2.3.	Heat	equation...................................... 44
2.3.1.	Fundamental solution.......................... 45
2.3.2.	Mean-value formula............................ 53
2.3.3.	Properties of solutions....................... 56
2.3.4.	Energy methods................................ 65
2.4.	Wave	equation...................................... 68
2.4.1.	Solution by spherical means................... 69
2.4.2.	Nonhomogeneous problem........................ 84
2.4.3.	Energy methods................................ 86
2.5.	Problems............................................ 89
2.6.	References.......................................... 93
3.	Nonlinear First-Order PDE................................ 94
3.1.	Complete integrals, envelopes....................... 95
3.1.1.	Complete integrals............................ 95
3.1.2.	New solutions from envelopes.................. 97
3.2.	Characteristics.................................... 100
3.2.1.	Derivation of characteristic ODE............. 100
3.2.2.	Examples..................................... 103
3.2.3.	Boundary conditions.......................... 107
3.2.4.	Local solution............................... 110
3.2.5.	Applications................................. 115
3.3.	Introduction to Hamilton-Jacobi equations.......... 121
3.3.1.	Calculus of variations, Hamilton’s ODE......	122
3.3.2.	Legendre transform, Hopf-Lax formula........	127
3.3.3.	Weak solutions, uniqueness................... 136
3.4.	Introduction to conservation laws.................. 143
3.4.1.	Shocks, entropy condition.................... 144
CONTENTS
xi
3.4.2.	Lax-Oleinik formula......................... 152
3.4.3.	Weak solutions, uniqueness.................. 157
3.4.4.	Riemann’s problem........................... 162
3.4.5.	Long time behavior.......................... 165
3.5.	Problems......................................... 171
3.6.	References....................................... 173
4.	Other Ways to Represent Solutions..................... 175
4.1.	Separation of variables.......................... 175
4.2.	Similarity solutions............................. 180
4.2.1.	Plane and traveling waves, solitons......... 180
4.2.2.	Similarity under scaling.................... 189
4.3.	Transform methods................................ 191
4.3.1.	Fourier transform........................... 191
4.3.2.	Laplace transform........................... 200
4.4.	Converting nonlinear into linear PDE............. 203
4.4.1.	Hopf-Cole transformation.................... 203
4.4.2.	Potential functions......................... 205
4.4.3.	Hodograph and Legendre transforms........... 207
4.5.	Asymptotics...................................... 209
4.5.1.	Singular perturbations...................... 209
4.5.2.	Laplace’s method............................ 214
4.5.3.	Geometric optics, stationary phase.......... 217
4.5.4.	Homogenization.............................. 229
4.6.	Power series..................................... 232
4.6.1.	Noncharacteristic surfaces.................. 232
4.6.2.	Real analytic functions..................... 237
4.6.3.	Cauchy-Kovalevskaya Theorem................. 239
4.7.	Problems......................................... 245
4.8.	References....................................... 247
PART II: THEORY FOR LINEAR PARTIAL
DIFFERENTIAL EQUATIONS
5.	Sobolev Spaces........................................ 251
xii
CONTENTS
5.1.	Holder spaces......................................... 252
5.2.	Sobolev spaces........................................ 254
5.2.1.	Weak derivatives................................ 254
5.2.2.	Definition of Sobolev spaces.................... 257
5.2.3.	Elementary properties........................... 259
5.3.	Approximation......................................... 262
5.3.1.	Interior approximation by smooth functions . . .	262
5.3.2.	Approximation by smooth functions............... 264
5.3.3.	Global approximation by smooth functions ....	265
5.4.	Extensions............................................ 267
5.5.	Traces................................................ 270
5.6.	Sobolev inequalities.................................. 275
5.6.1.	Gagliardo-Nirenberg-Sobolev inequality....... 275
5.6.2.	Morrey’s inequality............................. 280
5.6.3.	General Sobolev inequalities.................... 284
5.7.	Compactness........................................... 286
5.8.	Additional topics..................................... 289
5.8.1.	Poincare’s inequalities......................... 289
5.8.2.	Difference quotients............................ 291
5.8.3.	Differentiability a.e........................... 295
5.8.4.	Fourier transform methods....................... 297
5.9.	Other spaces of functions............................. 298
5.9.1.	The space Я-1................................... 298
5.9.2.	Spaces involving time........................... 300
5.10.	Problems............................................. 305
5.11.	References........................................... 308
6.	Second-Order Elliptic Equations........................... 309
6.1.	Definitions........................................... 309
6.1.1.	Elliptic equations.............................. 309
6.1.2.	Weak solutions.................................. 312
6.2.	Existence of weak solutions........................... 314
6.2.1.	Lax-Milgram Theorem............................. 314
CONTENTS
xiii
6.2.2.	Energy estimates................................ 316
6.2.3.	Fredholm alternative............................ 319
6.3.	Regularity........................................... 325
6.3.1.	Interior regularity............................. 326
6.3.2.	Boundary regularity............................. 334
6.4.	Maximum principles................................... 344
6.4.1.	Weak maximum principle.......................... 345
6.4.2.	Strong maximum principle........................ 348
6.4.3.	Harnack’s inequality............................ 352
6.5.	Eigenvalues and eigenfunctions....................... 353
6.5.1.	Eigenvalues of symmetric elliptic operators ....	353
6.5.2.	Eigenvalues of nonsymmetric elliptic operators .	359
6.6.	Problems............................................. 364
6.7.	References........................................... 367
7.	Linear Evolution Equations................................. 368
7.1.	Second-order parabolic equations..................... 368
7.1.1.	Definitions..................................... 369
7.1.2.	Existence of weak solutions..................... 372
7.1.3.	Regularity...................................... 377
7.1.4.	Maximum principles.............................. 387
7.2.	Second-order hyperbolic equations.................... 398
7.2.1.	Definitions..................................... 398
7.2.2.	Existence of weak solutions..................... 401
7.2.3.	Regularity...................................... 409
7.2.4.	Propagation of disturbances..................... 415
7.2.5.	Equations in two variables...................... 419
7.3.	Systems of first-order hyperbolic equations.......... 422
7.3.1.	Definitions..................................... 423
7.3.2.	Symmetric hyperbolic systems.................... 424
7.3.3.	Systems with constant coefficients.............. 431
7.4.	Semigroup theory..................................... 436
7.4.1.	Definitions, elementary properties.............. 436
xiv
CONTENTS
7.4.2.	Generating contraction semigroups................. 442
7.4.3.	Applications...................................... 445
7.5.	Problems............................................... 449
7.6.	References............................................. 452
PART III: THEORY FOR NONLINEAR PARTIAL
DIFFERENTIAL EQUATIONS
8.	The Calculus of Variations.................................. 456
8.1.	Introduction........................................... 456
8.1.1.	Basic ideas....................................... 456
8.1.2.	First variation, Euler-Lagrange equation....	457
8.1.3.	Second variation.................................. 461
8.1.4.	Systems........................................... 463
8.2.	Existence of minimizers................................ 468
8.2.1.	Coercivity, lower semicontinuity.................. 469
8.2.2.	Convexity......................................... 471
8.2.3.	Weak solutions of Euler-Lagrange equation . . .	476
8.2.4.	Systems........................................... 479
8.3.	Regularity............................................. 484
8.3.1.	Second derivative estimates....................... 485
8.3.2.	Remarks on higher regularity...................... 488
8.4.	Constraints............................................ 490
8.4.1.	Nonlinear eigenvalue problems..................... 490
8.4.2.	Unilateral constraints, variational inequalities .	494
8.4.3.	Harmonic maps..................................... 498
8.4.4.	Incompressibility................................. 500
8.5.	Critical points........................................ 505
8.5.1.	Mountain Pass Theorem............................. 505
8.5.2.	Application to semilinear elliptic PDE......	510
8.6.	Problems............................................... 516
8.7.	References............................................. 519
9.	Nonvariational Techniques................................... 520
9.1.	Monotonicity methods................................... 520
CONTENTS
xv
9.2.	Fixed point methods................................. 528
9.2.1.	Banach’s Fixed Point Theorem................... 528
9.2.2.	Schauder’s, Schaefer’s Fixed Point Theorems . .	532
9.3.	Method of subsolutions and supersolutions........... 537
9.4.	Nonexistence........................................ 541
9.4.1.	Blow-up........................................ 541
9.4.2.	Derrick-Pohozaev identity...................... 544
9.5.	Geometric properties of solutions................... 547
9.5.1.	Star-shaped level sets......................... 547
9.5.2.	Radial symmetry................................ 549
9.6.	Gradient flows...................................... 553
9.6.1.	Convex functions on Hilbert spaces............. 553
9.6.2.	Subdifferentials, nonlinear semigroups......... 559
9.6.3.	Applications................................... 565
9.7.	Problems............................................ 567
9.8.	References.......................................... 569
10.	Hamilton-Jacobi Equations............................... 571
10.1.	Introduction, viscosity solutions.................. 571
10.1.1.	Definitions................................... 573
10.1.2.	Consistency................................... 576
10.2.	Uniqueness......................................... 579
10.3.	Control theory, dynamic programming................ 583
10.3.1.	Introduction to control theory................ 583
10.3.2.	Dynamic programming........................... 585
10.3.3.	Hamilton-Jacobi-Bellman equation.............. 587
10.3.4.	Hopf-Lax formula revisited.................... 594
10.4.	Problems........................................... 596
10.5.	References......................................... 598
11.	Systems of Conservation	Laws........................ 599
11.1.	Introduction....................................... 599
11.1.1.	Integral solutions............................ 602
11.1.2.	Traveling waves, hyperbolic systems........... 605
xvi
CONTENTS
11.2.	Riemann’s problem................................ 612
11.2.1.	Simple waves................................ 612
11.2.2.	Rarefaction waves........................... 615
11.2.3.	Shock waves, contact discontinuities........ 616
11.2.4.	Local solution of Riemann’s problem......... 624
11.3.	Systems of two conservation laws................. 626
11.3.1.	Riemann invariants.......................... 627
11.3.2.	Nonexistence of smooth solutions............ 631
11.4.	Entropy criteria................................. 634
11.4.1.	Vanishing viscosity, traveling waves........ 634
11.4.2.	Entropy/entropy flux pairs.................. 638
11.4.3.	Uniqueness for a scalar conservation law...	642
11.5.	Problems......................................... 647
11.6.	References....................................... 648
APPENDICES
Appendix A:	Notation................................... 649
A.l. Notation for matrices............................. 649
A.2. Geometric notation................................ 650
A.3. Notation for functions............................ 651
A.4. Vector-valued functions........................... 655
A.5. Notation for estimates............................ 656
A.6. Some comments about notation...................... 657
Appendix B:	Inequalities............................... 657
B.l.	Convex functions.................................. 657
B.2.	Elementary inequalities........................... 658
Appendix C:	Calculus Facts............................. 663
C.l. Boundaries........................................ 663
C.2. Gauss-Green Theorem............................... 664
C.3. Polar coordinates, coarea formula................. 665
C.4. Convolution and smoothing......................... 666
C.5. Inverse Function Theorem.......................... 669
C.6. Implicit Function Theorem......................... 670
CONTENTS	xvii
C.7. Uniform convergence.............................. 672
Appendix D:	Linear Functional Analysis................ 673
D.l. Banach spaces.................................... 673
D.2. Hilbert spaces................................... 674
D.3. Bounded linear operators......................... 675
D.4. Weak convergence................................. 677
D.5. Compact operators, Fredholm theory............... 679
D.6. Symmetric operators.............................. 683
Appendix E:	Measure Theory............................ 684
E.l. Lebesgue measure................................. 685
E.2. Measurable functions and integration............. 686
E.3. Convergence theorems for integrals............... 687
E.4. Differentiation.................................. 688
E.5. Banach space-valued functions.................... 688
Bibliography............................................. 690
Index.................................................... 695
PREFACE
I present in this book a wide-ranging survey of many important topics in
the theory of partial differential equations (PDE), with particular emphasis
on various modern approaches. I have made a huge number of editorial
decisions about what to keep and what to toss out, and can only claim
that this selection seems to me about right. I of course include the usual
formulas for solutions of the usual linear PDE, but also devote large amounts
of exposition to energy methods within Sobolev spaces, to the calculus of
variations, to conservation laws, etc.
My general working principles in the writing have been these:
a.	PDE theory is (mostly) not restricted to two independent vari-
ables. Many texts describe PDE as if functions of the two variables (ж, у)
or (ж, t) were all that matter. This emphasis seems to me misleading, as
modern discoveries concerning many types of equations, both linear and
nonlinear, have allowed for the rigorous treatment of these in any number
of dimensions. I also find it unsatisfactory to “classify” partial differential
equations: this is possible in two variables, but creates the false impression
that there is some kind of general and useful classification scheme available
in general.
b.	Many interesting equations are nonlinear. My view is that overall
we know too much about linear PDE and too little about nonlinear PDE. I
have accordingly introduced nonlinear concepts early in the text and have
tried hard to emphasize everywhere nonlinear analogues of the linear theory.
xviii
PREFACE
xix
c.	Understanding generalized solutions is fundamental. Many of the
partial differential equations we study, especially nonlinear first-order equa-
tions, do not in general possess smooth solutions. It is therefore essential to
devise some kind of proper notion of generalized or weak solution. This is
an important but subtle undertaking, and much of the hardest material in
this book concerns the uniqueness of appropriately defined weak solutions.
d.	PDE theory is not a branch of functional analysis. Whereas
certain classes of equations can profitably be viewed as generating abstract
operators between Banach spaces, the insistence on an overly abstract view-
point, and consequent ignoring of deep calculus and measure theoretic esti-
mates, is ultimately limiting.
e.	Notation is a nightmare. I have really tried to introduce consistent
notation, which works for all the important classes of equations studied.
This attempt is sometimes at variance with notational conventions within a
subarea.
f.	Good theory is (almost) as useful as exact formulas. I incorporate
this principle into the overall organization of the text, which is subdivided
into three parts, roughly mimicking the historical development of PDE the-
ory itself. Part I concerns the search for explicit formulas for solutions, and
Part II the abandoning of this quest in favor of general theory asserting
the existence and other properties of solutions for linear equations. Part III
is the mostly modern endeavor of fashioning general theory for important
classes of nonlinear PDE.
Let me also explicitly comment here that I intend the development
within each section to be rigorous and complete (exceptions being the frankly
heuristic treatment of asymptotics in §4.5 and an occasional reference to a
research paper). This means that even locally within each chapter the topics
do not necessarily progress logically from “easy” to “hard” concepts. There
are many difficult proofs and computations early on, but as compensation
many easier ideas later. The student should certainly omit on first reading
some of the more arcane proofs.
I wish next to emphasize that this is a textbook, and not a reference
book. I have tried everywhere to present the essential ideas in the clearest
possible settings, and therefore have almost never established sharp versions
of any of the theorems. Research articles and advanced monographs, many
of them listed in the Bibliography, provide such precision and generality.
XX
PREFACE
My goal has rather been to explain, as best I can, the many fundamental
ideas of the subject within fairly simple contexts.
I have greatly profited from the comments and thoughtful suggestions
of many of my colleagues, friends and students, in particular: S. Antman,
J. Bang, X. Chen, A. Chorin, M. Christ, J. Cima, P. Colella, J. Cooper,
M. Crandall, B. Driver, M. Feldman, M. Fitzpatrick, R. Gariepy, J. Gold-
stein, D. Gomes, O. Hald, W. Han, W. Hrusa, T. Ilmanen, I. Ishii, I. Israel,
R. Jerrard, C. Jones, B. Kawohl, S. Koike, J. Lewis, T.-P. Liu, H. Lopes,
J. McLaughlin, K. Miller, J. Morford, J. Neu, M. Portilheiro, J. Ralston,
F. Rezakhanlou, W. Schlag, D. Serre, P. Souganidis, J. Strain, W. Strauss,
M. Struwe, R. Temam, B. Tvedt, J.-L. Vazquez, M. Weinstein, P. Wolfe,
and Y. Zheng.
I especially thank Tai-Ping Liu for many years ago writing out for me
the first draft of what is now Chapter 11.
I am extremely grateful for the suggestions and lists of mistakes from
earlier drafts of this book sent me by many readers, and I encourage others
to send me their comments, at evans@math.berkeley.edu. I have come to
realize that I must be more than slightly mad to try to write a book of
this length and complexity, but I am not yet crazy enough to think that I
have made no mistakes. I will therefore maintain a listing of errors
which come to light, and will make this accessible through the
math.berkeley.edu homepage.
Faye Yeager at UC Berkeley has done a really magnificent job typing
and updating these notes, and Jaya Nagendra heroically typed an earlier
version at the University of Maryland. My deepest thanks to both.
I have been supported by the NSF during much of the writing, most
recently under grant DMS-9424342.
LCE
August, 1997
Berkeley
Chapter 1
INTRODUCTION
1.1	Partial differential equations
1.2	Examples
1.3	Strategies for studying PDE
1.4	Overview
1.5	Problems
This chapter surveys the principal theoretical issues concerning the solv-
ing of partial differential equations.
To follow the subsequent discussion, the reader should first of all turn
to Appendix A and look over the notation presented there, particularly the
multiindex notation for partial derivatives.
1.1.	PARTIAL DIFFERENTIAL EQUATIONS
A partial differential equation (PDE) is an equation involving an unknown
function of two or more variables and certain of its partial derivatives.
Using the notation explained in Appendix A, we can write out symbol-
ically a typical PDE, as follows. Fix an integer к > 1 and let U denote an
open subset of Rn.
DEFINITION. An expression of the form
(1)	F(Dku(x),Dk~lu(x),... ,Du(x),u(x),x) = 0 (x € U)
is called a kth-order partial differential equation, where
F : Rn* x Rn*-1 x---xRnxRxU—
1
2
1. INTRODUCTION
is given, and
и : U —» R
is the unknown.
We solve the PDE if we find all и verifying (1), possibly only among those
functions satisfying certain auxiliary boundary conditions on some part Г
of dU. By finding the solutions we mean, ideally, obtaining simple, explicit
solutions, or, failing that, deducing the existence and other properties of
solutions.
DEFINITIONS.
(i)	The partial differential equation (1) is called linear if it has the form
^2 aa(x)Dau = f(x)
|a|<fc
for given functions aa (|a| < k), f. This linear PDE is homogeneous
if f =
(ii)	The PDE (1) is semilinear if it has the form
У2 aa(x)Dau + ao(Dk~1u,..., Du, u, x) = 0.
|a|=fc
(iii)	The PDE (1) is quasilinear if it has the form
aa(Dk~1u,..., Du, u, x)Dau + ao(Dk~1u,..., Du, u, x) = 0.
|a|=fc
(iv)	The PDE (1) is fully nonlinear if it depends nonlinearly upon the
highest order derivatives.
A system of partial differential equations is, informally speaking, a col-
lection of several PDE for several unknown functions.
DEFINITION. An expression of the form
(2)	F(Z>fcu(a;), Dfc-1u(z),...,Du(x), ufy), x) = О (x e J7)
is called a fcfh-order system of partial differential equations, where
F : Rmnfe x Rmn*-1 x  • • x Rmn x Rm x U Rm
is given and
u:l^Rm, u = (u\...,um)
is the unknown.
Here we are supposing that the system comprises the same number m
of scalar equations as unknowns (u1,..., um). This is the most common
circumstance, although other systems may have fewer or more equations
than unknowns.
Systems are classified in the obvious way as being linear, semilinear, etc.
1.2. EXAMPLES
3
Remark. We use “PDE” as an abbreviation for both “partial differential
equation” and “partial differential equations”.	□
1.2.	EXAMPLES
There is no general theory known concerning the solvability of all partial
differential equations. Such a theory is extremely unlikely to exist, given
the rich variety of physical, geometric, and probabilistic phenomena which
can be modeled by PDE. Instead, research focuses on various particular
partial differential equations that are important for applications within and
outside of mathematics, with the hope that insight from the origins of these
PDE can give clues as to their solutions.
Following is a list of many specific partial differential equations of in-
terest in current research. This listing is intended merely to familiarize the
reader with the names and forms of various famous PDE. To display most
clearly the mathematical structure of these equations, we have mostly set
relevant physical constants to unity. We will later discuss the origin and
interpretation of many of these PDE.
Throughout x 6 U, where U is an open subset of Rn, and t > 0. Also
Du = Dxu = (uX1,    ,uxn) denotes the gradient of и with respect to the
spatial variable x = (aq,..., zn).
1.2.1.	Single partial differential equations.
a. Linear equations.
1.	Laplace’s equation
n
Au =	' UXiXi = 0.
1=1
2.	Helmholtz’s (or eigenvalue) equation
—Xu = Xu.
3.	Linear transport equation
n
ut + 5?&uXi = 0.
1=1
4.	Liouville’s equation
n
ut -	= °-
i=i
4
1. INTRODUCTION
5.	Heat (or diffusion) equation
щ — Ди = 0.
6.	Schrodinger’s equation
iut + Ди = 0.
7.	Kolmogorov’s equation
n	n
& 3 'UxiXj “1“ 'U'Xj ~ 0*
i,j=l	i=l
8.	Fokker-Planck equation
n	n
ut~^2 (avu)XiXj - 22(blu)Zi = 0.
i,j=l	2=1
9.	Wave equation
utt - Ди = 0.
10.	Telegraph equation
Utt ”b dut uxx — 0.
11.	General wave equation
n	n
Utt a^uXiXj + b uXi = 0.
i,j=l	i=l
12.	Airy’s equation
U^ “I" Uxxx =
13.	Beam equation
Ut “h Uxxxx — 0-
1.2. EXAMPLES
5
div
b. Nonlinear equations.
1.	Eikonal equation
|Z>u| = 1.
2.	Nonlinear Poisson equation
-Xu = f(u).
3.	p-Laplacian equation
div(\Du\p~2Du) = 0.
4.	Minimal surface equation
Du
(1 + iDup)1/2
5.	Monge-Ampere equation
det(Z>2u) = f.
6.	Hamilton-Jacobi equation
ut + H(Du, x) = 0.
7.	Scalar conservation law
щ + divF(u) = 0.
8.	Inviscid Burgers’ equation
ut + uux — 0.
9.	Scalar reaction-diffusion equation
ut- Xu = f(u).
10.	Porous medium equation
щ — Д(и7) — 0.
11.	Nonlinear wave equations
utt - Xu = f(u),
utt — diva(Du) = 0.
12.	Korteweg-de Vries (KdV) equation
Ut + UUX ”1“ Uxxx — 0*
>
6
1. INTRODUCTION
1.2.2.	Systems of partial differential equations.
a.	Linear systems.
1.	Equilibrium equations of linear elasticity
/zAu + (Л + /z)D(divu) = 0.
2.	Evolution equations of linear elasticity
Utt — /zAu —	+ /z)-D(divu) = 0.
3.	Maxwell’s equations
Et = curl В
B( = — curl E
div В = div E = 0.
b.	Nonlinear systems.
1.	System of conservation laws
u4 + divF(u) = 0.
2.	Reaction-diffusion system
ut - Au = f(u).
3.	Euler’s equations for incompressible, inviscid flow
Г u* + u • Du = —Dp
[ div u = 0.
4.	Navier-Stokes equations for incompressible, viscous flow
Г и* + u • Du — Au = —Dp
[ div u = 0.
See Zwillinger [ZW] for a much more extensive listing of interesting
PDE.
1.3. STRATEGIES FOR STUDYING PDE
7
1.3.	STRATEGIES FOR STUDYING PDE
As explained in §1.1 our goal is the discovery of ways to solve partial differ-
ential equations of various sorts, but—as should now be clear in view of the
many diverse examples set forth in §1.2—this is no easy task. And indeed
the very question of what it means to “solve” a given PDE can be subtle,
depending in large part on the particular structure of the problem at hand.
1.3.1.	Well-posed problems, classical solutions.
The informal notion of a well-posed problem captures many of the desir-
able features of what it means to solve a PDE. We say that a given problem
for a partial differential equation is well-posed if
(a)	the problem in fact has a solution;
(b)	this solution is unique;
and
(c)	the solution depends continuously on the data given in the problem.
The last condition is particularly important for problems arising from
physical applications: we would prefer that our (unique) solution changes
only a little when the conditions specifying the problem change a little. (For
many problems, on the other hand, uniqueness is not to be expected. In
these cases the primary mathematical tasks are to classify and characterize
the solutions.)
Now clearly it would be desirable to “solve” PDE in such a way that
(a)-(c) hold. But notice that we still have not carefully defined what we
mean by a “solution”. Should we ask, for example, that a “solution” и must
be real analytic or at least infinitely differentiable? This might be desirable,
but perhaps we are asking too much. Maybe it would be wiser to require a
solution of a PDE of order k to be at least к times continuously differentiable.
Then at least all the derivatives which appear in the statement of the PDE
will exist and be continuous, although maybe certain higher derivatives will
not exist. Let us informally call a solution with this much smoothness a
classical solution of the PDE: this is certainly the most obvious notion of
solution.
So by solving a partial differential equation in the classical sense we mean
if possible to write down a formula for a classical solution satisfying (a)-(c)
above, or at least to show such a solution exists, and to deduce various of
its properties.
1.3.2.	Weak solutions and regularity.
But can we achieve this? The answer is that certain specific partial
differential equations (e.g. Laplace’s equation) can be solved in the classical
8
1. INTROD UCTION
sense, but many others, if not most others, cannot. Consider for instance
the scalar conservation law
ut + F(u)x = 0.
We will see in §3.4 that this PDE governs various one-dimensional phenom-
ena involving fluid dynamics, and in particular models the formation and
propagation of shock waves. Now a shock wave is a curve of discontinuity
of the solution u; and so if we wish to study conservation laws, and recover
the underlying physics, we must surely allow for solutions и which are not
continuously differentiable or even continuous. In general, as we shall see,
the conservation law has no classical solutions, but is well-posed if we allow
for properly defined generalized or weak solutions.
This is all to say that we may be forced by the structure of the par-
ticular equation to abandon the search for smooth, classical solutions. We
must instead, while still hoping to achieve the well-posedness conditions (a)
(c), investigate a wider class of candidates for solutions. And in fact, even
for those PDE which turn out to be classically solvable, it is often most
expedient initially to search for some appropriate kind of weak solution.
The point is this: if from the outset we demand that our solutions be very
regular, say fc-times continuously differentiable, then we are usually going
to have a really hard time finding them, as our proofs must then necessarily
include possibly intricate demonstrations that the functions we are building
are in fact smooth enough. A far more reasonable strategy is to consider as
separate the existence and the smoothness (or regularity) problems. The idea
is to define for a given PDE a reasonably wide notion of a weak solution, with
the expectation that since we are not asking too much by way of smoothness
of this weak solution, it may be easier to establish its existence, uniqueness,
and continuous dependence on the given data. Thus, to repeat, it is often
wise to aim at proving well-posedness in some appropriate class of weak or
generalized solutions.
Now, as noted above, for various partial differential equations this is
the best that can be done. For other equations we can hope that our weak
solution may turn out after all to be smooth enough to qualify as a classical
solution. This leads to the question of regularity of weak solutions. As we
will see, it is often the case that the existence of weak solutions depends
upon rather simple estimates plus ideas of functional analysis, whereas the
regularity of the weak solutions, when true, usually rests upon many intricate
calculus estimates.
Let me explicitly note here that once we are past Part I (Chapters 2-4),
our efforts will be largely devoted to proving mathematically the existence
1.4. OVERVIEW
9
of solutions to various sorts of partial differential equations, and not so much
to deriving formulas for these solutions. This may seem wasted or misguided
effort, but in fact mathematicians are like theologians: we regard existence
as the prime attribute of what we study. But unlike most theologians, we
need not always rely upon faith alone.
1.3.3.	Typical difficulties.
Following are some vague but general principles, which may be useful to
keep in mind:
(1)	Nonlinear equations are more difficult than linear equations; and,
indeed, the more the nonlinearity affects the higher derivatives, the
more difficult the PDE is.
(2)	Higher-order PDE are more difficult than lower-order PDE.
(3)	Systems are harder than single equations.
(4)	Partial differential equations entailing many independent variables
are harder than PDE entailing few independent variables.
(5)	For most partial differential equations it is not possible to write out
explicit formulas for solutions.
None of these assertions is without important exceptions.
1.4.	OVERVIEW
This textbook is divided into three major Parts.
PART I:	Representation Formulas for Solutions
Here we identify those important partial differential equations for which
in certain circumstances explicit or more-or-less explicit formulas can be had
for solutions. The general progression of the exposition is from direct formu-
las for certain linear equations, to far less concrete representation formulas,
of a sort, for various nonlinear PDE.
Chapter 2 is a detailed study of four exactly solvable partial differen-
tial equations: the linear transport equation, Laplace’s equation, the heat
equation, and the wave equation. These PDE, which serve as archetypes for
the more complicated equations introduced later, admit directly computable
solutions, at least in the case that there is no domain whose boundary geom-
etry complicates matters. The explicit formulas are augmented by various
indirect, but easy and attractive, “energy”-type arguments, which serve as
motivation for the developments in Chapters 6, 7 and thereafter.
Chapter 3 continues the theme of searching for explicit formulas, now
for general first-order nonlinear PDE. The key insight is that such PDE
10
1. INTROD UCTION
can, locally at least, be transformed into systems of ordinary differential
equations (ODE), the characteristic equations. We stipulate that once the
problem becomes “only” the question of integrating a system of ODE, it
is in principle solved, sometimes quite explicitly. The derivation of the
characteristic equations given in the text is very simple and does not require
any geometric insights. It is in truth so easy to derive the characteristic
equations that no real purpose is had by dealing with the quasilinear case
first.
We introduce also the Hopf-Lax formula for Hamilton-Jacobi equa-
tions (§3.3) and the Lax-Oleinik formula for scalar conservation laws (§3.4).
(Some knowledge of measure theory is useful here, but is not essential.)
These sections provide an early acquaintance with the global theory of these
important nonlinear PDE, and so motivate the later Chapters 10 and 11.
Chapter 4 is a grab bag of techniques for explicitly (or kind of explicitly)
solving various linear and nonlinear partial differential equations, and the
reader should study only whatever seems interesting. The section on the
Fourier transform is, however, essential. The Cauchy-Kovalevskaya Theorem
appears at the very end. Although this is basically the only general existence
theorem in the subject, and thus logically should perhaps be regarded as
central, in practice these power series methods are not so prevalent.
PART II:	Theory for Linear Partial Differential Equations
Next we abandon the search for explicit formulas and instead rely on
functional analysis and relatively easy “energy” estimates to prove the ex-
istence of weak solutions to various linear PDE. We investigate also the
uniqueness and regularity of such solutions, and deduce various other prop-
erties.
Chapter 5 is an introduction to Sobolev spaces, the proper setting for
the study of many linear and nonlinear partial differential equations via en-
ergy methods. This is a hard chapter, the real worth of which is only later
revealed, and requires some basic knowledge of Lebesgue measure theory.
However, the requirements are not really so great, and the review in Ap-
pendix E should suffice. In my opinion there is no particular advantage in
considering only the Sobolev spaces with exponent p = 2, and indeed in-
sisting upon this obscures the two central inequalities, those of Gagliardo-
Nirenberg-Sobolev (§5.6.1) and of Morrey (§5.6.2).
In Chapter 6 we vastly generalize our knowledge of Laplace’s equation to
other second-order elliptic equations. Here we work through a rather com-
plete treatment of existence, uniqueness and regularity theory for solutions,
including the maximum principle, and also a reasonable introduction to the
1.4. OVERVIEW
11
study of eigenvalues, including a discussion of the principal eigenvalue for
nonselfadjoint operators.
Chapter 7 expands the energy methods to a variety of linear partial
differential equations characterizing evolutions in time. We broaden our
earlier investigation of the heat equation to general second-order parabolic
PDE, and of the wave equation to general second-order hyperbolic PDE. We
study as well linear first-order hyperbolic systems, with the aim of motivat-
ing the developments concerning nonlinear systems of conservation laws in
Chapter 11. The concluding section 7.4 presents the alternative functional
analytic method of semigroups for building solutions.
(Missing from this long Part II on linear partial differential equations is
any discussion of distribution theory or potential theory. These are impor-
tant topics, but for our purposes seem dispensable, even in a book of such
length. These omissions do not slow us up much, and make room for more
nonlinear theory.)
PART III:	Theory for Nonlinear Partial Differential Equations
This section parallels for nonlinear PDE the development in Part II, but
is far less unified in its approach, as the various types of nonlinearity must
be treated in quite different ways.
Chapter 8 commences the general study of nonlinear partial differential
equations with an extensive discussion of the calculus of variations. Here
we set forth a careful derivation of the direct method for deducing the ex-
istence of minimizers, and discuss also a variety of variational systems and
constrained problems, as well as minimax methods. Variational theory is
the most useful and accessible of the methods for nonlinear PDE, and so
this chapter is fundamental.
Chapter 9 is, rather like Chapter 4 before, a gathering of assorted other
techniques of use for nonlinear elliptic and parabolic partial differential equa-
tions. We encounter here monotonicity and fixed-point methods, and a va-
riety of other devices, mostly involving the maximum principle. We study
as well certain nice aspects of nonlinear semigroup theory, to complement
the linear semigroup theory from Chapter 7.
Chapter 10 is an introduction to the modern theory of Hamilton-Jacobi
PDE, and in particular to the notion of “viscosity solutions”. We encounter
also the connections with the optimal control of ODE, through dynamic
programming.
12
1. INTROD UCTION
Chapter 11 picks up from Chapter 3 the discussion of conservation laws,
now systems of conservation laws. Unlike the general theoretical develop-
ments in Chapters 5-9, for which Sobolev spaces provide the proper abstract
framework, we are forced to employ here direct linear algebra and calculus
computations. We pay particular attention to the solution of Riemann’s
problem and to entropy criteria.
Appendices A-E provide for the reader’s convenience some background
material, with selected proofs, on inequalities, linear functional analysis,
measure theory, etc.
The Bibliography primarily provides a listing of interesting PDE books
to consult for further information. Since this is a textbook , and not a refer-
ence monograph, I have mostly not attempted to track down and document
the original sources for the myriads of ideas and methods we will encounter.
The mathematical literature for partial differential equations is truly vast,
but the books cited in the Bibliography should at least provide a starting
point for locating the primary sources.
1.5. PROBLEMS
1.	Classify each of the partial differential equations in §1.2 as follows:
(a) Is the PDE linear, semilinear, quasilinear or fully nonlinear?
(b) What is the order of the PDE?
The next exercises provide some practice with the multiindex notation
introduced in Appendix A.
2.	Prove the Multinomial Theorem
(x1 + ... + xn)k= V
|a|=fc
where ('“') := a! =	• • • <*n!, and xa = x^1 ... x&n. The sum
is taken over all multiindices a = (ац,..., an) with |a| = k.
3.	Prove Leibniz ’formula
Da(uv) = У ^D^uDa~^v,
where u, v : Rn —> R are smooth, :=	, and /3 < a means
Pi < (i = 1, • • • ,n).
4.	Assume that f : R" —> R is smooth. Prove
/(x) = £ ^Z>7(0)*Q + O(|x|fc+1) as x 0
|a|<fc
1.5. PROBLEMS
13
for each к = 1,2,.... This is Taylor’s formula in multiindex notation.
(Hint: Fix x G and consider the function of one variable g(t) :=
/(tx).)
Chapter 2
FOUR IMPORTANT
LINEAR PARTIAL
DIFFERENTIAL
EQUATIONS
2.1	Transport equation
2.2	Laplace’s equation
2.3	Heat equation
2.4	Wave equation
2.5	Problems
2.6 References
In this chapter we introduce four fundamental linear partial differen-
tial equations for which various explicit formulas for solutions are available.
These are
the transport equation	ut + b • Du = 0	(§2.1),
Laplace’s equation	Au = 0	(§2-2),
the heat equation	Ut — Au = 0	(§2.3),
the wave equation	utt - Au = 0	(§2.4).
Before going further, the reader should review the discussions of in-
equalities, integration by parts, Green’s formulas, convolutions, etc. in Ap-
pendices В and C, and later refer back to these as necessary.
17
18
2. FOUR IMPORTANT LINEAR PDE
2.1.	TRANSPORT EQUATION
Probably the simplest partial differential equation of all is the transport
equation with constant coefficients. This is the PDE
(1)	щ + b • Du = 0 in R" x (0, oo),
where b is a fixed vector in R", b = (bi,... , bn), and и : R x [0, oo) —► R
is the unknown, и = u(x,t). Here x = (aq,... ,xn) E R" denotes a typical
point in space, and t > 0 denotes a typical time. We write Du = Dxu =
(uX1, uXn) for the gradient of и with respect to the spatial variables x.
Which functions и solve (1)? To answer, let us suppose for the moment
we are given some smooth solution и and try to compute it. To do so, we
first must recognize that the partial differential equation (1) asserts that a
particular directional derivative of и vanishes. We exploit this insight by
fixing any point (x, t) € Rn x (0, oo) and defining
z(s) := u(x + sb, t + s) (s G R).
We then calculate
z(s) = Du(x + sb,t + s) • b + ut(x + sb,t + s) — 0	[’ = -7-| ,
\ /
the second equality holding owing to (1). Thus z(-) is a constant function of
s, and consequently for each point (x,t), и is constant on the line through
(x,t) with the direction (b, 1) G R"+1. Hence if we know the value of и at
any point on each such line, we know its value everywhere in Rn x (0,00).
2.1.1.	Initial-value problem.
For definiteness therefore, let us consider the initial-value problem
, .	( ut + b • Du = 0 in Rn x (0,00)
' '	(	и = g on Rn x {t = 0}.
Here b G R" and g : R" —> R are known, and the problem is to compute
u. Given (x, t) as above, the line through (x, t) with direction (b, 1) is
represented parametrically by (x + sb, t + s) (s G R). This line hits the
plane Г := R" x {t = 0} when s = — t, at the point (a; — tb, 0). Since и is
constant on the line and u(x — tb, 0) = g(x — tb), we deduce
(3)	u(x, t) = g(x — tb) (a;GR”,t>0).
So,	if (2) has a sufficiently regular solution u, it must certainly be given
by (3). And conversely, it is easy to check directly that if g is C1, then и
defined by (3) is indeed a solution of (2).

2.1. TRANSPORT EQUATION
19
Remark. If g is not C1, then there is obviously no C1 solution of (2). But
even in this case formula (3) certainly provides a strong, and in fact the
only reasonable, candidate for a solution. We may thus informally declare
u(x, t) = g(x — tb) (a; G R", t > 0) to be a weak solution of (2), even should g
not be C1. This all makes sense even if g, and thus u, are discontinuous. Such
a notion, that a nonsmooth or even discontinuous function may sometimes
solve a PDE, will come up again later when we study nonlinear transport
phenomena in §3.4.	□
2.1.2.	Nonhomogeneous problem.
Next let us look at the associated nonhomogeneous problem
. .	( ut + b  Du — f in R" x (0, oo)
(	и = g on R" x {t = 0}.
As before fix (x, t) G R"+1 and, inspired by the calculation above, set z(s) :=
u(x + sb,t + s) for s E R. Then
i(s) = Du(x + sb,t + s) • b + щ(х + sb,t + s) = f(x + sb,t + s).
Consequently
u(x, t) — g(x — bt) = z(0) — z(—t) = У
r°
= J f(x + sb,t + s)ds
= [ f(x + (s — t)b,s)ds-.
Jo
and so
(5)	u(x, t) — g{x — tb) + [ f(x + (s — t)b,s)ds (a; G Rn, t > 0)
Jo
solves the initial-value problem (4).
We will later employ this formula to solve the one-dimensional wave
equation, in §2.4.1.
Remark. Observe that we have derived our solutions (3), (5) by in effect
converting the partial differential equations into ordinary differential equa-
tions. This procedure is a special case of the method of characteristics,
developed later in §3.2.	□
20
2. FOUR IMPORTANT LINEAR PDE
2.2.	LAPLACE’S EQUATION
Among the most important of all partial differential equations are undoubt-
edly Laplace’s equation
(1)	Au = 0
and Poisson’s equation
(2)	—Au = /. *
In both (1) and (2), x 6 U and the unknown is и : U —> R, и = u(x),
where U C Rn is a given open set. In (2) the function f : U —> R is also
given. Remember from §A.3 that the Laplacian of и is Au = uXiXi.
DEFINITION. A C2 function и satisfying (1) is called a harmonic func-
tion.
Physical interpretation. Laplace’s equation comes up in a wide variety
of physical contexts. In a typical interpretation u denotes the density of
some quantity (e.g. a chemical concentration) in equilibrium. Then if V is
any smooth subregion within U, the net flux of u through dV is zero:
[ F i/dS = 0,
JdV
F denoting the flux density and v the unit outer normal field. In view of
the Gauss-Green Theorem (§C.2), we have
f div F dx = [ F • v dS = 0,
Jv	JdV
and so
(3)	divF = 0 in U,
since V was arbitrary. In many instances it is physically reasonable to as-
sume the flux F is proportional to the gradient Du, but points in the opposite
direction (since the flow is from regions of higher to lower concentration).
Thus
(4)	F=-aDu (a > 0).
*1 prefer to write (2) with the minus sign, to be consistent with the notation for general
second-order elliptic operators in Chapter 6.
2.2. LAPLACE’S EQUATION
21
Substituting into (3), we obtain Laplace’s equation
div(Du) = Au = 0.
If и denotes the
' chemical concentration
< temperature
. electrostatic potential,
equation (4) is
{Fick’s law of diffusion
Fourier’s law of heat conduction
Ohm’s law of electrical conduction.
See Feynman-Leighton-Sands [F-L-S, Chapter 12] for a discussion of the
ubiquity of Laplace’s equation in mathematical physics. Laplace’s equation
arises as well in the study of analytic functions and the probabilistic inves-
tigation of Brownian motion.	□
2.2.1.	Fundamental solution.
a. Derivation of fundamental solution.
One good strategy for investigating any partial differential equation is
first to identify some explicit solutions and then, provided the PDE is linear,
to assemble more complicated solutions out of the specific ones previously
noted. Furthermore, in looking for explicit solutions it is often wise to re-
strict attention to classes of functions with certain symmetry properties.
Since Laplace’s equation is invariant under rotations (Problem 2), it conse-
quently seems advisable to search first for radial solutions, that is, functions
of r = |x|.
Let us therefore attempt to find a solution и of Laplace’s equation (1)
in U = Rn, having the form
u(x) = v(r),
where r = |x| = (x^ + • • • + a;2)1/2 and v is to be selected (if possible) so
that Au = 0 holds. First note for i = 1,..., n that
= 2 (Ж1	* xn) %xi = ~ (x 0)-
We thus have
= v'(r)p uXiXi = v"(r)^ + v'(r) Q -
22
2. FOUR IMPORTANT LINEAR PDE
for i = 1,..., n, and so
Au = v"(r) + ——-v'(r).
r
Hence Au = 0 if and only if
(5)	v" + -—-v' = 0.
r
If v' 7^ 0, we deduce
, , v" 1 - n
log(u) = — =--------,
v' r
and hence u'(r) = for some constant a. Consequently if r > 0, we have
f Mogr + c (n = 2)
= 1	b ,	( ox
I + C	(n > 3),
where b and c are constants.
These considerations motivate the following
DEFINITION. The function
,R, (n = 2)
(6)	(Ж) := 1 , * i (n > 31
t n(n—2)a(n) |т|п—2	' —
defined forx 6Rn, x 0, is the fundamental solution of Laplace’s equation.
The reason for the particular choices of the constants in (6) will be
apparent in a moment. (Recall from §A.2 that o(n) denotes the volume of
the unit ball in Rn.)
We will sometimes slightly abuse notation and write Ф(х) = Ф(|а;|) to
emphasize that the fundamental solution is radial. Observe also that we
have the estimates
(7)	IWMI <	(^/0)
for some constant C > 0.
b. Poisson’s equation.
By construction the function x ь-> Ф(а?) is harmonic for x 0. If we shift
the origin to a new point y, the PDE (1) is unchanged; and so x ь-> ф(х — у)
is also harmonic as a function of x, x / y. Let us now take f : R" —> R and
note that the mapping x i-> Ф(х — y)f(y) (x y) is harmonic for each point
2.2. LAPLACE’S EQUATION
23
у GKn, and thus so is the sum of finitely many such expressions built for
different points y.
This reasoning might suggest that the convolution
и(ж)= futn Ф(х - y)f(y) dy
(8)
' jR2log(k-?/l)/(y)<fy (n = 2)
. n(n-2)a(n) Jr"	(n —
will solve Laplace’s equation (1). However, this is wrong: we cannot just
compute
(9)	Д«(х) = [ Д^Ф^т - y)/(y) dy = 0.
JR"
Indeed, as intimated by estimate (7), £>2Ф(х — у) is not summable near the
singularity at у = x, and so the differentiation under the integral sign above
is unjustified (and incorrect). We must proceed more carefully in calculating
Au.
Let us for simplicity now assume f E C2(R.n); that is, f is twice contin-
uously differentiable, with compact support.
THEOREM 1 (Solving Poisson’s equation). Define и by (8). Then
(i) ибС2(Вп)
and
(ii) —Ди = f inR".
We consequently see that (8) provides us with a formula for a solution
of Poisson’s equation (2) in Rn.
Proof. 1. We have
u(x) = / Ф(х - y)f(y) dy = / Ф(у)/(х - у) dy,
JW1	JW1
hence
u(x + hei) — u(x)
h
f(x + h,ej -y)~ f(x ~ У)
h
dy,
where h 0 and Cj = (0,..., 1,..., 0), the 1 in the tt/l-slot. But
f(x + ha ~y)~ f(x -y) df
UXi
h
24
2. FOUR IMPORTANT LINEAR PDE
uniformly on R” as h —> 0, and thus
ди	f	df	.
~(x)=	$(y)~(x-y)dy (? = l,...,n).
dXi	jRn	OXi
Similarly
fi2u	I d2f
(Ю) я = / ф^я-гЯг (x~y)dy (i,j = 1, •••,«).
OXiUXj	J^n	OXiUXj
As the expression on the right hand side of (10) is continuous in the variable
x, we see и G (72(Rra).
2.	Since Ф blows up at 0, we will need for subsequent calculations to
isolate this singularity inside a small ball. So fix £ > 0. Then
Au(i) = / $(y)Axf(x — y)dy + /	Ф(у)Д1/(я: - у) dy
(11)	J B(0,e)	JRn—B(0,e)
=: I£ + J£.
Now
„	f	( Ce2|loge| (n = 2)
(12)	|4| < C\\D2f\\L^ / |Ф(у)Иу < {	2
</B(0,e)	l ОС	> 3).
An integration by parts (see §C.2) yields
Л = [ ^(y)^yf(x-y)dy
JR"-B(0,e)
= -/	D$(y) • Dyf(x - y) dy
(13)	7Rn—B(0,e)
+ f	^v)%.{x~y^dS^
JdB(0,e) <JV
=-K£ + l£,
v denoting the inward pointing unit normal along <ЭВ(0, s). We readily check
Г	(	Cel log el	(n	= 2)
(14)	\L£\<\\Df\\LX(№nJ	\*(y)\dS(y)<\	'
JdB{G,e)	I	(n	>	3).
3.	We continue by integrating by parts once again in the term K£, to
discover
K£= [	A$(y)f(x-y)dy- [	^(y)f(x-y)dS(y)
J^-Btfi,e)	JdB(0,e)
f	ВФ
=	-	/	я-(!/№-!/)^(У),
JdB(0,e)
2.2. LAPLACE’S EQUATION
25
since Ф is harmonic away from the origin. Now £>Ф(у) = na^ jffir (j/ / 0)
and v =	on dB(Q,e). Consequently f£(y) = 1/ Г>Ф(у) = na(n)£n-i
on сШ(0,£). Since na(n)£n-1 is the surface area of the sphere <9.8(0, £), we
have
Ks =	[ f(x ~ dS^
/ir.	na(n)£n 1 Лш(о,е)
(15)	.
= ~t f(y)dS(y) -> -f(x) as e 0.
J dB(x,e)
(Remember from §A.3 that a slash through an integral denotes an average.)
4. Combining now (11)—(15) and letting £ —> 0, we find -Дф) = /(x),
as asserted.	□
Remarks, (i) We sometimes write
-ДФ = <50 in Rn,
<50 denoting the Dirac measure on Rn giving unit mass to the point 0. Adopt-
ing this notation, we may formally compute:
-Д«(х) = / -ДхФ(х - у) f(y)dy
JRn
= [ 6xf(y)dy = f(x) (zeir),
JRn
in accordance with Theorem 1. This corrects the erroneous calculation (9).
(ii) Theorem 1 is in fact valid under far less stringent smoothness re-
quirements for f: see Gilbarg-Trudinger [G-T].	□
2.2.2. Mean-value formulas.
Consider now an open set U C R” and suppose и is a harmonic function
within U. We next derive the important mean-value formulas, which declare
that u(x) equals both the average of и over the sphere dB(x, r) and the
average of и over the entire ball B(x,r), provided B(x,r) C U. These
implicit formulas involving и generate a remarkable number of consequences,
as we will momentarily see.
THEOREM 2 (Mean-value formulas for Laplace’s equation). If и € C2(U)
is harmonic, then
(16)	u(x) = + udS — + udy
J dB(x,r) J B(x,r)
for each ball B(x, r) C U.
26
2. FOUR IMPORTANT LINEAR PDE
Proof. 1. Set
</>(r) := -f u(y)dS(y) = 7 u(x + rz) dS(z).
J dB(x,r)	J dB(0,l)
Then
ф'(г)= + Du(x + rz) • zdS(z),
J ав(о,1)
and consequently, using Green’s formulas from §C.2, we compute
ф'(г) = / Du(y)  dS(y)
J dB(x,r)
I	9U A
= t	^r,dS^
J dB(x,r)
= - 7	Au(y) dy	= 0.
nJ B(x,r)
Hence ф is constant, and so
ф(г) = lim</>(t) = lim + u(y)dS(y) = u(x).
t-o	t—>o J dB(x,t)
2. Observe next that our employing polar coordinates, as in §C.3, gives
I udS j ds
dB(x,s) J
= u(x) I na(n)sn rds — a(n)rnu(x).
Jo
□
THEOREM 3 (Converse to mean-value property). IfuE C2(U) satisfies
u(x) = + udS
J dB(x,r}
for each ball B(x,r) C U, then и is harmonic.
Proof. If Au 0, there exists some ball B(x, r) CU such that, say, Au > 0
within B{x, r). But then for ф as above,
0 = ф'(г) = - 7 Au(y) dy > 0,
nJ B(x,r)
a contradiction.	□
2.2. LAPLACE’S EQUATION
27
2.2.3. Properties of harmonic functions.
We now present a sequence of interesting deductions about harmonic
functions, all based upon the mean-value formulas. Assume for the following
that U C is open and bounded.
a. Strong maximum principle, uniqueness.
THEOREM 4 (Strong maximum principle). Suppose и € C2(U) A C(U)
is harmonic within U.
(i) Then
max и = max u.
U	9U
(ii) Furthermore, if U is connected and there exists a point xq E U such
that
u(xq) = max u,
и
then
и is constant within U.
Assertion (i) is the maximum principle for Laplace’s equation and (ii) is
the strong maximum principle. Replacing и by — u, we recover also similar
assertions with “min” replacing “max”.
Proof. Suppose there exists a point xq E U with u(xq) = M := шаху u.
Then for 0 < r < dist(xo, dU), the mean-value property asserts
M = u(a?o) = 4- udy < M.
J B(x0,r)
As equality holds only if и = M within B(a?o,r), we see u(y) = M for all
у E B(x,r). Hence the set {x E U | и(х) = M} is both open and relatively
closed in U, and thus equals U if U is connected. This proves assertion (ii),
from which (i) follows.	□
Remark. The strong maximum principle asserts in particular that if U is
connected and и E C2(U) A C(W) satisfies
( Au = 0 in U
( и = g on dU,
where g > 0, then и is positive everywhere in U if g is positive somewhere
on dU.	□
An important application of the maximum principle is establishing the
uniqueness of solutions to certain boundary-value problems for Poisson’s
equation.
28
2. FOUR IMPORTANT LINEAR PDE
THEOREM 5 (Uniqueness). Let g E C(dU), f E C(U). Then there exists
at most one solution и E C2(U) П C'(U) of the boundary-value problem
(17)
—Au = f inU
и = g on dU.
Proof. If и and й both satisfy (17), apply Theorem 4 to the harmonic
functions w := ±(u — u).
b. Regularity.
Now we prove that if и E C2 is harmonic, then necessarily и E C°°.
Thus harmonic functions are automatically infinitely differentiable. This
sort of assertion is called a regularity theorem. The interesting point is that
the algebraic structure of Laplace’s equation Au — uXiXi = 0 leads to
the analytic deduction that all the partial derivatives of и exist, even those
which do not appear in the PDE.
THEOREM 6 (Smoothness). If и E 0(11) satisfies the mean-value prop-
erty (16) for each ball B(x,r) C U, then
и E
Note carefully that и may not be smooth, or even continuous, up to dU.
Proof. Let 7] be a standard mollifier, as described in §C.4, and recall that
7] is a radial function. Set u£ := t]£ * и in U£ = {x E U | dist(x, dU) > e}.
As shown in §C.4, u£ E C°°(Ue).
We will prove и is smooth by demonstrating that in fact и = u£ on U£.
Indeed if x E Ue, then
= u(x) / g£dy = u(x).
JB(O,e)
Thus u£ = и in U£, and so и E	for each e > 0.
2.2. LAPLACE’S EQUATION
29
c. Local estimates for harmonic functions.
Next we employ the mean-value formulas to derive careful estimates on
the various partial derivatives of a harmonic function. The precise structure
of these estimates will be needed below, when we prove analyticity.
THEOREM 7 (Estimates on derivatives). Assume и is harmonic in U.
Then
(18)	|Z>au(x0)| <	||w||£l(B(IO,r))
for each ball B(xq, r) C U and each multiindex a of order |o| = k.
Here
(19)	С°~а(пуС“~ a(n> f*"1’--)-
Proof. 1. We establish (18), (19) by induction on k, the case к = 0 being
immediate from the mean-value formula (16). For к = 1, we note upon
differentiating Laplace’s equation that uXi (i = 1,... ,n) is harmonic. Con-
sequently
=	uXidx\
J B(xg,r/2)
(20)	= I Д [ uvidS\
O(n)rn JdB(x0,r/2)
< ylhllL~(dB(x0,i))-
Now if x G dB(xo,r/2), then B(x,r/2) C B(a?o,r) C U, and so
1 /2\n
~ a(nj \г/ Ии11£1(В(®о,г))
by (18), (19) for к = 0. Combining the inequalities above, we deduce
|£>“u(a;o)| <
if = 1. This verifies (18), (19) for к = 1.
2.	Assume now к > 2 and (18), (19) is valid for all balls in U and each
multiindex of order less than or equal to к - 1. Fix B(xq,t) c U and let a
30
2. FOUR IMPORTANT LINEAR PDE
be a multiindex with |a| = k. Then Dau = (D@u)Xi for some i G {1,..., n},
|/3| = fc — 1. By calculations similar to those in (20), we establish that
|£»%(xo)| < у ||Л||^(^о^))-
If ж € дВ(хо, £), then B(x, V"7’) С B(a?o,r) C U. Thus (18), (19) for
к — 1 imply
, . , xl (2"+1n(fc — l))fe-1.. „
\D u(x)| <	чп+fc-l
a(n) (VO
Combining the two previous estimates yields the bound
(21)	|T>“Vo)| < Л+fc HUllL1(B(xo,r))-
This confirms (18), (19) for |a| = к.	□
d. Liouville’s Theorem.
Next we see that there are no nontrivial bounded harmonic functions on
all of R".
THEOREM 8 (Liouville’s Theorem). Suppose и : Rn —> R is harmonic
and bounded. Then и is constant.
Proof. Fix a?o G R", r > 0, and apply Theorem 7 on B(xq, r):
|Pu(xo)| < ^IMl£i(B(*o,r))
Cia(n)
<	-	||ulk°°(Rn) —
as r —> oo. Thus Du = 0, and so и is constant.	□
THEOREM 9 (Representation formula). Let f G C^R”), n > 3. Then
any bounded solution of
—Au = f in Rn
has the form
u(x) = [ Ф(х - y)f(y)dy + C (r G Rn)
JR"
for some constant C.
2.2. LAPLACE’S EQUATION
31
Proof. Since Ф(х) —> 0 as |ж| —> oo for n > 3, U(x) := Ф(х — y)f(y)dy
is a bounded solution of —Au = / in Rn. If и is another solution, w -.= u — u
is constant, according to Liouville’s Theorem.	□
Remark. If n — 2, Ф(х) = —^log|x| is unbounded as |ж| —> oo, and so
may be JR2 Ф(х - y)f(y) dy.	□
e. Analyticity.
Next we refine Theorem 6:
THEOREM 10 (Analyticity). Assume и is harmonic in U. Then и is
analytic in U.
Proof. 1. Fix any point a?o 6 U. We must show и can be represented by a
convergent power series in some neighborhood of xq-
Let r := i dist(x0, dU). Then M := ^j^||u||Li(B(a.Oi2r)) < oo.
2. Since B(x, r) c B(xo,2r) c U for each x E B(xo,r), Theorem 7
provides the bound
/2n+1n\|a|	, ,
IP4lL~(B(z0,r)) < M — J |а|К
Now Stirling’s formula ([RD, §8.22]) asserts lim^^oc — (2я-)1/г • Hence
|a||a| < Cel"l|a|!
for some constant C and all multiindices a. Furthermore, the Multinomial
Theorem implies
whence
Combining the previous inequalities now yields
/2n+1n2e\ |a|
(22)	||D"u||LOO(B(a;0jr)) < CM«!•
3. The Taylor series for и at a?o is
a
32
2. FOUR IMPORTANT LINEAR PDE
the sum taken over all multiindices. We assert this power series converges,
provided
(23)	lX-*01 < 2n+2n3e-
To verify this, let us compute for each N the remainder term:
о / x /X v-1 v Dau(xo)(x ~ ®o)“
RN(x) := u(x) - X 2^ -----------------------
k—0 |a|=fc
Dau(xo + t(x — а?о))(я: — a?o)Q
for some 0 < t < 1, t depending on x. We establish this formula by writing
out the first N terms and the error in the Taylor expansion about 0 for the
function of one variable g(f) := u{xq + t(x — xq)), at t = 1. Employing (22),
(23), we can estimate
|/Мж)| < cm ^2 Pn+ln2e>) ( J
1 v 71 “	\	r	) \2n+2n3eJ
|a|=N x	7
N 1	_	CM
- CMn (2n)'v	“	2'v
—> 0 as N —> oo.
□
See §4.6.2 for more on analytic functions and partial differential equa-
tions.
f. Harnack’s inequality.
Recall from §A.2 that we write V CC U to mean V с V C U and V is
compact.
THEOREM 11 (Harnack’s inequality). For each connected open set V
CC U, there exists a positive constant C, depending only on V, such that
sup и < C inf и
v v
for all nonnegative harmonic functions и in U.
Thus in particular
^u(y) < u(x) < Cu(y)
2.2. LAPLACE’S EQUATION
33
for all points x,y e V. These inequalities assert that the values of a non-
negative harmonic function within V are all comparable', и cannot be very
small (or very large) at any point of V unless и is very small (or very large)
everywhere in V. The intuitive idea is that since V is a positive distance
away from dU, there is “room for the averaging effects of Laplace’s equation
to occur”.
Proof. Let r := | dist(V, dUf Choose x,y eV, \x — y\ < r. Then
u(x) = -[ udz> - ,	-- [ udz
J B(x,2r) a(n)2nrn JBM
1	[	,	1	, X
= — + udz = —u(y).
^ПТв(у,г)
Thus 2nu(y) > u(x) > ^-u(y) if x,y e V, |ж — y\ < r.
Since V is connected and V is compact, we can cover V by a chain of
finitely many balls > each of which has radius r and Bi A B,_i 0
for i = 2,... , N. Then
u(x) > ^u(y)
for all x, у e V.	□
2.2.4. Green’s function.
Assume now U C IRn is open, bounded, and dU is C1. We propose
next to obtain a general representation formula for the solution of Poisson’s
equation
—Ди = f in U,
subject to the prescribed boundary condition
и = g on dU.
a. Derivation of Green’s function.
Suppose first of all и e CPfU) is an arbitrary function. Fix x e U,
choose s > 0 so small that B(x.e) c U, and apply Green’s formula from
§C.2 on the region V£ := U — B(x,e) to u(y) and Ф(у — x). We thereby
compute
/ и(у)ДФ(у - x) - Ф(у - x)Au(y) dy
(24) JVe	<ЭФ	a
= jдvU(y)^-(y-x)-Ф(y-x^y)dS(y),
34
2. FOUR IMPORTANT LINEAR PDE
v denoting the outer unit normal vector on dVe. Recall next ДФ(х — у) = О
for x у. We observe also
I / Ф(у - z) о"(у) dS(y)\ < C^-1 max |Ф| = o(l)
JdB(x,e)	UV	9B(0,e)
as £ —> 0. Furthermore the calculations in the proof of Theorem 1 show
[	u(y)^-(y-x)dS(y)= -[ u(y)dS(y)-> u(x)
JdB(x,e) °v	J dB(x,e)
as £ —> 0. Hence our sending £ —> 0 in (24) yields the formula:
/"	ди	дФ
u(x)= / Ф(у-х) — (у) -u(y) — (y-x)dS(y)
(25)	hu dv
- / Ф(у - ж)Ди(у) dy.
Ju
This identity is valid for any point x G U and any function и G C2(U).
Now formula (25) would permit us to solve for u(x) if we knew the
values of Ди within U and the values of и, ди/ди along dU. However for
our application to Poisson’s equation with prescribed boundary values for u,
the normal derivative ди/du along dU is unknown to us. We must therefore
somehow modify (25) to remove this term.
The idea is now to introduce for fixed x a corrector function фх — фх(у),
solving the boundary-value problem:
,9fn	f &ФХ = 0 in U
' '	| фх = Ф(у — x) on dU.
Let us apply Green’s formula once more, now to compute
- [ фх(у)Аи(у^у = f и(у)^-(у)-фх(у)^-(у^8(у>)
/97х Ju	Jdu vv
г	Qu
= ^у^у)-ф(у-х^у^8(у).
We introduce next this
DEFINITION. Green’s function for the region U is
G(x,y) := Ф(у — x) — фх(у) (x,yEU,x/y).
2.2. LAPLACE’S EQUATION
35
Adopting this terminology and adding (27) to (25), we find
(28)	u(ar) = - / u(y)^-(x,y)dS(y) - f G(x,y)&u(y) dy (xeU),
JdU ov	Ju
where
dG
-Q^(x,y') = DyG(x,y)  u(y)
is the outer normal derivative of G with respect to the variable y. Observe
that the term du/dv does not appear in equation (28): we introduced the
corrector фх precisely to achieve this.
Suppose now и e C2(t/) solves the boundary-value problem
(29)	( ~Au = f ln U
[ и = g on oU,
for given continuous functions f,g. Plugging into (28), we obtain
THEOREM 12 (Representation formula using Green’s function). If
и G O'2 (77) solves problem (29), then
(30)	и(ж) = - [ g(y)^-(x,y)dS(y)+ [ f(y')G(x,y)dy (xeU).
Jdu	Ju
Here we have a formula for the solution of the boundary-value problem
(29), provided we can construct Green’s function G for the given domain U.
This is in general a difficult matter, and can be done only when U has simple
geometry. Subsequent subsections identify some special cases for which an
explicit calculation of G is possible.
Remark. Fix x 6 U. Then regarding G as a function of y, we may sym-
bolically write
(~NG=6X in U
I G=0 ondU,
6X denoting the Dirac measure giving unit mass to the point x.	□
Before moving on to specific examples, let us record the general assertion
that G is symmetric in the variables x and y:
THEOREM 13 (Symmetry of Green’s function). For all x,y EU, x / у,
we have
G(y,x) = G(x,y).
36
2. FOUR IMPORTANT LINEAR PDE
Proof. Fix x,y E U, x y. Write
v(z) G(x,z), w(z) G(y,z) (z E U).
Then Au(z) = 0 (z x), Aw(z) = 0 (z y) and w = v = 0 on
dU. Thus our applying Green’s identity on V := U — [В(ж,е) U B(y,£)] for
sufficiently small e > 0 yields
f dv dw	f dw dv
(31)	/	— w - -~-vdS(z) = /	— v-— wdS(z),
JdB(x,e) dv dv	JdB(y,e) dv dv
v denoting the inward pointing unit vector field on dB(x, E)UdB(y, s). Now
w is smooth near ж; whence
\(	dS\ < CEn~r sup |v| = o(l) as e —> 0.
J9B(x,e) dv	9B(x,e)
On the other hand, t>(z) — Ф(г — ж) — <px(z'), where фх is smooth in U. Thus
Г	dv	f	d$
lim /	— wdS=lim /	—— (x — z)w(z) dS = w(x),
^°JdB(x^dv	^JdB^,e)dv
by calculations as in the proof of Theorem 1. Thus the left-hand side of (31)
converges to ад(ж) as e —> 0. Likewise the right hand side converges to u(y).
Consequently
G(y,x) = ю(ж) = v(y) = G(x,y).
□
b. Green’s function for a half-space.
In this and the next subsection we will build Green’s functions for two
regions with simple geometry, namely the half-space IR” and the unit ball
B(0,1). Everything depends upon our explicitly solving the corrector prob-
lem (26) in these regions, and this in turn depends upon some clever geo-
metric reflection tricks.
First let us consider the half-space
R” = {ж = (Ж1,... ,жп) e 1R” I xn > 0}.
Although this region is unbounded, and so the calculations in the previous
section do not directly apply, we will attempt nevertheless to build Green’s
function using the ideas developed before. Later of course we must check
directly that the corresponding representation formula is valid.
2.2. LAPLACE’S EQUATION
37
DEFINITION. If x = (®i,... ,xn--[,xn) € R”; its reflection in the plane
сЖ" is the point
x = (xi,...,xn_i,-a;n).
We will solve problem (26) for the half-space by setting
фх(у) :=	- x) = Ф(у! - Ж1,..., yn_i - жп_1, yn + xn) (x, у E R").
The idea is that the corrector фх is built from Ф by “reflecting the singular-
ity” from x G R" to x R”. We note
фх(у) = Ф(у - x) ifyEdWf,
and thus
Афх = 0	in R"
фх = Ф(у — ж) on dR",
as required.
DEFINITION. Green’s function for the half-space R" is
G(x, у) := Ф(у - x) - Ф(у - x) (x, у E R”, x / y).
Then
9G дФ	дФ
-1
na(n)
Уп xn yn + xn
\y — x\n \y — x\n
Consequently if у E <9R”,
9G.	. 9G .	. -2xn 1
= ~na{n)\x-vr
Suppose now и solves the boundary-value problem
. .	f Au = ° in R”
V >	\ u = 9 on <9R”.
Then from (30) we expect
(33)	и(ж) =	. 9^y\ dy (x e R")
na(n) JdRn \x -y\n У v +'
to be a representation formula for our solution. The function
K(x,y)-.= 2^n	1 (xeRX.ye^)
na(n) \x — y\n
is Poisson’s kernel for R", and (33) is Poisson’s formula.
We must now check directly that formula (33) does indeed provide us
with a solution of the boundary-value problem (32).
38
2. FOUR IMPORTANT LINEAR PDE
THEOREM 14 (Poisson’s formula for half-space). Assume g G C(R” J)A
and define и by (33). Then
(i)	zzgC00(R^)AL00(R”),
(ii)	Nu = 0 in R”,
and
(iii)	lim u(a?) = g(x°) for each point x° E SR”.
x~>x°
zdR^
Proof. 1. For each fixed x, the mapping у t—► G(x, y) is harmonic, except
for у = x. As G(x, y) = G(y,x) according to Theorem 13, x t—> G(x, y) is
harmonic, except for x = y. Thus x t—> — ^f-(x,y) = K(x,y) is harmonic
for x € R”, у E SR”.
2. A direct calculation, the details of which we omit, verifies
(34)	i= [ K(x,y)dy
for each x E R”. As g is bounded, и defined by (33) is likewise bounded.
Since x i—> K(x, y) is smooth for x y, we easily verify as well и E C'OO(R”),
with
Дф) = [ NxK(x,y)g(y)dy = 0 (x E R”).
Jd^
3. Now fix x° E SR”, e > 0. Choose 6 > 0 so small that
(35)	|^(y) - ^(x°)| < £ if |т/-ж0| < 6, у e SR”.
Then if I® — x°| < |, x E R” ,
I K(x,y)[g(y)-g(x°)]dy
Ж
(36)
I	K(x,y)\g(y)-g(x°)\dy
дК^ПВ(х°,6)
+ [	K(x,y)\g(y) - g(x°)\dy
Now (34), (35) imply
I <£ K(x,y)dy = £.
2.2. LAPLACE’S EQUATION
39
Furthermore if \x — x°| < and \y — x°| > 6, we have
\y - ®°| < |з/ - + 5 < |г/ - ж| + ||y - ж°|;
and so — ж| > ||y — x°|. Thus
J < 2||^||ь°° [ K(x,y)dy
JdR”-B(x°,S)
na(n)
I	\y — ж°| n dy
dRn-B(x°,6)
—> 0 as xra —> 0+.
Combining this calculation with estimate (36), we deduce |u(x)—^(x°)| < 2e,
provided \x — x°| is sufficiently small.	□
c. Green’s function for a ball.
To construct Green’s function for the unit ball B(0,1) we will again
employ a kind of reflection, this time through the sphere дВ(0,1).
DEFINITION. If x e IRn - {0}, the point
x
x' H
is called the point dual to x with respect to dB(0,1). The mapping x x
is inversion through the unit sphere dB(0, 1).
We now employ inversion through the sphere to compute Green’s func-
tion for the unit ball U = B°(0,1). Fix x e B°(0,1). Remember that we
must find a corrector function фх = фх(у) solving
(4^ = 0	in В0 (0,1)
[ ^ = Ф(у-ж) on 9В(0,1);
then Green’s function will be
(38)
G(x, у) = Ф(у - x) - фх(у).
The idea now is to “invert the singularity” from x 6 B°(0,1) to x ф
B(0,1). Assume for the moment n > 3. Now the mapping у t—► Ф(у — i) is
harmonic for у / x. Thus у t—► |ж|2-nФ(y — x) is harmonic for у / x, and so
(39)
ФЧу) := Ф(И(^-®))
40
2. FOUR IMPORTANT LINEAR PDE
is harmonic in U. Furthermore, if у € dB(0,1) and x / 0,
л-^=м2
= |x|2 - 2y  x + 1 = |ж-у|2.
Thus (|a;||y — x|)-(n-2) =	— y|-(n-2). Consequently
(40)	фх(у) = Ф(у - x) (ye dB(0.1)),
as required.
DEFINITION. Green’s function for the unit ball is
(41) G(x,y) := Ф(у - x) - Ф(|х|(у-ж)) (x,y e B(0,1), x / y).
The same formula is valid for n = 2 as well.
Assume now и solves the boundary-value problem
(42)
Ди = 0 inB°(0,l)
и = g in dB(0,1).
Then using (30), we see
(43)	u(x) = - f g(y)^Tx,y)dS(y).
According to formula (41),
dG.	. <ЭФ. . d ,.
But
дФ , _ . =	1 Xj-yj
dyi X У na(n) |t - y|n ’
and furthermore
дФ,,	= -1 Уг|ж|2-Жг =	1 yj\x\2-Xj
dyi X У X na(n) (|x||y — x|)n na(n) |ж — y\n
if у € dB(0,1). Accordingly
dG, , A dG( .
- ^n) Ar §	- X,) " K W2 + X,)
_ -1 1~и2
na(n) |ж — y\n'
2.2. LAPLACE’S EQUATION
41
Hence formula (43) yields the representation formula
1 - И2 Г g(y)
na(ri) JdB(o,i) \x ~ У\п
Suppose now instead of (42) и solves the boundary-value problem
f Au = 0 in 8°(0, r)
( '	[ u = g on 5.8(0, r)
for r > 0. Then u(x) = u(rx) solves (42), with g(x) = g(rx) replacing g.
We change variables to obtain Poisson’s formula
(45)	u(x) =	dS(y) (x e B°(0, r)).
na(n)r JdB(p,r) \x - y\n
The function
„2 _ |T|2 i
^(z,y):=-------r-т— ।-----rj- (ж G 8°(0,r), у G 38(0,r))
na(n)r \x — y\n
is Poisson’s kernel for the ball 8(0, r).
We have established (45) under the assumption that a smooth solution
of (44) exists. We next assert that this formula in fact gives a solution:
THEOREM 15 (Poisson’s formula for ball). Assume g € (7(38(0, r)) and
define и by (45). Then
(i)	и G C°°(80(0,r)),
(ii)	Au = 0 mB°(0, r),
and
(iii)	lim u(x) = g(x°) for each point x° G 38(0, r).
хев°(о,т)
The proof is similar to that for Theorem 14, and is left as an exercise.
2.2.5. Energy methods.
Most of our analysis of harmonic functions thus far has depended upon
fairly explicit representation formulas entailing the fundamental solution,
Green’s functions, etc. In this concluding section we illustrate some “en-
ergy” methods, which is to say techniques involving the T2-norms of various
expressions. These ideas foreshadow latter theoretical developments in Parts
II and III.
42
2. FOUR IMPORTANT LINEAR PDE
a. Uniqueness.
Consider first the boundary-value problem
(46)
-Au = /
и = g
in U
on dU.
We have already employed the maximum principle in §2.2.3 to show
uniqueness, but now set forth a simple alternative proof. Assume U is open,
bounded, and dU is C1.
THEOREM 16 (Uniqueness). There exists at most one solution и 6
C2(U) of (46).
Proof. Assume it is another solution and set w := и — й. Then Aw = 0 in
U, and so an integration by parts shows
0 = — / wAwdx = I \Dw\2dx.
Ju	Ju
Thus Dw - 0 in U, and, since w = 0 on dU, we deduce w = и — й = 0 in
U.	□
b. Dirichlet’s principle.
Next let us demonstrate that a solution of the boundary-value problem
(46) for Poisson’s equation can be characterized as the minimizer of an
appropriate functional. For this, we define the energy functional
f[w] = / -\Dw\2 — wfdx,
Ju *
w belonging to the admissible set
A = {w 6 C2(U") | w — g on dU}.
THEOREM 17 (Dirichlet’s principle). Assume и G C2(U) solves (46).
Then
(47)	7[u] = min/[w].
weA
Conversely, if и € A satisfies (47), then и solves the boundary-value problem
(46).
In other words if и G A, the PDE —Au = f is equivalent to the statement
that и minimizes the energy /[•].
2.2. LAPLACE’S EQUATION
43
Proof. 1. Choose w e A. Then (46) implies
0= / (—Au — f)(u — w)dx.
Ju
An integration by parts yields
0 = / Du  D(u — w) — f(u — w) dx,
Ju
and there is no boundary term since и — w = g — g = 0 on dU. Hence
/ ]Du\2 — uf dx = / Du-Dw — wfdx
Ju	Ju
< [ ^|Du|2dx + [ ^-\Dw\2 — wf dx,
Ju 2	Ju 2
where we employed the estimates
\Du  Pw| < |Du||Dw| < |Du|2 + ||Dw|2,
following from the Cauchy-Schwarz and Cauchy inequalities (§B.2). Rear-
ranging, we conclude
(48)	f[u] < 7[w] (w G Л).
Since и e A, (47) follows from (48).
2.	Now, conversely, suppose (47) holds. Fix any v G C^°(U) and write
г(т) := f[u + tv] (t G IR).
Since и + tv G A for each t, the scalar function г(-) has a minimum at zero,
and thus
i'(0) = ° (' = £),
\ d.T /
provided this derivative exists. But
i(r) = I -\Du + tDv\2 — (u + Tv)fdx
Ju 2
fl	T2
— I -\Du\2 + tDu  Dv +—\Dv\2 — (u + Tv)f dx.
Ju 2	2
Consequently
0 = г'(0) = I Du-Dv — vfdx= / (-Au-f)vdx.
Ju	Ju
This identity is valid for each function v G	and so —Au = f in
U.	□
Dirichlet’s principle is an instance of the calculus of variations applied
to Laplace’s equation. See Chapter 8 for more.
44
2. FOUR IMPORTANT LINEAR PDE
2.3.	HEAT EQUATION
Next we study the heat equation
(1)	щ — Au — 0
and the nonhomogeneous heat equation
(2)	ut - Au = /,
subject to appropriate initial and boundary conditions. Here t > 0 and
x E U, where U C IRn is open. The unknown is и : U x [0, oo) —> IR, и =
u(x, t), and the Laplacian A is taken with respect to the spatial variables x =
(®1,..., xn): Au = Axu = J3”=1 uXiXi  In (2) the function f : Ux [0, oo) —► IR
is given.
A guiding principle is that any assertion about harmonic functions yields
an analogous (but more complicated) statement about solutions of the heat
equation. Accordingly our development will largely parallel the correspond-
ing theory for Laplace’s equation.
Physical interpretation. The heat equation, also known as the diffusion
equation, describes in typical applications the evolution in time of the density
и of some quantity such as heat, chemical concentration, etc. If V C U is
any smooth subregion, the rate of change of the total quantity within V
equals the negative of the net flux through dV:
[ udx = ~ [ E vdS,
dt Jv Joy
F being the flux density. Thus
(3)	щ = — divF,
as V was arbitrary. In many situations F is proportional to the gradient
of u, but points in the opposite direction (since the flow is from regions of
higher to lower concentration):
F = —aDu (a > 0).
Substituting into (3), we obtain the PDE
щ = a div(Du) = a Au,
which for a = 1 is the heat equation.
The heat equation appears as well in the study of Brownian motion.
2.3. HEAT EQUATION
45
2.3.1. Fundamental solution.
a. Derivation of the fundamental solution.
As noted in §2.2.1 an important first step in studying any PDE is often
to come up with some specific solutions.
We observe that the heat equation involves one derivative with respect
to the time variable t, but two derivatives with respect to the space vari-
ables Xi (i = 1,... , n). Consequently we see that if и solves (1), then so
does u(Xx, A1 2t) for A e IR. This scaling indicates the ratio T (r = И) is
important for the heat equation and suggests that we search for a solution
of (1) having the form u(x,t) — v(y-) = v(^-) (t > 0, x E IRn), for some
function v as yet undetermined.
Although this approach eventually leads to what we want (see Problem
11), it is quicker to seek a solution и having the special structure
(4)	u(x,t) = ^v^ (a? € IRn, t > 0),
where the constants a, /3 and the function v : IRn —> IR must be found. We
come to (4) if we look for a solution и of the heat equation invariant under
the dilation scaling
u(x,t) t—► Xau(X/3x, At).
That is, we ask
u(x,t) = Xau(X^x, At)
for all A > 0, x E IRn, t > 0. Setting A = t-1, we derive (4) for v(y) :=
«(y,i)-
Let us insert (4) into (1), and thereafter compute
(5)	at-("+1)v(y) + f3t~(a+^y  Dv(y) + t-(“+2/3)Av(y) = 0
for у := t~@x. In order to transform (5) into an expression involving the
variable у alone, we take (3 = Then the terms with t are identical, and
so (5) reduces to
(6)
av + ^y  Dv + Av = 0.
JU
We simplify further by guessing v to be radial; that is, v(y) = w(|y|) for
some w : IR —> IR. Thereupon (6) becomes
1	,	.. n — 1 ,
aw + -rw + w -I----------w = 0,
2	r
46
2. FOUR IMPORTANT LINEAR PDE
for r = |y|, ' = Now if we set a = this simplifies to read
(r^w')'+ ^(rnw)'= 0.
Thus
rn~1w> + ^rnw = a
for some constant a. Assuming limr-.ooW, w' = 0, we conclude a = 0;
whence
, 1
W = —~rw.
Zi
But then for some constant b
r2
(7)	w = be~~* .
। д. 12
Combining (4), (7) and our choices for a,0, we conclude that e~ «
solves the heat equation (1).
This computation motivates the following
DEFINITION. The function
Ф(х,£) := <
1
(4тг«)"/2 e 4t
0
(x E Rn, t > 0)
t<0)
is called the fundamental solution of the heat equation.
Notice that Ф is singular at the point (0,0). We will sometimes write
Ф(х,£) = Ф(|х|,t) to emphasize that the fundamental solution is radial in
the variable x. The choice of the normalizing constant (4тг)-п/2 is dictated
by the following
LEMMA (Integral of fundamental solution). For each time t > 0,
= 1.
Proof. We calculate
1	_n_ r°°
□
A different derivation of the fundamental solution of the heat equation
appears in §4.3.2.
2.3. HEAT EQUATION
47
b. Initial-value problem.
We now employ Ф to fashion a solution to the initial-value (or Cauchy)
problem
(8)
( Ut — Au = 0 in Rn x (0, oo)
[ и = g on Rn x {t = 0}.
Let us note the function (x,t) >—> Ф(ж,£) solves the heat equation away
from the singularity at (0,0), and thus so does (x, t) н-> Ф(х — у, t) for each
fixed у E Rn. Consequently the convolution
u(z,t)=/ Ф(х — y,t}g(y}dy
(9)
1 ;	1	/	|z-y|2
= 777^2/ e~ " 9^dy *>0)
(4тгф/2 jRn
should also be a solution.
THEOREM 1 (Solution of initial-value problem). Assume g E C(Rn) П
L°°(IRn), and define и by (9). Then
(i)	uEC^R" x (0, oo)),
(ii)	Ut(x,t) — Au(x,t) = 0 (x E Rn, t > 0),
and
(iii)	lim u(x, t) = g(xQ} for each point x° E Rn.
(x,t)—»(a:o,0)
zeR'1, t>0
1 l*l2
Proof. 1. Since the function e « is infinitely differentiable, with uni-
formly bounded derivatives of all orders, on Rn x [<5, oo) for each 6 > 0, we
see that и E C'oo(Rn x (0, oo)). Furthermore
ut(x,t) - Au(x,t) = I [(Ф( - ДхФ)(х - y,t)]g(y)dy
(10)	JR"
= 0 (x € Rn, t > 0),
since Ф itself solves the heat equation.
2.	Fix x° E Rn, £ > 0. Choose 6 > 0 such that
(И)	Ш-5(т°)|<£ if |i/ —ж°|<<5, уеГ.
48
2. FOUR IMPORTANT LINEAR PDE
Then if \x — x°| < we have, according to the lemma,
|u(x, t) - 0(ж°)| = I [ Ф(ж - у, t)[g(y) - 5(x0)] dy\
JR"
<[	$(x-y,t)\g(y)-g(x°)\dy
JB(x°,6)
+ [	^(x-y,t)\g(y)-g(x°)\dy
J№-B(x°,6)
= '.I+J.
Now
I < e I Ф(х - у, t) dy = e,
JR"
owing to (11) and the lemma. Furthermore, if |x — x°| < | and \y — x°| > 6,
then
|У - ж°| < \У - x\ + < \y - x\ + ||y - ж°|.
Thus \y — ж| > ||y — ж°|. Consequently
J < 2||^||ь°° [	<&(x-y,t)dy
JR"-B(x°,6)
C [	1*-у|2 ,
- ^72 /	e и dy
t 1 J№-B(x0,6)
< 777/2 / e 16t dy
t 1 JR"—B(x°,5)
C	r2
= —7^ / e_i«rn_1dr —► 0	as t —>0+.
tn/2 Js
Hence if |ж — x°| < | and t > 0 is small enough, («(ж, t) — g(x°)| < 2e.	□
Remarks, (i) In view of Theorem 1 we sometimes write
( Ф< - ДФ =0	in Жп x (0, oo)
[ Ф --	on ]Rn x {t = 0},
6q denoting the Dirac measure on IRn giving unit mass to the point 0.
(ii)	Notice that if g is bounded, continuous, g > 0, g 0, then
is in fact positive for all points x G IRn and times t > 0. We interpret this
observation by saying the heat equation forces infinite propagation speed
2.3. HEAT EQUATION
49
for disturbances. If the initial temperature is nonnegative and is positive
somewhere, the temperature at any later time (no matter how small) is
everywhere positive. (We will learn in §2.4.3 that the wave equation in
contrast supports finite propagation speed for disturbances.)	□
c. Nonhomogeneous problem.
Now let us turn our attention to the nonhomogeneous initial-value prob-
lem
, .	( ut - Au = f in Rn x (0, oo)
' '	[ и = 0 on Rn x {t = 0}.
How can we produce a formula for the solution? If we recall the moti-
vation leading up to (9), we should note further that the mapping (x,t) >—>
Ф(х — у, t — s) is a solution of the heat equation (for given у E Rn, 0 < s < t).
Now for fixed s, the function
u = u(x, t;s)= / Ф(х — y,t — s)f(y, s)dy
JlRn
solves
,	.	J ut(-; s) — Au(-; s) = 0	in Rn x (s, oo)
s	[	u(-; s) = /(, s)	on Rn x {t = s},
which is just an initial-value problem of the form (8), with the starting time
t — 0 replaced by t = s, and g replaced by /(•, s). Thus u(-; s) is certainly
not a solution of (12).
However Duhamel’s principle* asserts that we can build a solution of
(12) out of the solutions of (12s), by integrating with respect to s. The idea
is to consider
u(x, t) = [ u(x,t;s)ds (x E Rn, t > 0).
Jo
Rewriting, we have
u(®,t)=/ [ Ф(х-y,t - s)f(y,s)dyds
(13)	Jo Jk"
/"*	1	f _ |д--у|2
= / 7777-------777 / e *(t } №>s) dyds’
J0 (47T(t - S))n/2 jRn
for x E Rn, t > 0.
To confirm that formula (13) works, let us for simplicity assume f E
Cf(Rn x [0, oo)) and f has compact support.
•Duhamel’s principle has wide applicability to linear ODE and PDE, and does not depend
on the specific structure of the heat equation. It yields, for example, the solution of the nonho-
mogeneous transport equation, obtained by different means in §2.1.2. We will invoke Duhamel’s
principle for the wave equation in §2.4.2.
50
2. FOUR IMPORTANT LINEAR PDE
THEOREM 2 (Solution of nonhomogeneous problem). Define и by (13).
Then
(i)	и e Cl(Rn x (0, oo)),
(ii)	щ(х, t) — Au(x, t) = f(x, t) (x € Rn, t > 0),
and
(iii)	lim u(x, t) = 0 for each point x° E Rn.
(x,t)-(xo,O)
xeIRn, t>o
Proof. 1. Since Ф has a singularity at (0,0), we cannot directly justify
differentiating under the integral sign. We instead proceed somewhat as in
the proof of Theorem 1 in §2.2.1.
First we change variables, to write
u(x,t) = [ [ $(y,s)f(x - y,t - sjdyds.
Jo JR"
As f 6 Cf(Rn x [0, oo)) has compact support and Ф = Ф(у, s) is smooth
near s = t > 0, we compute
Ut(x,f)= [ [ &(y,s)ft(x -y,t- s)dyds
Jo JR"
+ [ &(y,t)f(x — y,tydy
JR"
and
d2u	f* f	d2
~x—(x,t)= / / Ф(у,в)д—— f(x-y,t-s)dyds (i,j = l,...,n).
OXiOXj	Jo J^n	OXiOXj
Thus ut,D2u, and likewise u,Dxu, belong to C(Rn x (0, oo)).
2.	We then calculate
(14)
Ut(x,t) - Au(x,t) = / / Ф(у,в)[(^7 - Ах)/(ж - y,t - s)] dyds
Jo JR"
+ [ $(y,t)f(x — y,tydy
JR"
= [ [ Ф(у,з)[(-^~ - Ny)f(x-y,t-s)]dyds
Je JR"	os
+ [ [ Ф(з/,s)[(—- &y)f(x - y,t - s)]dyds
Jo JR"	os
+ [ $(y,t)f(x -y,0)dy.
JR"
=: Ie + Je + K.
г
2.3. HEAT EQUATION
51
Now
(15)	| J£| < (ll/tllboo + p2/||£oo) Г f $(y,s)dyds < eC,
J0 J^tn
by the lemma. Integrating by parts, we also find
Is = L	~У^~^ dyds
+ / <b(y,E)f(x-y,t-E)dy
(16)
- / $(y,t)f(x -y,ti)dy
JR"
= [ $(y,£)f(x-y,t-£)dy- K,
ж
since Ф solves the heat equation. Combining (14)-(16), we ascertain
ut(x,t) - Au(x,t) = lim / Ф(у,£)/(а: - y,t - s)dy
s—>0 JR"
= f(x, t) (x G R”, t > 0),
the limit as £ —> 0 being computed as in the proof of Theorem 1. Finally
note ||u(-,t)||£oo < t||/||£oo -> 0.	□
Remark. We can of course combine Theorems 1 and 2 to discover that
(17)	u(x,t) = [ Ф(х-y,t)y(y)dy + [ [ Ф(х — y,t - s)f(y,s)dyds
JR"	JO JR"
is, under the hypotheses on g and f as above, a solution of
.	( ut - Ди = f in R" x (0, oo)
[ и = g on R" x {t = 0}.
□
2.3.2. Mean-value formula.
First we recall some useful notation from §A.2. Assume U С R" is open
and bounded, and fix a time T > 0.
DEFINITIONS.
(i) We define the parabolic cylinder
UT := U x (0,T],
52
2. FOUR IMPORTANT LINEAR PDE
The region
(ii) The parabolic boundary of Ut is
Гт := Ut ~ Ut-
We interpret Ut as being the parabolic interior of U x [0,T]: note care-
fully that Ut includes the top U x {t = T}. The parabolic boundary Гт
comprises the bottom and vertical sides of U x [0, T], but not the top.
We want next to derive a kind of analogue to the mean-value property for
harmonic functions, as discussed in §2.2.2. There is no such simple formula.
However let us observe that for fixed x the spheres dB(x, r} are level sets of
the fundamental solution Ф(ж—у) for Laplace’s equation. This suggests that
perhaps for fixed (x, t) the level sets of fundamental solution Ф(ж — у, t — s)
for the heat equation may be relevant.
DEFINITION. For fixed x G R", t G R, r > 0, we define
E(x, t\r) := {(y, s) G Rn+1 | s < t, Ф(х — у, t — s) >	} .
This is a region in space-time, the boundary of which is a level set of
Ф(х—у, t—s). Note that the point (x, t) is at the center of the top. E(x, t; r)
is sometimes called a “heat ball”.
THEOREM 3 (A mean-value property for the heat equation). Let и G
Cj(?7t) solve the heat equation. Then
(19) u(x,t) =-^ ff	u(y,s)fe—^2dyds
4rn	{t-sY
2.3. HEAT EQUATION
53
for each E(x, t; r) C Ut-
Formula (19) is a sort of analogue for the heat equation of the mean-
value formulas for Laplace’s equation. Observe that the right hand side
involves only u(y, s) for times s < t. This is reasonable, as the value u(x, f)
should not depend upon future times.
Proof. We may as well assume upon translating the space and time coor-
dinates that x = 0 and t = 0. Write E(r) = E(0,0; r) and set
(20)
= [[	u(n/,r2s)^- dyds.
J Jew	s
We compute
ф'(г) = [[	+ 2rus— dyds
J Jew s s
=	/Jew § Uyiyi^ + 2“s V’ dyds
-.A + B.
Also, let us introduce the useful function
n	lr/12
(21)	V’ := log(-47rs) + +nlogr,
2	4s
and observe ip = 0 on dE(r), since Ф(у, — s) = r~n on dE(r). We utilize
(21) to write
r J Jew i=1
= —// 4nusip + 4 У2 изугУгФ
r J JEW	i=i
54
2. FOUR IMPORTANT LINEAR PDE
there is no boundary term since ip = 0 on ЭЕ(г). Integrating by parts with
respect to s, we discover
B = II ~inus^ + 4 V uyiynps dyds
r + J J E(r)	“
=^Я?„,г4”"*'4'+4Ь'-9- (-£  s
= 7^1 ff -Anusip - у 52 иУ^ dyds ~ A-
2\
tz I dyds
Consequently, since и solves the heat equation,
<//(r) = A + В
1
^n+1
[f л л ,	2n^
/ / -АпЬи-ф-----У uViyi dyds
JJE{r)	8
v- 1 ff .	. 2n
= > , Zn+i / / 4nuVi^yi - ~иУгУг dyds
i=i r JJe(t)	3
= 0, according to (21).
Thus ф is constant, and therefore
</>(r) = lim </>(t) = u(0,0) (lim	Ц-dyds) = 4u(0,0),
t^o	tn J JE(t) 3
as
1/7 '4w‘=ff
tn J JE(t) 3 J Jew 3
We omit the details of this last computation.	□
2.3.3. Properties of solutions.
a. Strong maximum principle, uniqueness.
First we employ the mean-value property to give a quick proof of the
strong maximum principle.
THEOREM 4 (Strong maximum principle for the heat equation). Assume
и G CI(Ut) A C(Ut) solves the heat equation in Ut-
(i) Then
max и = max u.
UT Гт
2.3. HEAT EQUATION
55
Strong maximum principle for the heat equation
(ii) Furthermore, ifU is connected and there exists a point (xo,M € Ut
such that
u(xo,to) = max ti,
UT
then
и is constant in Ut0.
Assertion (i) is the maximum principle for the heat equation and (ii)
is the strong maximum principle. Similar assertions are valid with “min”
replacing “max”.
Remark. So if и attains its maximum (or minimum) at an interior point,
then и is constant at all earlier times. This accords with our strong intuitive
interpretation of the variable t as denoting time: the solution will be constant
on the time interval [0, to] provided the initial and boundary conditions are
constant. However, the solution may change at times t > to, provided the
boundary conditions alter after to- The solution will however not respond
to changes in boundary conditions until these changes happen.
Take note that whereas all this is obvious on intuitive, physical grounds,
such insights do not constitute a proof. The task is to deduce such behavior
from the PDE.	□
Proof. 1. Suppose there exists a point (xq, to) G Ut with u(xo, to) = M :=
max^,;, u. Then for all sufficiently small r > 0, E(xo,to;r) C Ut', and we
56
2. FOUR IMPORTANT LINEAR PDE
employ the mean-value property to deduce
M = u(x0, to) = 7^ [[	u(y, s)£°—^2 dyds M'°
4rn J JE(xo,to-,r)	(to-s)2
since
i - 1 f f	lx° ~ У\2 Л J
4rn JJE(x0,t0!r) (to - s)2 У S'
Equality holds only if и is identically equal to M within E(xo,to',r}. Con-
sequently
it(y, s) = M for all (y, s) G E(xq, to; r).
Draw any line segment L in Ut connecting (xq, to) with some other point
(yo,so) £ Ut, with so < to- Consider
ro := min{s > sq | u(x, t) = M for all points (x,t) 6 L, s < t < to}.
Since и is continuous, the minimum is attained. Assume ro > sq. Then
14(20, ro) = M for some point (zq, ro) on LCiUt and so и = M on E(zq, ro', r)
for all sufficiently small r > 0. Since E(zq, ro; r) contains L П {ro — <r < t <
ro} for some small a > 0, we have a contradiction. Thus ro = $o, and hence
и = M on L.
2. Now fix any point x € U and any time 0 < t < to- There exist points
{xq,Xi,..., xm = x} such that the line segments in R" connecting Xj_i to Xj
lie in U for i = 1,..., m. (This follows since the set of points in U which can
be so connected to Xq by a polygonal path is nonempty, open and relatively
closed in U.) Select times to > ti > • • • > tm = t. Then the line segments in
Rn+1 connecting (xj_i,tj_i) to (a^ti) (г = l,...,m) lie in Ut- According
to Step 1, и = M on each such segment and so u(x, t) = M.	□
Remark. The strong maximum principle implies that if U is connected and
и G C2(f7r) П C(Ut) satisfies
ut — Ди = 0
и = 0
и = д
in Ut
on dU x [0, T]
on U x {t = 0}
where g > 0, then и is positive everywhere within Ut if g is positive some-
where on U. This is another illustration of infinite propagation speed for
disturbances.	□
An important application of the maximum principle is the following
uniqueness assertion.
2.3. HEAT EQUATION
57
THEOREM 5 (Uniqueness on bounded domains). Let g E С(Гт), f G
C'(Ur). Then there exists at most one solution и G Cf (Up) П C(Ut) of the
initial/boundary-value problem
(22)
щ — Au = f in Ut
и = g on Гр.
Proof. If и and й are two solutions of (22), apply Theorem 4 to w :=
±(li — U).
We next extend our uniqueness assertion to the Cauchy problem, that
is, the initial value problem for U — R". As we are no longer on a bounded
region, we must introduce some control on the behavior of solutions for large
M-
THEOREM 6 (Maximum principle for the Cauchy problem). Suppose
и G C?(Rn x (0,T]) П C(Rn x [0,T]) solves
(23)
ut — Au = 0 in R" x (0, T)
и = g on R" x {t = 0}
and satisfies the growth estimate
(24)
u(x,t) < Ae“lz|2
(i G r, 0 < t <T)
for constants A, a > 0. Then
sup и = sup g.
R" x [0,T] Rn
Proof. 1. First assume
(25)
4aT < 1;
in which case
(26)
4a(T + e) < 1
for some e > 0. Fix у G Rn, p > 0, and define
58
2. FOUR IMPORTANT LINEAR PDE
A direct calculation (cf. §2.3.1) shows
vt-Nv = 0 in R" x (О,Т].
Fix r > 0 and set U := B°(y,r), Ut = x (0,T], Then according to
Theorem 4,
(27)	maxv = maxv.
UT rT
2. Now if ж € Rn,
,	,	и	i*~yi2
v(x, 0) = u(x,0) —  -----——e4(T+e)
(28)	V k ’ (T + е)п/2
< u(x,0) = g(x);
and if |rr — y\ = r, 0 < t < T, then
и	r2
v(x,t) = u(x,t) — —---------——e4(T+e-t)
v v ' (T + e-t)nl2
<	Aea\x\2 - ----— е4(т+^Г) by (24)
(T + e - tyl2
<	_де“(1г/1+г)2 _ -
(T + e)n/2
Now according to (26),	= a+7 f°r some 7 > 0. Thus we may continue
the calculation above to find
(29) v(x, t) < Aea^+r^2 - p(4(a + 7))n/2e(a+^r'2 < sup g,
Rn
for r selected sufficiently large. Thus (27)-(29) imply
v(y,t) < supg
Rn
for all у G R", 0 < t < T, provided (25) is valid. Let p —> 0.
3. In the general case that (25) fails, we repeatedly apply the result
above on the time intervals [0, Ti], [Ti,27i,], etc., for T) =	□
THEOREM 7 (Uniqueness for Cauchy problem). Let g G C(R"), f G
C(Rn x [0,71]). Then there exists at most one solution и G Ci(Rn x (0,71]) П
C(Rn x [0, T]) of the initial-value problem
. .	( ut - An = f m Rn x (0, T)
( и = g on Rn x {t = 0}
satisfying the growth estimate
(31)	|u(x, t)| < Aea^2 (x G Rn, 0 < t < T)
for constants A, a > 0.
2.3. HEAT EQUATION
59
Proof. If и and й both satisfy (30), (31), we apply Theorem 6 to w
±(u — u).	□
Remark. There are in fact infinitely many solutions of
(32)
щ — Au = 0
и = 0
in R?l x (0,T)
on R” x {t = 0};
see for instance John [J, Chapter 7]. Each of the solutions besides и = 0
grows very rapidly as |x| —> oo.
There is an interesting point here: although и = 0 is certainly the “physi-
cally correct” solution of (32), this initial-value problem in fact admits other,
“nonphysical” solutions. Theorem 7 provides a criterion which excludes the
“wrong” solutions. We will encounter somewhat analogous situations in our
study of Hamilton-Jacobi equations and conservation laws, in Chapters 3,
10 and 11.	□
b. Regularity.
We next demonstrate that solutions of the heat equation are automati-
cally smooth.
THEOREM 8 (Smoothness). Suppose и G С2(Ст) solves the heat equa-
tion in Ut- Then
и G C°°(17t).
This regularity assertion is valid even if и attains nonsmooth boundary
values onfy.
Proof. 1. Recall from §A.2 that we write
C(x, t; r) = {(y, s) | |x — y\ < r, t — r2 < s <t}
to denote the closed circular cylinder of radius r, height r2, and top center
point (x, t).
Fix fyo, io) € Ut and choose r > 0 so small that C := C(xq, to; r) C Ut-
Define also the smaller cylinders C := C(a:o,to; |r), C" := C(xo,to;|r),
which have the same top center point fyo,to)-
Choose a smooth cutoff function £ = C(z,t) such that
0 < C < 1, C = 1 on C,
£ = 0 near the parabolic boundary of C.
Extend C = 0 in (Rn x [0, to]) — C.
60
2. FOUR IMPORTANT LINEAR PDE
2. Assume temporarily that и G C°°(Ur) and set
v(x, t) := ((x, t)u(x, t) (x E Rn, 0 < t < to).
Then
vt — (щ + Qu, Av = (Au + 2D(  Du + uA(.
Consequently
(33)	v = 0 onRn x {t = 0},
and
(34)	vt — Av = Qu — 2D(  Du — uA£ =: f
in R" x (0, to)- Now set
v(x,t):=[ [ Ф(х — y,t — s)f(y,s) dyds.
J0 JR"
According to Theorem 2
, ,	( vt - Av = f in Rn x (0, to)
( v = 0 on Rn x {t = 0}.
Since |v|, |v| < A for some constant A, Theorem 7 implies v = v; that is,
(36)	v(x,t) = [ [ Ф(х-y,t-s)f(y,s)dyds.
Jo JR"
Now suppose (x, t) G C". As £ = 0 off the cylinder C, (34) and (36) imply
u(x, t) = J J Ф(ж -y,t- s)[(G(y, s) - AC(y, s))u(y, s)
- 2D((y, s)  Du(y, s)] dyds.
2.3. HEAT EQUATION
61
u(x, t) = 11
(37 k ’ JJc
Note in this expression that the expression in the square brackets vanishes
in some region near the singularity of Ф. Integrate the last term by parts:
[Ф(х -y,t- s)(Cs(y, s) + N£(y, s))
+ 2£>г/Ф(ж — у, t — s) • D((y, s)]u(y, s) dyds.
We have proved this formula assuming и G C°°. If и satisfies only the
hypotheses of the theorem, we derive (37) with ue — r)£ * и replacing и,
being the standard mollifier in the variables x and t, and let £ —> 0.
3. Formula (37) has the form
(38) u(x, t) = J J K(x,t,y,s')u(y,s') dyds ((x, t) G C"),
where
K(x, t,y, s) = 0 for all points (y, s) G C",
since £ = 1 on C. Note also К is smooth on C — C. In view of expression
(38), we see и is C°° within C" = C(a:o,to; |r).	□
c. Local estimates for solutions of the heat equation.
Next we record some estimates on the derivatives of solutions to the
heat equation, paying attention to the differences between derivatives with
respect to Xj (i = 1,..., n) and with respect to t.
THEOREM 9 (Estimates on derivatives). There exists for each pair of
integers k,l = 0,1,..., a constant C^.i such that
Cki
~ rk+2l+n+2	(C(x,t;r))
for all cylinders C(x,t-,r/2) C C(x, t;r) C Ut, and all solutions и of the
heat equation in Ut-
Proof. 1. Fix some point in Ut- Upon shifting the coordinates, we may
as well assume the point is (0,0). Suppose first that the cylinder C(l) :=
C(0,0; 1) lies in Ut- Let C (|)	C(0,0;^). Then, as in the proof of
Theorem 8,
u(x,t) = Il K(x,fy,s)u(y,s)dyds ((x,t) G C(A))
62
2. FOUR IMPORTANT LINEAR PDE
for some smooth function K. Consequently
. .	l£)z£)tu(M)| < /7" \DtD*K(x,t,y,s)\\u(y,s)\dyds
(39)	J JC(i)
< C'fc/HwllL1(C(l))
for some constant Cki-
2. Now suppose the cylinder C(r) := C(0,0;r) lies in Ut- Let C(r/2) —
C(0,0;r/2). We rescale by defining
v(x,t} := u(rx,r2t}.
Then vt — Ди = 0 in the cylinder C(l). According to (39),
\DkDltv(x,t)\ < CfeZ||v||L1(c{1)) ((x,t) e C(i)).
But DkDltv(x,t) = r2l+kDkDltu(rx,r2t) and ||t>||Li(c(1)) = ^||u||L1(c(r)).
Therefore
- r2l+k+n+2 HullL1(C(r))-
'-A7/*)	1
□
Remark. If и solves the heat equation within Ut, then for each fixed time
0 < t < T, the mapping x i-> u(x, t) is analytic. (See Mikhailov [M].)
However the mapping 11—> u(x, t) is not in general analytic.	□
2.3.4. Energy methods.
a. Uniqueness.
Let us investigate again the initial/boundary-value problem
щ — Ди = f in Ut
и = g on Гт1.
(40)
We earlier invoked the maximum principle to show uniqueness, and
now—by analogy with §2.2.5—provide an alternative argument based upon
integration by parts. We assume as usual that U C R" is open, bounded
and that dU is C1. The terminal time T > 0 is given.
THEOREM 10 (Uniqueness). There exists at most one solution и
eC2(UT) of (40).
2.3. HEAT EQUATION
63
Proof. 1. If u is another solution, w := и — й solves
(41)
wt — Aw = 0
w = 0
in Ur
on Гт-
2. Set
e(t) := f w2(x,t)dx (0 < t < T).
Ju
Then
f ( d
k(t) = 2 / wwt dx I = —
Ju \ dt
— 2 I wAw dx
Ju
= —2 I ]Dw]2dx < 0,
Ju
and so
e(t) < e(0) = 0 (0 < t < T).
Consequently w = и — й = Q m Ut-
Observe that the foregoing is a time-dependent variant of the proof of
Theorem 16 in §2.2.5.
b. Backwards uniqueness.
A rather more subtle question concerns uniqueness backwards in time
for the heat equation. For this, suppose и and й are both smooth solutions
of the heat equation in Ut, with the same boundary conditions on dU:
(42)
Ut — Au = 0
u = g
in Ut
on dU x [0,T],
(43)
(Ut — Ай - 0
t й = 9
in Ut
on dU x [0,T],
for some function g. Note carefully that we are not supposing и = й at time
t = 0.
64
2. FOUR IMPORTANT LINEAR PDE
THEOREM 11 (Backwards uniqueness). Suppose u,u G C2(J7r) solve
(42), (43). If
u(x, T) = й(х, T) (x G U),
then
и = й within Ut-
In other words, if two temperature distributions on U agree at some time
T > 0, and have had the same boundary values for times 0 < t < T, then
these temperatures must have been identically equal within U at all earlier
times. This is not at all obvious.
Proof. 1. Write w := и — й and, as in the proof of Theorem 10, set
e(t) := / w2(x,t)dx (0<t<T).
Ju
As before
(44)	e(t) = —2 [ \Dw\2dx [ = — ) .
Ju	\
Furthermore
e(t) = —4 / Dw - Dwt dx
Ju
(45)	= 4 / Awwj dx
Ju
= 4 I (Aw)2dx by (41).
Ju
Now since w = 0 on dU,
I \Dw\2dx = — I wAwdx
Ju	Ju
( г \1/2 / г \1/2
— [I w2dx\ (j (Aw)2dxj
Thus (44) and (45) imply
(e(t))2 = 4 ^Dw^dxj
£ СЬЯ (4/.,дш)2dx
= e(t)e(t).
2.4. WAVE EQUATION
65
Hence
(46)	g(f)e(f) > (e(i))2 (0<t<T).
2.	Now if e(i) = 0 for all 0 < t < T, we are done. Otherwise there exists
an interval [ti,t2] C [0,T], with
(47)	e(t) > 0 for ti < t < t2, e(t2) = 0.
3.	Now write
(48)	f(t) := loge(t) (h < t < t2).
Then
and so f is convex on the interval (ti,t2). Consequently if 0 < т < 1,
ti < t < t2, we have
/((1 - Tfa + rt) < (1 - r)/(ti) + r/(t).
Recalling (48), we deduce
e((l - r)ti + rt) < e(ti)1-'re(t)'r,
and so
0 < e((l - r)ti + rt2) < e(ti)1-'re(t2)T (0 < т < 1).
But in view of (47) this inequality implies e(t) = 0 for all times ti < t < t2,
a contradiction.	□
2.4. WAVE EQUATION
In this section we investigate the wave equation
(1)	utt - Au = 0
and the nonhomogeneous wave equation
(2)	utt - A = f,
subject to appropriate initial and boundary conditions. Here t > 0 and
x G U, where U C R" is open. The unknown is и : U x [0, oo) —> R,
и = u(x, t), and the Laplacian A is taken with respect to the spatial variables
66
2. FOUR IMPORTANT LINEAR PDE
x = (xi,, xn). In (2) the function f : U x [0, oo) —> R is given. A common
abbreviation is to write
□u = utt — Au.
We shall discover that solutions of the wave equation behave quite differ-
ently than solutions of Laplace’s equation or the heat equation. For example,
these solutions are generally not C°°, exhibit finite speed of propagation, etc.
Physical interpretation. The wave equation is a simplified model for a
vibrating string (n = 1), membrane (n = 2), or elastic solid (n = 3). In
these physical interpretations u(x, t) represents the displacement in some
direction of the point x at time t > 0.
Let V represent any smooth subregion of U. The acceleration within V
is then
and the net contact force is
F-vdS,
where F denotes the force acting on V through dV and the mass density
is taken to be unity. Newton’s law asserts the mass times the acceleration
equals the net force:
This identity obtains for each subregion V and so
Utt = ~ div F.
For elastic bodies, F is a function of the displacement gradient Du; whence
utt + div F(Du) — 0.
For small Du, the linearization F(Du) & —aDu is often appropriate; and so
Utt — aNu = 0.
This is the wave equation if a = 1.
This physical interpretation strongly suggests it will be mathematically
appropriate to specify two initial conditions, on the displacement и and the
velocity ut, at time t = 0.
2.4. WAVE EQUATION
67
2.4.1. Solution by spherical means.
We began §§2.2.1 and 2.3.1 by searching for certain scaling invariant
solutions of Laplace’s equation and the heat equation. For the wave equation
however we will instead present the (reasonably) elegant method of solving
(1) first for n = 1 directly and then for n > 2 by the method of spherical
means.
a. Solution for n = 1, d’Alembert’s formula.
We first focus our attention on the initial-value problem for the one-
dimensional wave equation in all of R:
( Utt - uxx = 0 in R x (0, oo)
' '	[ и = g, Ut = h on R x {t = 0},
where g, h are given. We desire to derive a formula for и in terms of g and
h.
Let us first note the PDE in (3) can be “factored”, to read
Write
Then (4) says
Vt(x, t) + vx(x, t) = 0 (x e R, t > 0).
This is a transport equation with constant coefficients. Applying formula
(3) from §2.1.1 (with n = 1, b = 1), we find
(6)	v(x,t) = a(x — t)
for a(x) := v(x, 0). Combining now (4)-(6), we obtain
Ut(x, t) — ux(x, t) = a(x — t) in R x (0, oo).
This is a nonhomogeneous transport equation; and so formula (5) from §2.1.2
(with n = 1, b = -1, f(x,t) = a(x - t)) implies
u(x, t) = / a(x + (t — s) — s) ds + b(x + t)
(7)
= ~ a(y) dy + b(x + t),
z Jx-t
68
2. FOUR IMPORTANT LINEAR PDE
where we have f>(x) := u(x, 0).
We lastly invoke the initial conditions in (3) to compute a and b. The
first initial condition in (3) gives
b(x) = g(x) (x G R);
whereas the second initial condition and (5) imply
a(x) = v(x, 0) — Ut(x, 0) — ux(x, 0) = h(x) — g'(x) (i 6 R).
Our substituting into (7) now yields
I rx+t
u(x, t) = - h(y) - g'(y) dy + g(x + t).
z Jx-t
Hence
1	1 fx+t
(8)	u(x, t) = -[g(x +1) + g(x - t)] + - J h(y) dy (xgR, t > 0).
This is d’Alembert’s formula.
We have derived formula (8) assuming и is a (sufficiently smooth) solu-
tion of (3). We need to check that this really is a solution.
THEOREM 1 (Solution of wave equation, n = 1). Assume g G C2(R),
h G CJ(R), and define и by d’Alembert’s formula (8). Then
(i)	и G C2(Rn x [0, oo)),
(ii)	Utt — uxx = 0 in R x (0, oo),
and
(iii)	lim u(x, f) = g(x°), lim щ(х, f) — hlx0)
»(x°,0)	(x,t)—>(xo,0)
t>0	i>0
for each point x° G R.
The proof is a straightforward calculation.
Remarks, (i) In view of (8), our solution и has the form
u(x, f) = F(x +1) + G(x — f)
for appropriate functions F and G. Conversely any function of this form
solves Utt — uxx = 0. Hence the general solution of the one-dimensional wave
equation is a sum of the general solution of ut — ux = 0 and the general
solution of ut + ux = 0. This is a consequence of the factorization (4).
2.4. WAVE EQUATION
69
(ii) We see from (8) that if g G Ck and h G C'fe~1, then и G Ck, but is not
in general smoother. Thus the wave equation does not cause instantaneous
smoothing of the initial data, as does the heat equation.	□
A reflection method. To illustrate a further application of d’Alembert’s
formula, let us next consider this initial/boundary-value problem on the
half-line R+ = {x > 0}:
(9)
Utt UXx — 0
u = g, Ut = h
и = 0
in R+ x (0, oo)
on x {t = 0}
on {x = 0} x (0, oo),
where g, h are given, with <?(0) = h(0) = 0.
We convert (9) into the form (3) by extending u, g, h to all of К by odd
reflection. That is, we set
й{х, t) := <	S' S' Al Al S S Al VI *» 1 's' 3 1 \	 	/
P(®) :=	f g(x)	(x > 0) I -g(-x) & < o),
h(x) :=	( h(x)	(x > 0) ( — h(—x) (x < 0).
( ^tt — 'U'XX
( й = g, v,t = h
Then (9) becomes
in R x (0, oo)
onRx {t = 0}.
Hence d’Alembert’s formula (8) implies
1	1 fx+t ~
й(х, t) = -	+ t) + g(x - t)] + - / h(y) dy.
4	1 Jx-t
Recalling the definitions of й, g, h above, we can transform this expression
to read for x > 0, t > 0:
(10)
u(x, t) — <
I [5(x + t) + g(x - t)] + 1	h(y) dy
1[5(ж + t) - g(t - я)] + I f^t h(^ dy
if x > t > 0
if 0 < x < t.
If h = 0, we can understand formula (10) as saying that an initial dis-
placement g splits into two parts, one moving to the right with speed one
and the other to the left with speed one. The latter then reflects off the
point x = 0, where the vibrating string is held fixed.	□
70
2. FOUR IMPORTANT LINEAR PDE
b. Spherical means.
Now suppose n > 2, m > 2, and и G Cm(Rn x [0, oo)) solves the initial-
value problem
(11)
J Utt —	= 0 in R" x (0, oo)
I и = g, Ut = h on Rn x {t = 0}.
We intend to derive an explicit formula for и in terms of g, h. The plan
will be to study first the average of и over certain spheres. These averages,
taken as functions of the time t and the radius r, turn out to solve the
Euler-Poisson-Darboux equation, a PDE which we can for odd n convert
into the ordinary one-dimensional wave equation. Applying d’Alembert’s
formula, or more precisely its variant (10), eventually leads us to a formula
for the solution.
Notation, (i) Let x G Rn, t > 0, r > 0. Define
(12)	U(x;r,f) := + u(y,t)dS(y),
J дВ{х,т)
the average of u(-, t) over the sphere dB(x, r).
(ii) Similarly,
(13)
G(x-,r):=f	g(y)dS(y)
J dB(x,r)
H(x-,r) := /	h(y)dS(y).
.	J 9B(x,r)
For fixed x, we hereafter regard U as a function of r and t, and discover
a partial differential equation U solves:
LEMMA 1 (Euler-Poisson-Darboux equation). Fix x G R", and let и
satisfy (11). Then U G Cm(R+ x [0,oo)) and
(Utt- Urr ~ ^Ur = 0 in R+ x (0, oo)
(14'	t U = G, Ut = H on»Lx{t = 0}.
The partial differential equation in (14) is the Euler-Poisson-Darboux
equation. (Note that the term Urr + ~±Ur is the radial part of the Laplacian
Д in polar coordinates.)
2.4. WAVE EQUATION
71
Proof. 1. As in the proof of Theorem 2 in §2.2.2 we compute for r > 0
(15)	Ur(x-, r,t) = --f	&u(y, t) dy.
nJ B(x,r)
From this equality we deduce limr_>o+ Ur(x;r, t) = 0. We next differentiate
(15), to discover after some computations that
(16) Urr(x-,r, t) = + NudS + ( — — 1 ] + Nudy.
J ЭВ(х,г)	\П / J B(x,r)
Thus limr^(jT Urr(x;r, t) = ^Au(a?,t). Using formula (16) we can similarly
compute Urrr, etc., and so verify that U 6 C'm(R+ x [0, <x>)).
2. Continuing the calculation above, we see from (15) that
Ur = -+	uttdy by (11)
nJ B(x,r)
1	1 f
= —/ Uttdy-
na(n) rn 1
Thus
rn-1Ur = —у-r [ uttdy,
na(n)
and so
(r^Ur), = —[ uttdS
na(n) JdB(x,r)
= rn~1-[ uttdS = rn-lUtt.
J dB(x,r)
□
c. Solution for n = 3,2, Kirchhoff’s and Poisson’s formulas.
The overall plan in the ensuing subsections will be to transform the
Euler-Poisson-Darboux equation (14) into the usual one-dimensional wave
equation. As the full procedure is rather complicated, we pause here to
handle the simpler cases n = 3,2, in that order.
Solution for n = 3. Let us therefore hereafter take n = 3, and suppose
и 6 <72(R3 x [0, oo)) solves the initial-value problem (11). We recall the
definitions (12), (13) of U,G,H, and then set
(17)
U := rU,
72
2. FOUR IMPORTANT LINEAR PDE
(18)
G := rG, H := rH.
We now assert that U solves
(19)
Utt — Urr	=	0	in R+ x (0, oo)
<	U = G, Ut	=	H	on IR+ x {t = 0}
U	=	0	on {r = 0} x (0, oo).
Indeed
Utt = rUtt
= r
2
U^tt “h Uj
r
by (14), with n = 3
= rUrr + 2l7r = (U + rUr)r
— U^T.
Applying formula (10) to (19), we find for 0 < r < t
1 _	_	i гг+* _
(20)	U(x-,r,t) = ^[G(r + t)-G(t-r)] + -J H(y)dy.
Since (12) implies u(x,t) = limr_>0+ U(x-,r, t), we conclude from (17), (18),
(20) that
.	.	U(x-,r,t)
u(x,t) = hm ----------
r—»o+ r
\G^t + r)-G(t-r) , 1 [t+r T~Tt 4 J '
= Л [-----------2r---------+ Tr L Л d\
Owing then to (13), we deduce
(21)	u(x,t) =	\t-l- gdS]+t-l- hdS.
ot \ J dB(x,t) ) J 9B(x,t)
But
-[ g(y) ds(y) =-[ g(x + tz)dS(z)-,
J dB(x,t)	J ав(о,1)
and so
^7 I / gdS] = -[ Dg(x + tz)  z dS(z)
ut \J dB(x,t) ] J dB(0,l)
= 1 DM  (^) dS(y).
J 9B(x,t)	\ t /
2.4. WAVE EQUATION
73
Returning to (21), we therefore conclude
(22)	u(z,t) = /	th(y)+g(y)+Dg(y)-(y-x)dS(y) (x € R3, t > 0).
J dB(x,t)
This is Kirchhoff’s formula for the solution of the initial-value problem (11)
in three dimensions.
Solution for n = 2. No transformation like (17) works to convert the
Euler-Poisson-Darboux equation into the one-dimensional wave equation
when n = 2. Instead we will take the initial-value problem (11) for n = 2
and simply regard it as a problem for n = 3, in which the third spatial
variable x3 does not appear.
Indeed, assuming и G C*2(R2 x [0, oo)) solves (11) for n = 2, let us write
(23)
u(a?i,X2,X3,t) := u(xi,a?2,t).
Then (11) implies
(24)
utt - Au = 0
й = g, ut = h
in R3 x (0, oo)
on R3 x {t — 0},
for
g(xi,X2,x3) := g(xi,X2), h(xi,X2,x3) := h(xi,X2)-
If we write x = (xi,X2) G R2 and x = (xi,a?2,0) G R3, then (24) and
Kirchhoff’s formula (in the form (21)) imply
(25)
u(a?,t) — u(x,t)
=	| t -/	gds] +1 [ hdS,
vt \ J	J J 9B(x,t)
where B(x,t) denotes the ball in R3 with center x, radius t > 0, and dS
denotes two-dimensional surface measure on dB(x,t). We simplify (25) by
observing
/	gdS=—~—^[	gdS
J dB(x,t) *7ГГ JQB(x,t)
=	[	^(y)(l + l-E>7(y)|2)1/2dy,
47rt2
where 7(1/) = (t2 — \y — ж)2)1/2 for у € B(x,t). The factor “2” enters
since dB(x,t) consists of two hemispheres. Observe that (1 + ID7I2)1/2 =
74
2. FOUR IMPORTANT LINEAR PDE
t(t2 — \y — x|2) x/2. Therefore
/ gdS =
J dB(x,t)
g(y) ,
— \y — a?!2)1/2
s(y)

Consequently formula (25) becomes
z .	19
«(«.<) = 2ft
(26)
ft2/
\ J (f2 - |y - z|2)1/2
+ 2/s(;c,()(t2-|y-x|2)i/2dy-
But
(21	______Ш_______dV = tI S(X + IZ)
7b(i,o(«2-I»-^I2)1/2 J B(o,n(i-M2)1/2
and so
u x2 I ___________У\У)_____ . 1
/B(l,t)(t2-|y-x|2)l/2 У)
= f 9(x + tz) f Dg(x + tz)-z
J В(О,1)(1-И)1/2	/в(0,1) (1-Н2)1/2
= t I	, x f Da(y) -(y-x) ,
J B(X,t) (t2 - \y - XI2)I/2 dy + 7B(1|i) (t2 - \y - XI2) 1/2 ay-
Hence we can rewrite (26) and obtain the relation
(97}	D-1 -I t9^ + t2fl^ + tD9^ -(У-^И
(27) u(x,<) - -y------------------------------------------“У
for x € R2, t > 0. This is Poisson’s formula for the solution of the initial-
value problem (11) in two dimensions.
This trick of solving the problem for n = 3 first and then dropping to
n = 2 is the method of descent.
d. Solution for odd n.
In this subsection we solve the Euler-Poisson-Darboux PDE for odd
n > 3. We first record some technical facts.
2.4. WAVE EQUATION
75
LEMMA 2 (Some useful identities). Let ф : R —* R be Ck+1. Then for
к = 1,2,...:
(0 (£)	Ш
(ii)	(r2fc-^(r)) =
where the constants 3k(.j = 0,..., к — 1) are independent of ф.
Furthermore,
(iii)	/$ = 1-3-5.(2k	— 1).
The proof by induction is left as an exercise.
Now assume
n > 3 is an odd integer
and set
n = 2k + 1 (k > 1).
Henceforth suppose и G Cfc+1(Rn X [0, oo)) solves the initial-value prob-
lem (11). Then the function U defined by (12) is (7fc+1.
Notation. We write
(28)
' U(r,t) := (^)k X(r2k ^(x-r,^)
< G(r) := (if )fc“1 (r2fc-1G(x;r))	(r > 0, t > 0)
- W) := (^^(r^H&r)).
Then
(29)	U(r, 0) = G(r), Ut(r, 0) = Я(г).
Next we combine Lemma 1 and the identities provided by Lemma 2 to
demonstrate that the transformation (28) of U into U in effect converts the
Euler-Poisson-Darboux equation into the wave equation.
LEMMA 3 (U solves the one-dimensional wave equation). We have
Utt — Urr = 0 in R+ x (0, oo)
< U = G, Ut = H on R+ x {t = 0}
U — 0 on {r = 0} x (0, oo).
76
2. FOUR IMPORTANT LINEAR PDE
Proof. If r > 0,

к— 1
(r2fc-xU)
к
(r2kUr) by Lemma 2,(i)
fc-i
2k 1Urr + 2kr2k 2
1
r dr J
1 л \ i
r dr )
r2fc-l
n — 1
r
(n = 2k + 1)
the next-to-last equality holding according to (14). Using Lemma 2,(ii) we
conclude as well that U = 0 on {r = 0}.	□
In view of Lemma 3, (29), and formula (10), we conclude for 0 < r < t
that
(30)
U(r,t) = |[G(r + t)-G(t-
£
1 rt+r
r)] + 2jt_ &{y}dy
for all r e R, t > 0. But recall u(x,t) = Итг_>о^(я;;лО- Furthermore
Lemma 2,(ii) asserts
U(r,t) =	(r2fc-1U(x; r,t))
\r dr/
fc—1
= 52/?^+1^-l/(x;r,t);
j=o
and so
lim = lim U(x; r, t) = u(x, t).
r^o $r r-o v	v '
Thus (30) implies
u(x, t) = —г lim
4 --»
6(t+r)-6(t-r)l
2r	2r Jt_r
= ^IG'(i) + A(t)].
Po
2.4. WAVE EQUATION
77
Finally then, since n = 2k + 1, (30) and Lemma 2,(iii) yield this repre-
sentation formula:
. ч 1
U(X,t) = —
7n
(31)
) \tdtj у J dB(x,t) J
n — 3 /	\ “I
2	hds\
tat; у	j dB(x,t) )
k where n is odd and 7n = 1 • 3 • 5....(n — 2),
for x G Rn, t > 0.
We note that 73 = 1, and so (31) agrees for n — 3 with (21) and thus
with Kirchhoff’s formula (22).
It remains to check that formula (31) really provides a solution of (11).
THEOREM 2 (Solution of wave equation in odd dimensions). Assume n
is an odd integer, n > 3, and suppose also g G C"n+1(Rn), h G C'm(Rn), for
m = Define и by (31). Then
(i) uGC2(Rnx [0,00)),
(ii) utt — Au = 0 m Rn x (0, oo),
and
(iii)
lim u(x,t) = g(x°), lim ut(x,t) = h(x°)
(z,Q—>(x°,0)	»(x°,0)
xefr1, t>o	xeR", t>o
for each point x° g Rn.
Proof. 1. Suppose first g = 0; so that
n-3
= 2-(1 * ) 2	.
*Уп \	/
Then Lemma 2,(i) lets us compute
(32)
4 (Is)2	
From the calculation in the proof of Theorem 2 in §2.2.2, we see as well that
Ht =	Ah dy.
'	nJ B(x,t)
Consequently
1
^tt / \ ’
na(n)yn
1
na(n)7n
n—3
78
2. FOUR IMPORTANT LINEAR PDE
On the other hand,
t) - Дж -Z h(x + y)dS(y)
J
= / AhdS.
J dB(x,t)
Consequently (32) and the calculations above imply иц = &u in Rn x (0, oo).
A similar computation works if h = 0.
2. We leave it as an exercise to confirm, using Lemma 2,(ii)-(iii), that и
takes on the correct initial conditions.	□
Remarks, (i) Notice that to compute u(x, t) we need only have information
on g,h and their derivatives on the sphere dB(x,t), and not on the entire
ball B(x,t).
(ii)	Comparing formula (31) with d’Alembert’s formula (8) (n = 1), we
observe that the latter does not involve the derivatives of g. This suggests
that for n > 1, a solution of the wave equation (11) need not for times t > 0
be as smooth as its initial value g: irregularities in g may focus at times
t > 0, thereby causing и to be less regular. (We will see later in §2.4.3 that
the “energy norm” of и does not deteriorate for t > 0.)
(iii)	Once again (as in the case n = 1) we see the phenomenon of finite
propagation speed of the initial disturbance.
(iv)	A completely different derivation of formula (31) (using the heat
equation!) is in §4.3.2.	□
e. Solution for even n.
Assume now
n is an even integer.
Suppose и is a Cm solution of (11), m = We want to fashion a rep-
resentation formula like (31) for u. The trick, as above for n = 2, is to
note
(33)
u(a?i,... ,a?n+i,t) := -u(a?i,... ,xn,t)
solves the wave equation in Rn+1 x (0, oo), with the initial conditions
й = g, щ = h on Rn+1 x {t = 0},
2.4. WAVE EQUATION
79
where
(34)
p(a?i,... ,zn+i) := g(xi, ...,zn)
h(zj,...,xn_|-i) := /i(zi,...,xn^.
As n + 1 is odd, we may employ (31) (with n + 1 replacing n) to secure
a representation formula for й in terms of g, h. But then (33) and (34) yield
at once a formula for и in terms of g, h. This is again the method of descent.
To carry out the details, let us fix x € Rn, t > 0, and write x =
(zi,... , xn, 0) € R"+1. Then (31), with n + 1 replacing n, gives
u(z,f) = —б”1/
7n+i cnytat/ \ J dB(x,t) /
(35)	L ,	'
+ (|D)2 /us) ,
\totj у j дв(х,е> J
B(x,t) denoting the ball in Rn+1 with center x and radius t, and dS n-
dimensional surface measure on dB(x,t). Now
(36)	/ gdS = -,-----------—Д------— f gdS.
J dB(x,t) (n + l)a(n + l)£n JdB(x,t)
Note that dB(x,t) П {yn+i > 0} is the graph of the function 7(7/) :=
(t2 — \y — яр)1/2 for у e B(x,t) c R". Likewise dB(x,t) П {yn+i < 0}
is the graph of —7. Thus (36) implies
(37)	-[ gdS = , J [	6?(y)(l + |£>7(у)|2)1/2<*У,
J dB(x,t) (n + l)a(n + l)t JB(x,t)
the factor “2” entering because dB(x, t) comprises two hemispheres. Note
that (1 + |Z?7(t/)|2)1/2 = t(t2 — \y — z|2)-1/2. Our substituting this into (37)
yields
(n + l)a(n + l)tn 1	(t2 - \y - zl2)1/2 dy
2ta(n) [ g(y) d
(n + l)a(n + 1) J В(Х^ (t2 - \y - zj2)1/2
We insert this formula and the similar one with h in place of g into (37),
and find
u(z,t) =
1	2«(^)	1 (tn f g(y)
7n+i (n + l)a(n + 1) dt\tdt) JB(x,t) (t2 - |y - z|2)i/2
+	1A f	Ky) '
+ \tdt) Г /B(it)(f2_|y_xl2)i/2dy
80
2. FOUR IMPORTANT LINEAR PDE
Since 7n+i = 1 • 3 • 5 • • • (n — 1) and a(n) =	we may compute
7n = 2  4 • •  (n - 2) • n.
Hence the resulting representation formula for even n is:
(38)
/ x 1
U(X,t) = —
7n
2£V2 2	f
dt)\tdt) JB^t)^-\y-x\^/2
>	n-2 /	\
run cl _______h-^__av
where n is even and -fn = 2  4    (n — 2) • n,
for x e Rn, t > 0.
Since 72 = 2, this agrees with Poisson’s formula (27) if n = 2.
THEOREM 3 (Solution of wave equation in even dimensions). Assume n
is an even integer, n >2, and suppose also g e Cm+1(Rn), h e C'm(Rn),
for m =	. Define и by (38). Then
(i)	и G C2(R" x [0, oo)),
(ii)	иц — Ди = 0 in Rn x (0, oo),
and
(iii)	lim u(x,t) = g(x°), lim uAx,t) = h(x°)
(z,t)—»(z°,0)	(z,t)—»(x°,0)
zeR”, t>o	zeR", t>o
for each point x° G Rn.
This follows from Theorem 2.
Remarks, (i) Observe, in contrast to formula (31), that to compute u(x, t)
for even n we need information on и = g, Ut = h on all of B(x, t), and not
just on dB(x, t).
(ii) Comparing (31) and (38) we observe that if n is odd and n > 3,
the data g and h at a given point x € R" affect the solution и only on
the boundary {(y,t) | t > 0, |a? — y| = t} of the cone C = {(y,t) I I > 0,
— y\ < t}. On the other hand, if n is even the data g and h affect и within
all of C. In other words, a “disturbance” originating at x propagates along
a sharp wavefront in odd dimensions, but in even dimensions continues to
have effects even after the leading edge of the wavefront passes. This is
Huygens ’ principle.	□
2A. WAVE EQUATION
81
2.4.2. Nonhomogeneous problem.
We next investigate the initial-value problem for the nonhomogeneous
wave equation
( uu — Au = f in Rn x (0, oo)
( )	t и = 0, Ut = 0 on Rn x {t = 0}.
Motivated by Duhamel’s principle (introduced earlier in §2.3.1), we define
и = u(x, t\ s) to be the solution of
z ч f utt(-; s) - Au(-; s) = 0 in R" x ($, oo)
3	[u(-;s) = 0, Ut(-;s) =	on Rn x {t = $}.
Now set
(41)	u(x,t) := [ u(x,t',s)ds (x E Rn, t > 0).
Jo
Duhamel’s principle asserts this is a solution of
f utt - Au = f in x (0, oo)
' '	| и = 0, ut = 0 on Rn x {t = 0}.
THEOREM 4 (Solution of nonhomogeneous wave equation). Assume n >
2 and f e cIn/2l+1(IRn x [0, oo)). Define и by (41). Then
(i)	и E C2(Rn x [0, oo)),
(ii)	utt —	= f in Rn x (0, oo),
and
(iii)	lim u(x,t) = 0, lim uAx, t) = 0 for each point x° E Rn.
(z,t)—*(z°,0)	(z,t)—>(z°,0)
zeR", t>o	zelR", t>o
Proof. 1. If n is odd, ^] + 1 — ^2^- According to Theorem 2 u(-, •; s) e
(72(R" x [0, oo)) for each s > 0, and so и € C2(Rn x [0, oo)). If n is even,
+ 1 = Hence и E C2(Rn x [0, oo)), according to Theorem 3.
2. We then compute:
ut(x, t) = u(x,t;t) + / щ(х, s) ds = I ut(x,t; s) ds,
Jo	Jo
utt(x,t) = ut(x,t;t) + / utt(x,t; s) ds
Jo
= f(x,t)+ / utt(x,t-s)ds.
Jo
82
2. FOUR IMPORTANT LINEAR PDE
Furthermore
Au(x,t) = / Au(x,t; s') ds = / иц(х,Т, s) ds.
Jo	Jo
Thus
иц(х, t) — Ди(х, t) = f(x, t) (x e Rn, t > 0),
and clearly u(x, 0) = Ut(x, 0) = 0 for x e R".	□
Remark. The solution of the general nonhomogeneous problem is conse-
quently the sum of the solution of (11) (given by formulas (8), (31) or (38))
and the solution of (42) (given by (41)).	□
Examples, (i) Let us work out explicitly how to solve (42) for n = 1. In
this case d’Alembert’s formula (8) gives
I rx+t—s	| rt rx+t—s
u(x,t-,s) = - f(y,s)dy,u(x,t) = -l / f(y,s)dyds.
“ Jx—t+s	“ JO Jx—t+s
That is,
I rt rx+s
(43) u(x,t) = - / / f(y,t — s)dyds (x € R, t > 0).
2 Jo Jx-s
(ii)	For n = 3, Kirchhoff’s formula (22) implies
u(a?,t;s) = so that u(x,t) — [ (i Jo 4% J 4% J Therefore (44)	u(x,t) = -^- [ 47r J B(x,t)	--(t-s)-f	f(y,s)dS-, J 9B(x,t—s) ~ s') \ -[	f(y, s)dS ) ds 9B(x,t—s)	J П 0 J 9B(x,t—s) \J s) r f f 1		abar. (ieR3 0) |y-z|
solves (42) for n = 3. The integrand on the right is called a retarded potential.
□
2.4. WAVE EQUATION
83
2.4.3. Energy methods.
The explicit formulas (31) and (38) demonstrate the necessity of making
more and more smoothness assumptions upon the data g and h to ensure
the existence of a C2 solution of the wave equation for larger and larger
n. This suggests that perhaps some other way of measuring the size and
smoothness of functions may be more appropriate. Indeed we will see in this
section that the wave equation is nicely behaved (for all n) with respect to
certain integral “energy” norms.
a.	Uniqueness.
Let U C Rn be a bounded, open set with a smooth boundary dU, and
as usual set Ut = Ux (О, T], Гт = Ut — Ut, where T > 0.
We are interested in the initial/boundary-value problem
(45)
' Utt - Ди = f
’	и = g
ut = h
in Ut
on Гт
on U x {t = 0}.
THEOREM 5 (Uniqueness for wave equation). There exists at most one
function и G C2(Ut) solving (45).
Proof. If й is another such solution, then w и — й solves
' Wtt — Aw = 0
< w = 0
wt = 0
in Ut
on Гт
on U x {t = 0}.
Define the “energy”
e(t) [ w2(x,t) + \Dw(x,tyfdx (0 < f < T).
2 Ju
We compute
e(t) = / WtWtt + Dw  Dwt dx
Ju
= / Wt(wtt — &w)dx = 0.
Ju
dt)
There is no boundary term since w = 0 , and hence Wt = 0, on dU x [0,T],
Thus for all 0 < t < T, e(t) = e(0) = 0, and so wt, Dw = 0 within Ut- Since
w = 0 on U x {t — 0}, we conclude w — и — й = 0 in Ut-	□
84
2. FOUR IMPORTANT LINEAR PDE
Cone of dependence
b.	Domain of dependence.
As another illustration of energy methods, let us examine again the
domain of dependence of solutions to the wave equation in all of space. For
this, suppose и G C2 solves
utt — A^u = 0 in Rn x (0, oo).
Fix a?o € Rn, t0 > 0 and consider the cone
C = {(a?,t) | 0 < t < to, |z — a?o| < to - t}.
THEOREM 6 (Finite propagation speed). If и = Ut = 0 on B{xo,to'),
then и = 0 within the cone C.
In particular, we see that any “disturbance” originating outside B(xq, to)
has no effect on the solution within C, and consequently has finite propaga-
tion speed. We already know this from the representation formulas (31) and
(38), at least assuming g = и and h = щ on Rn x {t = 0} are sufficiently
smooth. The point is that energy methods provide a much simpler proof.
Proof. Define
e(t) := - [	u2(x,t) + \Du(x,t)\2dx (0 < t < to).
2 JB(xo,to-t)
2.5. PROBLEMS
85
Then
(46)
I Uf + |Z>u|2dS
9B(x0,t0-t)
e(t) = / ututt + Du - Du
JB(x0,t0-t)
= / Ut(utt - Au)dx
J Bfxojo-t')
+ [ <^UtdS-^l u2 + \Du\2dS
JdB(xo,to-t)	JdB(xo,to—t)
[	du	In 1,	12
= I	gj* ~ 2«< - 2|Du| dS-
dB(x0,t0-t)
Now
(47)
du
diul
< l«.l|D«l < |«? + ||O«|2,
by the Cauchy-Schwarz and Cauchy inequalities (§B.2). Inserting (47) into
(46), we find e(t) < 0; and so e(t) < e(0) = 0 for all 0 < t < to- Thus Ut,
Du = 0, and consequently и = 0 within the cone C.	□
A generalization of this proof to more complicated geometry appears
later, in §7.2.4.
2.5.	PROBLEMS
In the following exercises, all given functions are assumed smooth, unless
otherwise stated.
1.	Write down an explicit formula for a function и solving the initial-
value problem
( ut + b • Du + си = 0 in Rn x (0, oo)
[	и = g on Rn x {t = 0}.
Here c 6 IR and b G Rn are constants.
2.	Prove that Laplace’s equation Au = 0 is rotation invariant; that is, if
О is an orthogonal n x n matrix and we define
v(x) := u(Qx) (x e Rn),
then Av = 0.
3.	Modify the proof of the mean value formulas to show for n > 3 that
u(0) = -f	gdS-]—-------r—— [	(7—:—x-------n I fdx,
Tав(о,г)	n(n - 2)a(n) /в(0,г) \ИП 2 rn 2 J J
86
2. FOUR IMPORTANT LINEAR PDE
provided
-Au = f
u = g
in 5°(0,r)
on 35(0, r).
4.	We say v G C2(U) is subharmonic if
—Au < 0
in U.
(a)	Prove for subharmonic v that
v(z) < + v dy for all B(x,r) C U.
J B(x,r)
(b)	Prove that therefore maxp v = max.du v.
(c)	Let ф : R —» R be smooth and convex. Assume и is harmonic
and v := </>(u). Prove v is subharmonic.
(d)	Prove v := |5u|2 is subharmonic, whenever и is harmonic.
5.	Prove that there exists a constant C, depending only on n, such that
max |u| < C( max lol + max 1/1)
B(o,i) ав(од) b(o,i)
whenever и is a smooth solution of
(-Au = / in 5°(0,1)
( и = g on 35(0,1).
6.	Use Poisson’s formula for the ball to prove
rn-2 Г-|Ж|	2 Г+И
Г (r + \x|)n-i- Г (r - |x|)-i W
whenever и is positive and harmonic in 5°(0, r). This is an explicit
form of Harnack’s inequality.
7.	Prove Theorem 15 in §2.2.4. (Hint: Since и = 1 solves (44) for g = 1,
the theory automatically implies
[ K(x,y)dS(y) = 1
for each x G 5°(0,1).)
8.	Let и be the solution of
Г Au = 0 in R”
( и = g on 3R”
2.5. PROBLEMS
87
given by Poisson’s formula for the half-space. Assume g is bounded
and g(x) = |z| for x G <9R”, |z| < 1. Show Du is not bounded near
x = 0. (Hint: Estimate ц(Ле"Р~ц(°) )
9.	Let U+ denote the open half-ball {x G Rn | |ж| < 1, xn > 0}. Assume
и G C(U+~) is harmonic in U+, with и = 0 on dU+ П {xn = 0}. Set
f u(x)	if xn > 0
v(x) := <
t --u(xi,...,zn_i,-zn) if xn < 0
for x G U = 5°(0,1). Prove v is harmonic in U.
10.	Suppose и is smooth and solves щ — Au = 0 in Rn x (0, oo).
(i)	Show u\(x,t) := u(Ax,A2t) also solves the heat equation for
each A £ R.
(ii)	Use (i) to show v(x, t) := x  Du(x, t) + 2tut(x, t) solves the heat
equation as well.
11.	Assume n = 1 and u(a?,t) =
(a)	Show
= 'U'XX
if and only if
(*)	4zv"(z) + (2 + z)v'(z) = 0 (z > 0).
(b)	Show that the general solution of (*) is
v(z) = c [ e~3^is~1^2ds + d.
Jo
(c)	Differentiate v (qr) with respect to x and select the constant c
properly, so as to obtain the fundamental solution Ф for n = 1.
12.	Write down an explicit formula for a solution of
( щ — Au + cu = f in R" x (0, oc)
[	и — g on Rn x {t — 0},
where c G R.
13.	Given g : [0, oo) —> R, with g(0) — 0, derive the formula
x ft 1	— x2
u(x,t) = -=/ -----------— e^^g(s)ds
V4tf Jo (t - sp/2
88
2. FOUR IMPORTANT LINEAR PDE
for a solution of the initial/boundary-value problem
Uf Uxx — 0
< и — 0
и = g
in R+ x (0, oo)
on R+ x {t = 0},
on {x = 0} x [0, oo).
(Hint: Let v(x,t) := u(x,t) — g(t) and extend v to {x < 0} by odd
reflection.)
14.	We say v G C2(Ut) is a subsolution of the heat equation if
Vt — Au <0 in Ut-
(a)	Prove for a subsolution v that
u(x,t) < ft	v(y, s)y|—dyds
^rn JJE(x,t-T) U - «I
for all E(x, t\ r) C Ut-
(b)	Prove that therefore max^ v = maxrT v.
(c)	Let ф : R —» R be smooth and convex. Assume и solves the heat
equation and v := ф(и). Prove v is a subsolution.
(d)	Prove v := |Eu|2 + u2 is a subsolution, whenever и solves the
heat equation.
15.	(a) Show the general solution of the PDE uxy = 0 is
= F(a?) + G(y)
for arbitrary functions F, G.
(b)	Using the change of variables £ = x + t, g = x — t, show
utt — uxx = 0 if and only if = 0.
(c)	Use (a) and (b) to rederive d’Alembert’s formula.
16.	Assume E = (E1, E2, E3) and В = (В1, В2, В3) solve Maxwell’s equa-
tions (§1.2.2). Show
utt - Au = 0
where и = Ег or В1 (г = 1,2,3).
17.	(Equipartition of energy). Let и G C2(R x [0, oo)) solve the initial-
value problem for the wave equation in one dimension:
utt — uxx = 0 in R x (0, oo)
и = g,ut = h on R x {t = 0}.
2.6. REFERENCES
89
Suppose g, h have compact support. The kinetic energy is k(t) :=
| Uf(x, t) dx and the potential energy is p(t) := |	и£(х, t) dx.
Prove
(i)	k(t) + p(t) is constant in t,
(ii)	k(t) = p(t) for all large enough times t.
18.	Let и solve	,	,
( utt — An = 0 in RJ x (0, oo)
[ и = g, Ut = h on R3 x {i = 0},
where g, h are smooth and have compact support. Show there exists
a constant C such that
|u(x,t)| < C/t (x e R3, t > 0).
2.6.	REFERENCES
Section 2.2	A good source for more on Laplace’s and Poisson’s equations is Gilbarg-Trudinger [G-T, Chapters 2-4]. The proof of an- alyticity is from Mikhailov [М]. J. Cooper helped me with Green’s functions.
Section 2.3	See John [J, Chapter 7] or Friedman [FR2] for further infor- mation concerning the heat equation. Theorem 3 is due to N. Watson [W], as is the proof of Theorem 4. Theorem 6 is taken from John [J], and Theorem 8 follows Mikhailov [М]. Theorem 11 is from Payne [PA, §2.3].
Section 2.4	See Antman [A] for a careful derivation of the one-dimension- al wave equation as a model for a vibrating string. The solution of the wave equation presented here follows Folland [Fl], Strauss [ST].
Section 2.5	J. Goldstein suggested Problem 17.
Chapter 3
NONLINEAR
FIRST-ORDER PDE
3.1	Complete integrals, envelopes
3.2	Characteristics
3.3	Introduction to Hamilton-Jacobi equations
3.4	Introduction to conservation laws
3.5	Problems
3.6	References
In this chapter we investigate general nonlinear first-order partial differ-
ential equations of the form
F{Du,u, x) = 0,
where x G U and U is an open subset of Rn. Here
F:RnxRxC?^R
is given, and и : U R is the unknown, и = u(x).
Notation. Let us write
F = F(p,z,x) = F(pi,... ,pn,z,X!,... ,xn)
for p G Rn, z € R, x G U. Thus “p” is the name of the variable for which
we substitute the gradient Du(x), and “г” is the variable for which we
substitute u(x). We also assume hereafter that F is smooth, and set
DpF = (FPl,..., FPn)
’ dzf = fz
< DXF = (FX1,... ,FXn).
□
91
92
3. NONLINEAR FIRST-ORDER PDE
We are concerned with discovering solutions и of the PDE F(Du, u, x~) =
0 in U, usually subject to the boundary condition
и = g on Г,
where Г is some given subset of dU and g : Г —* R is prescribed.
Nonlinear first-order partial differential equations arise in a variety of
physical theories, primarily in dynamics (to generate canonical transforma-
tions), continuum mechanics (to record conservation of mass, momentum,
energy, etc.) and optics (to describe wavefronts). Although the strong
nonlinearity generally precludes our deriving any simple formulas for so-
lutions, we can, remarkably, often employ calculus to glean fairly detailed
information about solutions. Such techniques, discussed in §§3.1 and 3.2,
are typically only local. In §§3.3 and 3.4 we will for the important cases
of Hamilton-Jacobi equations and conservation laws derive certain global
representation formulas for appropriately defined weak solutions.
3.1.	COMPLETE INTEGRALS, ENVELOPES
3.1.1.	Complete integrals.
We begin our analysis of the nonlinear first-order PDE
(1)	F(Du, и, x) = 0
by describing some simple classes of solutions and then learning how to build
from them more complicated solutions.
Suppose first A c Rn is an open set. Assume for each parameter a =
(ai,..., an) G A we have a C2 solution и = u(x-,a) of the PDE (1).
Notation. We write
(^<ii UXiai . . . UXnai \
I
uan UXian . . . UXnan / nx(n+l)
□
DEFINITION. A C2 function и = u(x; a) is called a complete integral in
U x A provided
(i)	u(x; a) solves the PDE (1) for each a & A
and
(ii)	rank(Dait, D2au) = n (x E.U, a G A).
3.1. COMPLETE INTEGRALS, ENVELOPES
93
Remark. Condition (ii) ensures u(x; a) “depends on all the n independent
parameters ai,..., an”. To see this, suppose В C R”-1 is open, and for each
b G В assume v = v(x;b) (x G U) is a solution of (1). Suppose also there
exists a C1 mapping : A —> B,	,..., such that
(3)	u{x-, a) = v(x-, ^>(a)) (x G U, a € A).
That is, we are supposing the function u(x; a) “really depends only on the
n — 1 parameters bi,..., bn-i”• But then
n—1
uXiaj (ж; a) = 22 v^bk (x; г/>(а)Уф*. (a) (г, j = 1,..., n).
fc=i
Consequently
n-i	/С1
det(T>2au) = 22 vxibkl   • vXnbkn det •..	1=0,
fci,...,fcn~l	\ 'ilfan	J
4 Vai	 • •	ran '
since for each choice of fci,..., fcn G {1,..., n — 1}, at least two columns in
the corresponding matrix are equal. As
n—1
uaj (x; a) = 22 vbk (X1 *K<iy№aj («) O' = 1, • • •, n),
fc=i
a similar argument shows the determinant of each n x n submatrix of
(Dau, Dj.au) equals zero, and thus this matrix has rank strictly less than
n.	□
Example 1.	Clairaut’s equation from differential geometry is the PDE
(4)	x  Du + f(Du) — u,
where f : Rn —* R is given. A complete integral is
(5)	u(x\a) = a • x + f(a) (xeU)
for a e Rn.	□
Example 2.	The eikonal* equation from geometric optics is the PDE
(6)	|Du| = 1.
A complete integral is
(7)	u(x;a,ty = a • x + b (x G J7)
for x G U, ae дВ(0,1), b G R.	□
= image (Greek)
94
3. NONLINEAR FIRST-ORDER PDE
Example 3.	The Hamilton-Jacobi equation from mechanics is in its sim-
plest form the partial differential equation
(8)
щ + H(Du) = 0,
where H : Rn —> R. Here и depends on x = (aq,... ,жп) G Rn and t € R.
As before we have set t — xn+i and written Du = Dxu = (uX1,. ., uXn). A
complete integral is
(9)	u(x, t; a,b) = a • x — tH(a) + b (x G Rn, t > 0)
where a e R", b G R.
□
3.1.2.	New solutions from envelopes.
We next demonstrate how to build more complicated solutions of our
nonlinear first-order PDE (1), solutions which depend on an arbitrary func-
tion of n — 1 variables, and not just on n parameters. We will construct
these new solutions as envelopes of complete integrals or, more generally, of
other m-parameter families of solutions.
DEFINITION. Let и = u(x;a) be a C1 function of x G U, a G A, where
U C R" and A C Rm are open sets. Consider the vector equation
(10)	Dau{x-,af — 0 (x G U, a G A).
Suppose that we can solve (10) for the parameter a as a C1 function of x,
(11)	a = </>(a:);
thus
(12)	Dau(x-, ф(х)) =0 (x G U).
We then call
(13)	v(af) := u(x; ф(хУ) (x G U)
the envelope of the functions {it(-; a)}aeq.
By forming envelopes we can build new solutions of our nonlinear first-
order partial differential equation:
3.1. COMPLETE INTEGRALS, ENVELOPES
95
THEOREM 1 (Construction of new solutions). Suppose for each a € A
as above that и — it(-; a) solves the partial differential equation (1). Assume
further that the envelope v, defined by (12) and (13) above, exists and is a
C1 function. Then v solves (1) as well.
The envelope v defined above is sometimes called a singular integral of
(1)-
Proof. We have v(x) = u(x; ф(х)); and so for i = 1,..., n
n
vXi (®) = uXi (ж; ф(ж)) + u<h
i=l
= uXi(x; </>(xf), according to (12).
Hence for each x G U,
F(Dv(x),v(x),x) = F(Du(x; dfixf), u(x-, </>(ж)), x) = 0.
□
Remark. The geometric idea is that for each x G U, the graph of v is
tangent to the graph of u(-; a) for a = ф(х). Thus Dv = Dxu(-; a) at x, for
a = ф(х).	□
Example 4. Consider the PDE
(14)	u2(l +|Du|2) = 1.
A complete integral is
u(x,a) = ±(1 — \x — al2)1/2 (|r — a| < 1).
We compute
provided a = ф(х) = x. Thus v = ±1 are singular integrals of (14).	□
To generate still more solutions of the PDE (1) from a complete integral,
we vary the above construction. Choose any open set A' C Rn-1 and any
C1 function h : A' —* R, so that the graph of h lies within A. Let us write
a = (ai,..., an) = (a', an) for a1 = (ai,..., an-i) G R"-1.
96
3. NONLINEAR FIRST-ORDER PDE
DEFINITION. The general integral (depending on h) is the envelope v' —
v'(x) of the functions
u'(x-,a') = u(x;a',Ь(а'У) (x &U, a E A!},
provided this envelope exists and is C1.
In other words, in computing the envelope we are now restricting only
to parameters a of the form a = (а/,/г(а/)), for some explicit choice of the
function h.
Remarks, (i) Thus from a complete integral, which depends upon n ar-
bitrary constants щ,..., an, we build (whenever the foregoing construction
works) a solution depending on an arbitrary function h of n — 1 variables.
(ii) It is tempting to believe that once we can find as above a solution
of (1) depending on an arbitrary function h we have found all the solutions
of (1). This need not be so, however. Suppose our PDE has the structure
F(Du, u,x) = Fi(Du,u,x)F2(Du,u,x) — 0.
If iti (re, a) is a complete integral of the PDE Fi (Du, u, x) = 0, and we succeed
in finding a general integral corresponding to any function h, we will still
have missed all the solutions of the PDE Fz(Du,u,x) = 0.	□
Example 5. An alternative form for a complete integral of the eikonal
equation |Dit| = 1 for n = 2 is
(15)	u(x-, a) = xi cos ai + x? sin ai + a? (x, aEK2).
We set h = 0, so that
u(x\ai) = xi cosai + X2 sinai
represents the subfamily of planar solutions of |Dit| — 1, whose graphs pass
through the point (0,0,0) 6 R3. We then compute the envelope by writing
Dai u' = —xi sin щ + X2 cos щ = 0.
Thus ai = arctan , and consequently
u'(x) = xi cos(arctan —) + X2 sin(arctan —) = ±\xI (x G R2)
X1	X1
solves |Dit| = 1 for x ф 0.	□
Example 6. Let H(p) — |p|2, h = 0 in Example 3 above. Then
u'(x, t',d) = x • a — t|a|2.
We calculate the envelope by setting Dau' = x — 2ta — 0. Hence a =
and so
u'(x,t} = x-^-t\^ = ^- (re G Rn, t > 0)
7 ’ 7	24	12t I 4t	7
solves the Hamilton-Jacobi equation щ + |Z?it|2 = 0.	□
3.2. CHARACTERISTICS
97
3.2.	CHARACTERISTICS
3.2.1.	Derivation of characteristic ODE.
We return to our basic nonlinear first-order PDE
(1)	F(Du, и, x) = 0 in U,
subject now to the boundary condition
(2)	и = g on Г,
where Г C dU and g : Г —* R are given. We hereafter suppose that F, g are
smooth functions.
We develop next the method of characteristics, which solves (1), (2) by
converting the PDE into an appropriate system of ODE. This is the plan.
Suppose и solves (1), (2) and fix any point x e U. We would like to calculate
u(x) by finding some curve lying within U, connecting x with a point x° G Г
and along which we can compute u. Since (2) says и = g on Г, we know
the value of и at the one end x°. We hope then to be able to calculate и all
along the curve, and so in particular at x.
Finding the characteristic ODE. How can we choose the curve so all
this will work? Let us suppose it is described parametrically by the function
x(s) = (a;1(s),... ,a:n(s)), the parameter s lying in some subinterval of R.
Assuming it is a C2 solution of (1), we define also
(3)	z(s) := u(x(s)).
In addition, set
(4)	p(s) := P«(x(s));
that is, p(s) = (p1^),... ,pn(s)), where
(5)	p’(s) = ttZi(x(s)) (i = l,...,n).
So z(-) gives the values of и along the curve and p(-) records the values of
the gradient Du. We must choose the function x(-) in such a way that we
can compute z(-) and p(-).
For this, first differentiate (5):
(6)	Pl(«) = ^«х^(х(в))^'(в)
j=i	'	'
98
3. NONLINEAR FIRST-ORDER PDE
This expression is not too promising, since it involves the second derivatives
of u. On the other hand, we can also differentiate the PDE (1) with respect
tO Xp.
n ' QP	QP	Qp
(7)	— (Du,u,x)uXjXi + — (Du,u,x)uXi + 7— (Du,u,x) = 0.
dpi	3 dz	oxi
j=i J
We are able to employ this identity to get rid of the “dangerous” second
derivative terms in (6), provided we first set
dF
(8)	^(s) = —(p(s),z(s),x(s)) (j = l,...,n).
c/pj
Assuming now (8) holds, we evaluate (7) at x = x(s), obtaining thereby
from (3), (4) the identity:
52 |“(P(S)> z(s)’	(x(s))
• т &Pj
J = 1	J
dF	dF
+ fl-(p(sM(s),x(s)Ms) +	(p(s),*(s),x(s)) = 0.
(J %	OXi
Substitute this expression and (8) into (6):
dF
= -Д—(p(s),*(s),x(s))
(9)	Sz' №
- -^-(p(s),^(s),x(s))pl(s) (г =
Finally we differentiate (3):
(10)	z(s) =	|^(x(s))ij(s) = ^pJ'(s)|^(p(s),z(s),x(s)),
, dxj	opj
j=i 3	j=i	J
the second equality holding by (5) and (8).
We summarize by rewriting equations (8)-(10) in vector notation:
(a) p(s) = -Z?a:F(p(s),z(s),x(s)) - Z?zF(p(s),z(s),x(s))p(s)
(11)	(b) z(s) = DpF(p(s),z(s),x(s)) -p(s)
k (c) x(s) = DpF(p(s),z(s),x(s)).
This important system of 2n + 1. first-order ODE comprises the char-
acteristic equations of the nonlinear first-order PDE (1). The functions
p(-) = (pW    ,pn(-)), z(-), x(-) = (r1(-),... , rn(-)) are called the charac-
teristics. We will sometimes refer to x(-) as the projected characteristic: it
is the projection of the full characteristics (p(-),z(-),x(-)) C IR2n+1 onto the
physical region U C IRn.
3.2. CHARACTERISTICS
99
We have proved:
THEOREM 1 (Structure of characteristic ODE). Let и G C2(17) solve
the nonlinear, first-order partial differential equation (1) in U. Assume x(-)
solves the ODE (ll)(c), where p(-) = 2?u(x(-)), z(-) = u(x(-)). Then p(-)
solves the ODE (H)(a) and z(-) solves the ODE (ll)(b), for those s such
that x(s) g U.
We still need to discover appropriate initial conditions for the system
of ODE (11), in order that this theorem be useful. We accomplish this in
§3.2.3 below.
Remark. The characteristic ODE are truly remarkable in that they form
a closed system of equations for x(-), г(-) = it(x(-)), and p(-) = Dit(x(-)),
whenever it is a smooth solution of the general nonlinear PDE (1). The
key step in the derivation is our setting x = DPF, so that—as explained
above—the terms involving second derivatives drop out. We thereby obtain
closure, and in particular are not forced to introduce ODE for the second
and higher derivatives of u.	□
3.2.2.	Examples.
Before continuing our investigation of the characteristic equations (11),
we pause to consider some special cases for which the structure of these
equations is especially simple. We illustrate as well how we can sometimes
actually solve the characteristic ODE and thereby explicitly compute solu-
tions of certain first-order PDE, subject to appropriate boundary conditions.
a.	F linear.
Consider first the situation that our PDE (1) is linear and homogeneous,
and thus has the form
(12)	F(Du, u, x) = b(a:) • Du(x) + c(r)u(r) = 0 (r G 17).
Then F(p, z, x) = b(x) • p + c(x)z, and so
(13)	DPF = b(x).
In this circumstance equation (11) (c) becomes
(14)	x(s) = b(x(s)),
an ODE involving only the function x(-). Furthermore equation (ll)(b)
becomes
(15)
z(s) = b(x(s)) -p(s).
100
3. NONLINEAR FIRST-ORDER PDE
Since p(-) = Zht(x(-)), equation (12) simplifies (15), yielding
(16)
z(s) = -c(x(s))z(s).
This ODE is linear in z(-), once we know the function x(-) by solving (14).
In summary,
(17)
(a)	x(s) = b(x(s))
(b)	z(s) = -c(x(s))z(s)
comprise the characteristic equations for the linear, first-order PDE (12).
(We will see later that the equation for p(-) is not needed.)	□
Example 1. We demonstrate the utility of equations (17) by explicitly
solving the problem
(18)
— aqu^ = u in U
и = g on Г,
where U is the quadrant {aq > 0, aq > 0} and Г = {aq > 0,aq = 0} C dU.
The PDE in (18) is of the form (12), for b = (—aq,aq) and c = —1. Thus
the equations (17) read
(19)
Accordingly we have
a:1 = —x2, x2 = x1
z = z.
a:1(s) = a;°coss, x2(s) = x° sin s
z(s) = z°es = д(ж°) es,
where a:0 > 0, 0 < s < |. Fix a point (aq,aq) 6 U. We select s > 0,
x° > 0 so that (aq,aq) = (a:1(s), a;2(s)) = (ж0 cos s, x° sin s). That is, x° =
(x2 + a^)1/2, s = arctan f j2) • Therefore
it(aq,aq) = u(a:1(s),r2(s)) = z(s) = g(x0) es
= 9((х1 + ^)1/2)е“я“<Й).
□
3.2. CHARACTERISTICS
101
b.	F quasilinear.
The partial differential equation (1) is quasilinear should it have the
form
(20)	F(J)u, u, x) = b(ac, u(x)) • Du(x) + c(x, u(x)) = 0.
In this circumstance F(p, z, x~) = b(rc, z) • p + c(x, z); whence
DPF = b(rc, z).
Hence equation (ll)(c) reads
x(s) = b(x(s), z(s)),
and (ll)(b) becomes
z(s) = b(x(s), z(s)) • p(s)
=-c(x(s),z(s)), by (20).
Consequently
,2П	Г (a) x(s) = b(x(s),z(s))
((b) z(s) =-c(x(s),z(s))
are the characteristic equations for the quasilinear first-order PDE (20).
(Once again the equation for p(-) is not needed.)	□
Example 2. The characteristic ODE (21) are in general difficult to solve,
and so we work out in this example the simpler case of a boundary-value
problem for a semilinear PDE:
uxi + uX2 = it2 in U
и = g on Г.
Now U is the half-space {x2 > 0} and Г = {x2 = 0} = dU. Here b = (1,1)
and c = — z2. Then (21) becomes
x1 =1, x2 = 1
.	9
z = z .
Consequently
a:1(s) = x® + s, x2(s) = s
= z° „ = g(l0)„
' '	1—sz° 1—sg(T°) ’
where x° G R, s > 0, provided the denominator is not zero.
102
3. NONLINEAR FIRST-ORDER PDE
Fix a point (ж1, x2) G U. We select s > 0 and x° G R so that (rci, x2) =
(a:1(s),a:2(s)) = (a:0 + s, s); that is, rc° — rci — x2, s — x2. Then
u(x1,x2>) = u(x1(s),x2(s)) = z(s) =
i sg\x )
= д(хг -x2)
1 - 3:29(2:1 - x2) ’
This solution of course makes sense only if 1 — x2g(xi — x2)	0.	□
c.	F fully nonlinear.
In the general case, the full characteristic equations (11) must be inte-
grated, if possible.
Example 3. Consider the fully nonlinear problem
J ^1^2 ~ u in U
1	11 — r2 nn Г
I	U — To Oil 1 ,
where U = {aq > 0}, Г = {rci = 0} = dU. Here F(p, z,x) = pip2 — z, and
hence the characteristic ODE (11) become
.1	1	.9	9
P = P > P — P
< z = 2p1p2
i1 = p2, x2 = p1.
We integrate these equations to find
' a:1^) = p2(es — 1), x2(s) = x° + p®(es — 1)
<	z(s) = z° + plp^e2s - 1)
k p\s) = P]*esp2(s) = p2es,
where x° G R, s G R, and z° = (re0)2.
We must determine p° = (р^р^). Since и = a:2 on Г, p® — иЖ2(О,а:0) =
2ar°. Furthermore the PDE uT1uT2 = it itself implies p^p® = z° = (a:0)2, and
so P| =	. Consequently the formulas above become
( ar1(s) = 2x°(es — l),a:2(s) = ^-(es + 1)
<	z(s) = (a:°)2e2s
( p\s) = es, p2(s) = 2x°es.
Fixapoint (xi,x2) G U. Select s and a:0 so that (xi,x2) = (ar1(s),a:2(s))
= (2ar°(es — 1), ^r(es + 1)). This equality implies x° = 4x2~xi , es =	;
and so
11(2:1,3:2) = u(a:1(s),a:2(s)) = z(s) = (a:°)2e2s
_ (a:i + 4x2)2
16
□
3.2. CHARACTERISTICS
103
(27)
3.2.3.	Boundary conditions.
We return now to developing the general theory.
a.	Straightening the boundary.
We intend in the section following to invoke the characteristic ODE
(11) actually to solve the boundary-value problem (1), (2), at least in a
small region near an appropriate portion Г of dU. In order to simplify
the relevant calculations, it is convenient first to change variables, so as to
“flatten out” part of the boundary dU. To accomplish this, we first fix
any point x° G dU. Then utilizing the notation from §C.l, we find smooth
mappings Ф, Ф : R” —» Rn such that Ф = Ф-1 and Ф straightens out dU
near x°. (See the illustration in §C.l.)
Given any function и : U —> R, let us write V := Ф(17) and set
(24)	v(y) := к(Ф(у)) (у € V).
Then
(25)	u(x) — и(Ф(х)) (x G IT).
Now suppose that it is a C1 solution of our boundary-value problem (1), (2)
in U. What PDE does v then satisfy in V?
According to (25), we see
n
= £^(Ф(.))Ф^(^) (i = l,...,n);
fc=i
that is,
Du(x) = _Рг|(1/)1)Ф(;г).
Thus (1) implies
0 = F(Du(x),u(x),x
(26)	= Г(2?г;(г/)РФ(Ф(г/)), v(y), Ф(у)).
This is an expression having the form
G(£>v(y),v(y),y) = 0 in V.
In addition v = h on A, where A := Ф(Г) and h(y) := д(Ф(у)).
In summary, our problem (1), (2) transforms to read
G(Dv,v,y) = 0 in V
v = h on A,
for G, h as above. The point is that if we change variables to straighten out
the boundary near x°, the boundary-value problem (1), (2) converts into a
problem having the same form.
104
3. NONLINEAR FIRST-ORDER PDE
b.	Compatibility conditions on boundary data.
In view of the foregoing computations, if we are given a point x° € Г we
may as well assume from the outset that Г is flat near x°, lying in the plane
{xn = 0}.
We intend now to utilize the characteristic ODE to construct a solution
(1), (2), at least near rc°, and for this we must discover appropriate initial
conditions
(28)	p(0) = p°, z(0) = z°, x(0) = ж0.
Now clearly if the curve x(-) passes through x°, we should insist that
(29)	z° = g(x°).
What should we require concerning p(0) = p°? Since (2) implies
u(xi... xn-i, 0) = g(xi... xn~i) near ж0, we may differentiate to find
иХг(х°) = 9xi(x°) (г = 1,..., n — 1).
As we also want the PDE (1) to hold, we should therefore insist p° =
(Pi,... ,p0) satisfies these relations:
Г Pi=9zi(x0) (i = l,...,n-l)
( J	\ F(p°,z°,x°) = 0.
These identities provide n equations for the n quantities p° = (p?,..., p„).
We call (29) and (30) the compatibility conditions. A triple (р°,г°,ж°) 6
R2n+1 verifying (29), (30) is admissible. Note z° is uniquely determined
by the boundary condition and our choice of the point x°, but a vector p°
satisfying (30) may not exist or may not be unique.
c.	Noncharacteristic boundary data.
So now assume as above that re0 6 Г, that Г near ж0 lies in the plane
{xn = 0}, and that the triple (p°, z°, ж0) is admissible. We are planning to
construct a solution и of (1), (2) in U near x° by integrating the character-
istic ODE (11). So far we have ascertained x(0) = a:0, z(0) = z°, p(0) = p°
are appropriate boundary conditions for the characteristic ODE, with x(-)
intersecting Г at re0. But we will need in fact to solve these ODE for nearby
initial points as well, and must consequently now ask if we can somehow
appropriately perturb (р°,г°,ж°), keeping the compatibility conditions.
3.2. CHARACTERISTICS
105
In other words, given a point у = (j/i,... ,yn--[,0) G Г, with у close to
x°, we intend to solve the characteristic ODE
(a) p(s) = -Z?IF(p(s),z(s),x(s)) - Z?zF(p(s),z(s),x(s))p(s)
(31)	(b) z(s) = PpF(p(s),z(s),x(s))  p(s)
. (c) x(s) = Z?pF(p(s),z(s),x(s)),
with the initial conditions
(32)	p(0) = q(y), z(0) = g(y), x(0) = y.
Our task then is to find a function q(-) = (g1^),... , <7n(-))> so that
(33)	q(x°) = p°
and (q(l/))9(l/), ?/) is admissible; that is, the compatibility conditions
,34x	f ql(y) = 9хАу) (i = i,...,n-i)
I F(q(y),g(y),y) = 0
hold for all у G Г close to
LEMMA 1 (Noncharacteristic boundary conditions). There exists a unique
solution q(-) of (33), (34) for all у g Г sufficiently close to x°, provided
(35)	FPn(p°,zo,x°^0.
We say the admissible triple (p°, z°, x°) is noncharacteristic if (35) holds.
We henceforth assume this condition.
Proof. To simplify notation, let us now temporarily write у = (yi,..., yn) G
Rn. We apply the Implicit Function Theorem (§C.6) to the mapping
G:F хГ-, Rn, G(p,y) = (G\p,y),.. .,Gn(p,y)),
where
Г Gi(p,y)=pi-gXi(y) (i = 1,..., n — 1)
I Gn(p,y) = F(p,g(y),y).
Now G(p°,a;0) = 0, according to (29), (30). Also
and thus
det DpG(p°, r°) = FPn (p°, z°, x°)	0,
in view of the noncharacteristic condition (35). The Implicit Function Theo-
rem thus ensures we can uniquely solve the identity G(p, y) = 0 for p = q(y),
provided у is close enough to x°.	□
106
3. NONLINEAR FIRST-ORDER PDE
Remark. If Г is not flat near x°, the condition that Г be noncharacteristic
reads
(36)	DpF(p°,zo,x°)-^xo)/O,
v(x°) denoting the outward unit normal to dU at x°.	□
3.2.4.	Local solution.
Remember that our aim is to use the characteristic ODE to build a
solution и of (1), (2), at least near Г. So as before we select a point x° G Г
and, as shown in §3.2.3, may as well assume that near x° the surface Г is flat,
lying in the plane {xn = 0}. Suppose further that (p°, z°, re0) is an admissible
triple of boundary data, which is noncharacteristic. According to Lemma 1
there is a function q(-) so that p° = q(a?°) and the triple (q(p),p(p),y) is
admissible, for all у sufficiently close to x°.
Given any such point у = (yi,... ,pn-i,0), we solve the characteristic
ODE (31), subject to initial conditions (32).
Notation. Let us write
' P(«) = Р(.У, «) = Р(У1, • • , Уп-i, s)
< z(s) = z(y,s) = z(ylt . . . ,yn-!,s)
. x(s) = x(y,s) = x(yi,...,yn-i,s)
to display the dependence of the solution of (31), (32) on s and y. □
LEMMA 2 (Local invertibility). Assume we have the noncharacteristic
condition FPn(p0,z0,x0) Ф 0. Then there exist an open interval I C R
containing 0, a neighborhood W of x° in Г c Rn-1, and a neighborhood V
of x° in Rn, such that for each x gV there exist unique s g I, у G W such
that
x = *(y,s).
The mappings x > s, у are C2.
Proof. We have x(rr°, 0) = x°. Consequently the Inverse Function Theorem
(§C.5) gives the result, provided det Z?x(a;0,0) ф 0. Now
х(у,0) = (у,0) (уеГ),
and so if i = 1,..., n — 1,
дУг ' о (j = n).
3.2. CHARACTERISTICS
107
Furthermore equation (31)(c) implies
(ж0,0) = FPj(p°, z°, x°).
Thus
7?х(ж°,0) =
/1	0FP1(p°,z°,z°)\
0	1	:
\0 • •-0 ЕРп(р0,г°,ж0)/nxn
whence det _Dx(rr°, 0)	0 follows from the noncharacteristic condition (35).
□
In view of Lemma 2 for each r 6 V, we can locally uniquely solve the
equation
( x = x.(u, s),
1 for у = у(ж), S = S(X).
Finally, let us define
(38)	Г u(x) := г(у(ж),в(ж))
I р(ж) ••= р(у(ж),в(ж))
for x G V and s,y as in (37).
We come finally to our principal assertion, namely, that we can locally
weave together the solutions of the characteristic ODE into a solution of the
PDE.
THEOREM 2 (Local Existence Theorem). The function и defined above
is C2 and solves the PDE
F(Du(x), u(x), x) = 0 (ж G V),
with the boundary condition
u(x) = g(x) (жбГПУ).
Proof. 1. First of all, fix у G Г close to ж0 and, as above, solve the charac-
teristic ODE (31), (32) for p(s) = p(y, s), z(s) = z(y, s), and x(s) = x(y, s).
2.	We assert that if у G Г is sufficiently close to ж0, then
(39)	/(y,s) := F(p(y,s),z(y,s),x(y,s)) = 0 (s G R).
108
3. NONLINEAR FIRST-ORDER PDE
To see this, note
f(y, 0) = F(p(y, 0), z(y, 0), x(y, 0))
= ^(q(y),5(y),y) = 0,
by the compatibility condition (34). Furthermore
ds ’	“ dpj dz	dxj
j=i J	j=i J
This calculation and (40) prove (39).
3.	In view of Lemma 2 and (37)-(39), we have
F(p(x),it(r),a:) = 0 (x £ V).
To conclude, we must therefore show
(41)	p(a;) = Du(x) (x e V).
4.	In order to prove (41), let us first demonstrate for s £ I, у £ W that
z.ox	dz	\дх31	\
(42)	ds^y,S^ =
j=i
and
(43)	= ]^ (</,*)$-0/,*) (i = l,...,n-l).
cfyi j—1	&1И
These formulas are obviously consistent with the equality (41) and will later
help us prove it. The identity (42) results at once from the characteristic
ODE (31)(b),(c). To establish (43), fix у £ Г, i £ {1,... ,n — 1}, and set
z. .4	if , dz	• dx3
(44)	r (sy.=—(y,s}—
3.2. CHARACTERISTICS
109
We first note r’(0) = gXi(y) ~ Ql(y) = 0 according to the compatibility
condition (34). In addition, we can compute
(45)
. . _ d2z ул dpi dxi , d2x3
dyids	ds dyi + dyids
j=i
In order to simplify this expression, let us first differentiate the identity (42)
with respect to yi:
d2z \^\dp> dx3 - d2xj ’
(46)	= У. w—+	•
usuyi	j—i	dsdyi
Substituting (46) into (45), we discover
.	ул	dpi	dx3	dpi	dxi
Г S	.дУг	ds	9s	дУг
(47)	3
- v	\dp>	(—} -	- —nA —
“ dyt \dpj) \ dxj dz ) dyi
Now differentiate (39) with respect to уг:
Л, dF dp>	dF dz	-A dF dxi n
dpj dyi	dz dyi dxj dyi
We employ this identity in (47), thereby obtaining
by (31)(a).
(48)
Hence rl(-) solves the linear ODE (48), with the initial condition гг(0) = 0.
Consequently rl(s) = 0 (s 6 R, i = 1,... ,n — 1); and so identity (43) is
verified.
5. We finally employ (42), (43) in proving (41). Indeed, if j = 1,..., n,
dv^ = dzds_ y4 dz d?/
dxj ds dxj dyi dxj
(71	jk \	71 “ 1 / 71 о Ic У\ о э
I — + V I I
ds I dxj “ dyi I dxj
«=1	/ J г=1 \/с=1	/ J
k (dxk ds <--i dxk dy1 \
\ ds dxj dyi dxj I
K = 1	\	J 1=1	J /
n n
k=l •* k=l
by (42), (43)
110
3. NONLINEAR FIRST-ORDER PDE
This assertion at last establishes (41), and so finishes up the proof. □
3.2.5.	Applications.
We turn now to various special cases, to see how the local existence
theory simplifies in these circumstances.
a.	F linear.
Recall that a linear, homogeneous, first-order PDE has the form
(49)	F(Du,u,x) = b(r) • Du(x) + c(x)u(x) = 0 (x e U).
Our noncharacteristic assumption (36) at a point x° e Г as above becomes
(50)	b(a:°) • i/(a:0) / 0,
and thus does not involve z° or p° at all. Furthermore if we specify the
boundary condition
(51)	и = g on Г,
we can uniquely solve equation (34) for q(y) if у e Г is near rc°. Thus we
can apply the Local Existence Theorem 2 to construct a unique solution
of (49), (51) in some neighborhood V containing x°. Note carefully that
although we have utilized the full characteristic equations (31) in the proof
of Theorem 2, once we know the solution exists, we can use the reduced
equations (17) (which do not involve p(-)) to compute the solution. Observe
also the projected characteristics x(-) emanating from distinct points on Г
cannot cross, owing to uniqueness of solutions of the initial-value problem
for the ODE (17)(a).
Example 4. Suppose the trajectories of the ODE
(52)	x(s) = b(x(s))
are as drawn for Case 1.
We are thus assuming the vector field b vanishes within U only at one
point, which we will take to be the origin 0, and b • v < 0 on Г := dU. Can
we solve the linear boundary-value problem
.	f b • Du = 0 in U
(53)	<	T-, 9
v	[ и = g on Г '
Invoking Theorem 2 we see that there exists a unique solution и defined
near Г, and indeed that u(x(s)) = u(x(0)) = g(xQ) for each solution of the
3.2. CHARACTERISTICS
111
Case 2: flow across a domain
ODE (52), with the initial condition x(0) =/J £ Г, However, this solution
cannot be smoothly continued to all of U (unless g is constant): any smooth
solution of (53) is constant on trajectories of (52), and thus takes on different
values near x = 0.
On the other hand, now suppose the trajectories of the ODE (52) look
like the illustration for Case 2. We are consequently now assuming that each
trajectory of the ODE (except those through the characteristic points A, B)
enters U precisely once, somewhere through the set
Г := {x € dU | b(r) • v(x) < 0},
and exits U precisely once. In this circumstance we can find a smooth
solution of (53) by setting и to be constant along each flow line.
112
3. NONLINEAR FIRST-ORDER PDE
Case 3: flow with characteristic points
Assume finally the flow looks like Case 3. We can now define и to be
constant along trajectories, but then и will be discontinuous (unless g(B) =
9(D)).
Note that the point D is characteristic and that the local existence theory
fails near D.	□
b.	F quasi linear.
Should F be quasilinear, the PDE (1) becomes
(54)	F(Du, u, x) = b(x, u)  Du + c(x, u) = 0.
The noncharacteristic assumption (36) at a point x° G Г reads b(x°, z°) 
v(x°)	0, where z° = g(x°). As in the preceding example, if we specify the
boundary condition
(55)	и — g on Г,
we can uniquely solve the equations (34) for q(y) if у G Г near x°. Thus
Theorem 2 yields the existence of a unique solution of (54), (55) in some
neighborhood V of x°. We can compute this solution in V using the reduced
characteristic equations (21), which do not explicitly involve p(-).
In contrast to the linear case, however, it is possible that the projected
characteristics emanating from distinct points in Г may intersect outside V;
such an occurrence usually signals the failure of our local solution to exist
within all of U.
Example 5 (Characteristics for conservation laws). As an instance of a
quasilinear first-order PDE, we turn now to the scalar conservation law
G(Du, щ, u, x, t) = щ + div F(u)
= щ + Fz(u) • Du = 0
3.2. CHARACTERISTICS
113
in U = R" x (0, oo), subject to the initial condition
(57)	и = g on Г — Rn x {t = 0}.
Here F : R —> Rn, F = (F1,... , Fn), and, as usual, we have set t =
xn+i. Also, “div” denotes the divergence with respect to the spatial variables
(xi,.. .,xn), and Du = Dxu = (uX1,.. .,uXn).
Since the direction t = xn+i plays a special role, we appropriately modify
our notation. Writing now q = (p,pn+i) and у = (x, t), we have
G(q, z, y) = pn+i + F'(z) • p,
and consequently
DqG = (F'(z),l), DyG = Q, DzG = F"(z)-p.
Clearly the noncharacteristic condition (35) is satisfied at each point y° =
(x°,0) G Г. Furthermore equation (21)(a) becomes
Г x*(s) = F^'fyfy)) (i = l,...,n)
|xn+1(s) = l.
Hence xn+1(s) = s, in agreement with our having written xn+i = t above.
In other words, we can identify the parameter s with the time t.
Equation (21)(b) reads z(s) = 0. Consequently
(59)	z(s) = z° = g(x°);
and (58) implies
(60)	x(s) = F/fy(x0))s+x0.
Thus the projected characteristic y(s) = (x(s),s) = (F'(<7(x°))s + x°, s)
(s > 0) is a straight line, along which и is constant.
Crossing characteristics. But suppose now we apply the same reasoning
to a different initial point z° E Г, where <?(х0)	g(z°). The projected char-
acteristics may possibly then intersect at some time t > 0. Since Theorem 1
tells us и = <?(х°) on the projected characteristic through x° and и = g(z°)
on the projected characteristic through z°, an apparent contradiction arises.
The resolution is that the initial-value problem (56), (57) does not in general
have a smooth solution, existing for all times t > 0.	□
We will discuss in §3.4 the interesting possibility of extending the local
solution (guaranteed to exist for short times by Theorem 2) to all times
t > 0, as a kind of “weak” or “generalized” solution.
114
3. NONLINEAR FIRST-ORDER PDE
Remark. Let us also note we can eliminate s from equations (59), (60) to
obtain an implicit formula for u. Indeed given x € Rn and t > 0, we see
that since s = t,
u(x(t), t) = z(t) = tf(x(t) - tF'(z0))
= ff(x(t) -tF/(u(x(t),t))).
Hence
(61)	и = g(x — tF'tuY).
This implicit formula for и as a function of x and t is a nonlinear analogue
of equation (3) in §2.1. It is easy to check (61) does indeed give a solution,
provided
1 + tDg(x -	• F"(u) / 0.
In particular if n = 1, we require
1 + tg'(x - tF'(u))F"(u) / 0.
Note that if F" > 0, but д' < 0, then this will definitely be false at some
time t > 0. This failure of the implicit formula (61) reflects also the failure
of the characteristic method.	□
c. F fully nonlinear.
The form of the full characteristic equations can be quite complicated for
fully nonlinear first-order PDE, but sometimes a remarkable mathematical
structure emerges.
Example 6 (Characteristics for the Hamilton-Jacobi equation). We look
now at the general Hamilton-Jacobi PDE
(62)	G(Du, щ,и,х, t) = щ + H(Du,x) = 0,
where Du = Dxu = (uX1,... ,uXn). Then writing q = (p,pn+i), у = (x, t),
we have
G(q, z,y) =Pn+i +Я(р,х);
and so
DqG = (DpH(p,x), 1), DyG = (DxH(p,x),0), DZG = 0.
Thus equation (ll)(c) becomes
f ^(s) = fg(p(s),x(s)) (i = l,...,n)
t±n+1(s) = l.
3.3. INTRODUCTION TO HAMILTON-JACOBI EQUATIONS
115
In particular we can identify the parameter s with the time t.
Equation (11) (a) for the case at hand reads
Г p\s) = -fj(pfy),x(s)) (i = l,...,n)
|pn+1(s) = 0;
the equation (ll)(b) is
i(s) = РрЯ(р(«),х(«)) • p(s) +pn+1(s)
= Dpff(p(s),x(s)) • p(s) - Я(р(«),х(«)).
In summary, the characteristic equations for the Hamilton-Jacobi equation
are:
(64)
'(a) p(s) = -Dx#(p(s),x(s))
< (b) z(5) = Ppff(p(s),x(s))-p(s)-ff(p(5),x(s))
< (c) x(s) = ЯрЯ(р(в),х(в))
for p(-) = (p1(-),---,Pn(-)), 4-)> and x(-) = (x1(-),---,^n(-))-
The first and third of these equalities,
Г х = Г>рЯ(р,х)
I P = -ДгЯ(р,х),
are called Hamilton’s equations. We will discuss these ODE and their rela-
tionship to the Hamilton-Jacobi equation in much more detail, just below
in §3.3. Observe that the equation for z(-) is trivial, once x(-) and p(-) have
been found by solving Hamilton’s equations.	□
As for conservation laws (Example 5), the initial-value problem for the
Hamilton-Jacobi equation does not in general have a smooth solution и
lasting for all times t > 0.
3.3. INTRODUCTION TO HAMILTON-JACOBI
EQUATIONS
In this section we study in some detail the initial-value problem for the
Hamilton-Jacobi equation:
. .	( щ + H(Du) = 0 in Rn x (0, oo)
' '	[	и = g on Rn x {t = 0}.
Here и : Rn x [0, oo) —> R is the unknown, и = u(x, t), and Du = Dxu =
(uX1,..., uXn). We are given the Hamiltonian H : Rn —> R and the initial
function g : R” —> R.
116
3. NONLINEAR FIRST-ORDER PDE
Our goal is to find a formula for an appropriate weak or generalized
solution, existing for all times t > 0, even after the method of characteristics
has failed.
3.3.1. Calculus of variations, Hamilton’s ODE.
Remember from §3.2.5 that two of the characteristic equations associated
with the Hamilton-Jacobi PDE
щ + H{Du,x) = 0
are Hamilton’s ODE
( x = DpH (p, x)
I P = -DxH(p,ii},
which arise in the classical calculus of variations and in mechanics. (Note
the x-dependence in H here.) In this section we recall the derivation of
these ODE from a variational principle. We will then discover in §3.3.2 that
this discussion contains a clue as to how to build a weak solution of the
initial-value problem (1).
a. The calculus of variations.
Assume that L : Rn x R" —> R is a given smooth function, hereafter
called the Lagrangian.
Notation. We write
L = L(q,x) = Lfa, ...,qn,x\,...,xn) (q,x e Rn)
and
f DqL = (Lqi    Lqn)
I DXL = (Ex, • • • LXn).
Thus in the formula (2) below “g” is the name of the variable for which
we substitute w(s), and “x” is the variable for which we substitute w(s).
□
Now fix two points x, у G Rn and a time t > 0. We introduce then the
action functional
/•*	/ d \
(2)	J[w(-)] = уо L(w(s),w(s))ds = — j ,
defined for functions w(-) = (w'(-), w2(-),... ,w"(-)) belonging to the ad-
missible class
Д = {w(-) G C2([0, t]; Rn) | w(0) — y, w(t) = x}.
3.3. INTRODUCTION TO HAMILTON JACOBI EQUATIONS
117
A problem in the calculus of variations
Thus a C2 curve w(-) lies in A if it starts at the point у at time 0, and
reaches the point x at time t. A basic problem in the calculus of variations
is then to find a curve x(-) G A satisfying
(3)	Z[x(-)] = min Z[w(-)].
w(-)eA
That is, we are asking for a function x(-) which minimizes the functional
/[•] among all admissible candidates w(-) G A.
We assume next that there in fact exists a function x(-) G A satisfying
our calculus of variations problem, and will deduce some of its properties.
THEOREM 1 (Euler-Lagrange equations). The function x(-) solves the
system of Euler-Lagrange equations
(4)	(£><7L(x(s),x(s)))+ D3;L(x(s),x(s)) = 0 (0 < s < t).
This is a vector equation, consisting of n coupled second-order equations.
Proof. 1. Choose a smooth function v : [0,t] —> Rn, v = (v1,... ,vn),
satisfying
(5)	v(0) - v(t) = 0,
and define for т G R
(6)	w(-) := x(-) + rv(-).
Then w(-) G A and so
Z[x(-)] < Z[w(-)].
Thus the real-valued function
118
3. NONLINEAR FIRST-ORDER PDE
has a minimum at т = 0, and consequently
(7)	?'(0) = 0	=
provided i'(0) exists.
2.	We explicitly compute this derivative. Observe
i(r) — f L(ic(s) +	ds,
Jo
and so
ft n
i'(r) = /	+ rv,x + rv)vl + LxJx +rv,x + rv)v’ds.
Jo i=i
Set т = 0 and remember (7):
ftn
0 = i'(0) = / У2 (x, x)i>’+	(x, х)пг ds.
Jo i=i
We recall (5) and then integrate by parts in the first term inside the integral,
to discover
0 =
-^(Шх)) + Шх)
vl ds.
This identity is valid for all smooth functions v = (n1,..., vn) satisfying the
boundary conditions (5), and so
-	CMk,x)) + -M*, x) = 0
for 0 < s < t, i = 1,..., n.
□
Remark. We have just demonstrated that any minimizer x(-) G A of /[•]
solves the Euler-Lagrange system of ODE. It is of course possible that a
curve x(-) G A may solve the Euler-Lagrange equations without necessarily
being a minimizer: in this case we say x(-) is a critical point of /[•]. So every
minimizer is a critical point, but a critical point need not be a minimizer.
□
3.3. INTRODUCTION TO HAMILTON-JACOBI EQUATIONS
119
Example. If L(q, x) = |m|g|2 — 0(x), where m > 0, the corresponding
Euler-Lagrange equation is
mx(s) = f(x(s))
for f := —Оф. This is Newton’s law for the motion of a particle of mass m
moving in the force field f generated by the potential <f>. (See Feynman-
Leighton-Sands [F-L-S, Chapter 19].)	□
b. Hamilton’s ODE.
We now convert the Euler-Lagrange equations, a system of n second-
order ODE, into Hamilton’s equations, a system of 2n first-order ODE.
We hereafter assume the C2 function x(-) is a critical point of the action
functional, and thus solves the Euler-Lagrange equations (4).
First we set
(8)	p(s) := DgL(x(s),x(s)) (0 < s < t);
p(-) is called the generalized momentum corresponding to the position x(-)
and velocity x(-). We next make this important hypothesis:
I Suppose for all x,p G R" that the equation
p = DqL(q,x)
can be uniquely solved for q as a smooth
function of p and x, q = q(p, x).
We will examine this assumption in more detail later: see §3.3.2.
DEFINITION. The Hamiltonian H associated with the Lagrangian L is
H(p, x) .= p  cfip, x) — L(c[(p, x),x) (j>,x G Rn),
where the function q(-, •) is defined implicitly by (9).
Example (continued). The Hamiltonian corresponding to the Lagrangian
L(q, x) = |m|g|2 — </>(x) is
H(p,x) = 2m +
The Hamiltonian is thus the total energy, the sum of the kinetic and potential
energies; the Lagrangian is the difference between the kinetic and potential
energies.	□
Next we rewrite the Euler-Lagrange equations in terms of p(-),x(-):
120
3. NONLINEAR FIRST-ORDER PDE
THEOREM 2 (Derivation of Hamilton’s ODE). The functions x(-) and
p(-) satisfy Hamilton’s equations:
x(s) = DpH(p(s),x(s))
p(s) = -DxH(p(s),x(s))
for 0 < s < t. Furthermore,
the mapping s i—► H(p(s),x(s)) is constant.
(10)
Remark. The equations (10) comprise a coupled system of 2n first-order
ODE for x(-) = (x1(-),... ,xn(-)) and p(-) = (p1(-)> • • • ,Pn(')) (defined by
(8))-	□
Proof. First note from (8) and (9) that x(s) = q(p(s),x(s)).
Let us hereafter write q(-) = (q1(-), , (?”(•))• We then compute for
i = 1,... ,n:
g(p,x) =
&	*	A»v	V	v
=	by (9),
and
= fyp.x) + ^р^(р,х) -
k=l
= q\p,x), again by (9).
Thus
ДО*
— (p(s),x(s)) = g’(p(s),x(s)) = ±‘(s);
&Pi
and likewise
|^(p(s),x(s)) = -^(q(p(s),x(s)),x(s)) = -||(x(s),x(s))
= -p\s).
Finally, observe
d rrz . .	. ^dH.i ЭН .i
-Я(р(»),хМ) = ^^-р +^-x
г=1
П nrr / nrr\
= 0.
□
3.3. INTRODUCTION TO HAMILTON-JACOBI EQUATIONS
121
Remark. See Arnold [AR, Chapter 9] for more on Hamilton’s ODE and
Hamilton-Jacobi PDE in classical mechanics. We are employing here differ-
ent notation than is customary in mechanics: our notation is better overall
for PDE theory.	□
3.3.2. Legendre transform, Hopf—Lax formula.
Now let us try to find a connection between the Hamilton-Jacobi PDE
and the calculus of variations problem (2)-(4). To simplify further, we also
drop the ^-dependence in the Hamiltonian, so that afterwards H = H(p).
We start by reexamining the definition of the Hamiltonian in §3.3.1.
a. Legendre transform.
We hereafter suppose the Lagrangian L : Rn —► R satisfies these condi-
tions:
(11)	the mapping q i—> L(g) is convex
and
The convexity implies L is continuous.
DEFINITION. The Legendre transform of L is
(13)	L*(p) = sup {p  q - L(q)} (p G Rn).
</eRn
Motivation for Legendre transform. Why do we make this definition?
For some insight let us note in view of (12) that the “sup” in (13) is really
a “max”; that is, there exists some q* G Rn for which
L*(p)=P-9*-Mg*)
and the mapping q t-+ p • q — L(g) has a maximum at q = q*. But then p =
DL(q*f provided L is differentiable at q*. Hence the equation p — DL(q)
is solvable (although perhaps not uniquely) for q in terms of p, q* = q(p).
Therefore
L*(p) = p-q(p) -L(q(p)).
However, this is almost exactly the definition of the Hamiltonian H asso-
ciated with L in §3.3.1 (where, recall, we are now assuming the variable x
does not appear). We consequently henceforth write
(14)	H = L*.
Thus (13) tells us how to obtain the Hamiltonian H from the Lagrangian
L.	□
Now we ask the converse question: given H, how do we compute L?
122
3. NONLINEAR FIRST-ORDER PDE
THEOREM 3 (Convex duality of Hamiltonian and Lagrangian). Assume
L satisfies (11), (12) and define H by (13), (14).
(i) Then
the mapping p > H(p) is convex
and
(ii) Furthermore
(15)
L = H*.
Remark. Thus H is the Legendre transform of L, and vice versa:
L = H*, H = L*.
We say H and L are dual convex functions.
Proof. 1. For each fixed q, the function p i—> p  q — L(q) is linear, and
consequently the mapping
p i-> Я(р) = L*(p) = sup {p • q - L(q)}
q<=S.n
is convex. Indeed, if 0 < г < 1, p,p E Rn,
H(rp + (1 - т)р) = sup{(rp + (1 - т)р) • q - L(q)}
< rswp{p-q- L(q)}
ч
+ (1-t) sup{p • q - L(q)}
= тН(р) + (1 -т)Я(р).
2. Fix any A > 0, p 0. Then
Я(р) = sup {p -q-L(q)}
</eRn
>аН-цаА) (, = aA)
> Alni — max L.
~	B(0,A)
Thus liminfippoo > A for all A > 0.
3.3. INTRODUCTION TO HAMILTON-JACOBI EQUATIONS
123
3. In view of (14)
Я(р) + £(д) >p q
for all p, q G Rn, and consequently
L(g) > sup {p • q - H(p}} = Я*(д).
peR”
On the other hand
Я*(д) = sup {p - q — sup {p  r - L(r)}}
PeR" reR"
= sup inf {p -(q-r) + L(r)}.
PeR" reR’
Now since q >—► L(g) is convex, according to §B.l there exists s G Rn such
that
L(r) > L(q) + s  (r - q) (r G Rn).
(If L is differentiable at q, take s = DL{q).') Taking p = s in (16), we
compute
Я*(д) > inf {s • (q - r) + L(r)} = L(q).
rEliv1
□
b. Hopf—Lax formula.
Let us now return to the initial-value problem (1). Recall that the
calculus of variations problem with Lagrangian L, discussed in §3.3.1, led to
Hamilton’s ODE for the associated Hamiltonian Я. Since these ODE are
also the characteristic equations of the Hamilton-Jacobi PDE, we conjecture
there is probably a direct connection between this PDE and the calculus of
variations.
So if x G Rn and t > 0 are given, we should presumably try to minimize
the action
[ L(-w(s))ds
Jo
over functions w : [0, t] —> Rn satisfying w(t) = x. But what should we take
for w(0)? As we must somehow take into account the initial condition for
our PDE, let us try modifying the action to include the function g evaluated
at w(0):
[ E(w(s)) ds + g(w(0)).
Jo
124
3. NONLINEAR FIRST-ORDER PDE
Next let us construct a candidate for a solution to the initial-value prob-
lem (1), in terms of a variational principle entailing this modified action.
We accordingly set
(17) u(x, t) := inf L(w(s)) ds + g(y) | w(0) = y, w(t) = x} ,
the infimum taken over all C1 functions w(-) with w(t) = x. (Better justi-
fication for this guess will be provided much later, in Chapter 10.)
We propose now to investigate the sense in which и so defined by (17)
actually solves the initial-value problem for the Hamilton-Jacobi PDE:
(18)
щ + H(Du) = 0
и = g
in Rn x (0, oo)
on Rn x {t = 0}.
Recall we are assuming H is smooth,
(19)
H is convex and
lim = +00.
, |p|—too
We henceforth suppose also
(20)
g : Rn —> R	is Lipschitz continuous;
this means Lip(#) := supx>2/eRn | |g(u f < °°-
x*y 1	1 gl J
First we note formula (17) can be simplified:
THEOREM 4 (Hopf-Lax formula). If x G Rn and t > 0, then the solution
и = u(x,t) of the minimization problem (17) is
(21)
u(x, t) = min $ tL
yeR" I
DEFINITION. We call the expression on the right hand side of (21) the
Hopf-Lax formula.
Proof. 1. Fix any у G Rn and define w(s) := у + f (x — y) (0 < s < t).
Then the definition (17) of и implies
+ s(y),
3.3. INTRODUCTION TO HAMILTON-JACOBI EQUATIONS
125
and so
u(x, t) < inf < tL I -—- ) + g(y) > .
2. On the other hand, if w(-) is any C1 function satisfying w(t) = x, we
have
l(- f w(s)ds] <- [ L(w(s))ds
\t Jo J t Jo
by Jensen’s inequality (§B.l). Thus if we write у = w(0), we find
']+д(.У)< [ b(w(s)) ds + g(y)-,
/	Jo
and consequently
. „ ( r f x — y\ . . I
1tL ~T~ + Г - u(x’e>'
убК" f \ г /	J
3. We have so far shown
u(x, t)
= inf
x-y
t
+ s(y)
and leave it as an exercise to prove the infimum above is really a minimum.
□
We now commence a study of various properties of the function и defined
by the Hopf-Lax formula (21). Our ultimate goal is showing this formula
provides a reasonable weak solution of the initial-value problem (18) for the
Hamilton-Jacobi equation.
First, we record some preliminary observations.
LEMMA 1 (A functional identity). For each x G Rn and 0 < s < t, we
have
(22)	u(x, t) = mm {(t — s)L	+ u(y, s)} .
In other words, to compute u(-, t), we can calculate и at time s and then
use u(-,s) as the initial condition on the remaining time interval [s, t].
Proof. 1. Fix у G Rn, 0 < s < t and choose z G Rn so that
(23)	u(y, s) = sL (У $ Z\ + g(z).
126
3. NONLINEAR FIRST-ORDER PDE
Now since L is convex and = (1 — f) 7—^ + f we have
L	\ t / l <э L о
\ t J \ t / \t — s / t \ s
Thus
x — y\ r (У — z\ , .
----- I + sL I ----j + g(z)
t — S )	\ S )
+ u(y,s),
by (23). This inequality is true for each у G Rn. Therefore, since у н-> u(y, s)
is continuous (according to Lemma 2 below), we have
+ u(y, s)
2. Now choose w such that
(25)	u(x, t) = tL	+ g(w),
and set у := jx + (1 - f) w. Then 7^ =	= jL^L. Consequently
\	I /	VO	V	О
+ u(y, s)
, xr(^ — w\	(y — w\ . .
< (t — s)L I -----j + sL I -----j + </(w)
\	t J \	s J
= tL	+ g(w) = u(x, t),
by (25). Hence
+ u(y, s)| < u(x, t).
(24)
yeR"
v -s J
(26)
□
LEMMA 2 (Lipschitz continuity). The function и is Lipschitz continuous
in Rn x [0,00), and
и = g on Rn x {t - 0}.
3.3. INTRODUCTION TO HAMILTON-JACOBI EQUATIONS
127
Proof. 1. Fix t > 0, x,x G Rn. Choose у G R" such that
(27)	tL +	= u(x, 0-
Then
{r (x — z\ . . ) r (x — y\ , .
tL —-— + g(z) > — tL I —-— - g(y)
\ t j J \ t j
<g(x-x + y)~ g(y) < Lip(^)|x - x|.
Hence
u(x,t) — u(x, t) < Lip(<?)|x — x|;
and, interchanging the roles of x and x, we find
(28)	|u(x, t) — u(x, t)\ < Lip(<?)|x — x|.
2.	Now select x G Rn, t > 0. Choosing у = x in (21), we discover
(29)	u(x, t) < tL(0) + g(x).
Furthermore,
u(x, t) = min tL (	- ) + g(y)
3/€л&п I \ t /	J
> g(x) + minJ - Lip(5)|x - y\ + tL (
убК"	\ i / j
= g(x) - t max{Lip(0)H - L(z)} (z =
z€	I
= g(x) — t max max{w • z — L(z)}
w€B(0,Lip(g)) zeRn
= o(x) — t max H.
B(0,Lip(g))
This inequality and (29) imply
|u(x, t) — <?(x)| < Ct
for
(30)	C := max(|L(0)|, max |Я|).
B(0,Lip(g))
3.	Finally select x G Rn, 0 < t < t. Then Lip(u(-,t)) < Lip(<?) by (28)
above. Consequently Lemma 1 and calculations like those employed in step
2 above imply
|u(x, t) — u{x,t)| < C\t — t|
128
3. NONLINEAR FIRST-ORDER PDE
for the constant C defined by (30).
□
Now Rademacher’s Theorem (which we will prove later, in §5.8.3) asserts
that a Lipschitz function is differentiable almost everywhere. Consequently
in view of Lemma 2 our function и defined by the Hopf-Lax formula (21)
is differentiable for a.e. (x,t) G R" x (0, oo). The next theorem asserts и in
fact solves the Hamilton-Jacobi PDE wherever и is differentiable.
THEOREM 5 (Solving the Hamilton-Jacobi equation). Suppose x G Rn,
t > 0, and и defined by the Hopf-Lax formula (21) is differentiable at a point
(x,t) gR”x (0,oo). Then
uffxff) + H(Du(x,t)) = 0.
Proof. 1. Fix q G Rn, h > 0. Owing to Lemma 1,
r (x + hq — у
hL ----------
\ h
< hL(ff} + u(x, t).
Hence
u(x + hq,t + h) — u(x,t)
< L{q).
h
Let h —> 0+, to compute
q  Du(x, t) + щ(х, f) < L{q}.
This inequality is valid for all q G Rn, and so
(31) uffx,f) + H{Du{x, t)) = ut(x,t) + max{g • Du(x, t) — L(<?)} < 0.
qeRn
The first equality holds since H = L*.
2. Now choose z such that u(x, t) = tL (^f^) + g(z). Fix h > 0 and set
s = t — h, у = ^x + (1 — |) z. Then	and thus
и(ж, t) — u(y, s) > tL	—J + 5(г)
(x — z\
—j~J 
That is,
sL (-—-'j + g(z)
\ s /
u(x, t) - u((l - y) x + t - h) (x — z\
h	\ t J
3.3. INTRODUCTION TO HAMILTON-JACOBI EQUATIONS
129
Let h —> 0+ to compute
x — z	r I x - z
---- Du(x, t) + ufix, t) > L ( —-—
t---\ t
Consequently
ut(x, i) + H(Du(x, t)) = ut(x, t) + max{g • Du(x, t) — L{q)}
geRn
.	. x — z	. r (x — z
> ut(x, t) H----- Du(x, t) — L I —-—
t	\ t
0.
This inequality and (31) complete the proof.
□
We summarize:
THEOREM 6 (Hopf-Lax formula as solution). The function и defined by
the Hopf-Lax formula (21) is Lipschitz continuous, is differentiable a.e. in
Rn x (0, oo), and solves the initial-value problem
ut + H(Du) = 0 a.e. in R" x (0, oo)
и = g on Rn x {t = 0}.
3.3.3. Weak solutions, uniqueness.
a. Semiconcavity.
In view of Theorem 6 above it may seem reasonable to define a weak
solution of the initial-value problem (18) to be a Lipschitz function which
agrees with g on R” x {t = 0}, and solves the PDE a.e. on Rn x (0, oo). How-
ever, this turns out to be an inadequate definition, as such weak solutions
would not in general be unique.
Example. Consider the initial-value problem
(33)
( Ut + |Ux|2 = 0
( и = 0
in R x (0, oo)
on R x {t = 0}.
One obvious solution is
However the function
щ(х, t) = 0.
0
u2(x,t) := <
x — t
if |x| > t
if 0 < x < t
if — t < x < 0
130
3. NONLINEAR FIRST-ORDER PDE
is Lipschitz continuous and also solves the PDE a.e. (everywhere, in fact,
except on the lines x = 0, ±t). It is easy to see that actually there are
infinitely many Lipschitz functions satisfying (33).	□
This example shows we must presumably require more of a weak solution
than merely that it satisfy the PDE a.e. We will look to the Hopf-Lax
formula (21) for a further clue as to what is needed to ensure uniqueness.
The following lemma demonstrates that и inherits a kind of “one-sided”
second-derivative estimate from the initial function g.
LEMMA 3 (Semiconcavity). Suppose there exists a constant C such that
(34)	g(x + zj - 2g(x) + g(x - z) < С|г|2
for all x, z G Rn. Define и by the Hopf-Lax formula (21). Then
u(x + z,t) — 2u(x. t) + u(x — z,t) < C|z|2
for all x, z G Rn, t > 0.
Remark. We say g is semiconcave provided (34) holds. It is easy to check
(34) is valid if g is C2 and supj^n |D2^| < oo. Note that g is semiconcave
if and only if the mapping x i—> g(x) — j|x|2 is concave for some constant
C.	□
Proof. Choose у G Rn so that u(x, t) = tL (^^) + g(y)- Then, putting
у + z and у — z in the Hopf-Lax formulas for u(x + г, t) and u(x — z, t), we
find
+ g(y + z) -2 tL
+ tL
u(x + z,t) — 2u(x, t) + u{x — z, t)
(x — у
------
t
x — y\
~j~j +g(y-z)
= 9(y + z) - 2g(y) -I- g{y - z)
< С|г|2, by (34).
□
As a semiconcavity condition for и will turn out to be important, we
pause to identify some other circumstances under which it is valid. We will
no longer assume g to be semiconcave, but will suppose the Hamiltonian H
to be uniformly convex.
3.3. INTRODUCTION TO HAMILTON-JACOBI EQUATIONS
131
DEFINITION. A C2 convex function H : Rn —> R is called uniformly
convex (with constant 0 > 0) if
n
(35)	22 HPiP. (p)^ > 0|£|2 for all p,£ e Rn
m=i
We now prove that even if g is not semiconcave, the uniform convexity
of H forces и to become semiconcave for times t > 0: this is a kind of mild
regularizing effect for the Hopf-Lax solution of the initial-value problem
(18).
LEMMA 4 (Semiconcavity again). Suppose that H is uniformly convex
(with constant 0) and и is defined by the Hopf-Lax formula (21). Then
u(x + z,t) — 2u(x, t) + u(x — z,t) < ^-|z|2
Ot
for all x,z G Rn, t > 0.
Proof. 1. We note first using Taylor’s formula that (35) implies
11/9
-Я(Р1) + -Я(р2) - g|pi -P2p.
Z	Z	о
(36)
Next we claim that for the Lagrangian L we have the estimate
^b(gi) + -L(g2) < L
z	z
2
(37)
for all qi,q2 G Rn. Verification is left as an exercise.
2.	Now choose у so that u(x,t) = tL (^f2) + g(y). Then using the
same value of у in the Hopf-Lax formulas for u(x + z, t) and u(x — z,t), we
calculate
u(x + z,t) — 2u(x, t) + u(x — z, t)
< (i(£±i^)+9(ri]_2[1L
+ [iL(^p!)+9W]
x + z-y\ I (x-z-y\ (x~y
—Lt { ----------- I — ij I -
2 \ t )	\ t
x — y\ , .
——) +g(y)
= 2t
<2(± -
“ 86» t
t 7
ut
2
the next-to-last inequality following from (37).
132
3. NONLINEAR FIRST-ORDER PDE
b. Weak solutions, uniqueness.
In this section we show that semiconcavity conditions of the sorts dis-
covered for the Hopf-Lax solution и in Lemmas 3 and 4 can be utilized as
uniqueness criteria.
DEFINITION. We say that a Lipschitz continuous function и : Rn x
[О, сю) —> R is a weak solution of the initial-value problem:
, .	( щ + H(Du) = 0 in Rn x (0, сю)
' '	|	и = g on Rn x {t = 0}
provided
(a)	u(x, 0) = g(x) (x € Rn),
(b)	utfX’t') + H(Du(x,tf) — 0 for a.e. (x,t) € Rn x (0, сю),
and
(c)	u(x + z,t) — 2u(x, t) + u(x — z, t) < C (1 + j) |z|2
for some constant C > 0 and all x,z E Rn, t > 0.
Next we prove that a weak solution of (38) is unique, the key point being
that this uniqueness assertion follows from the inequality condition (c).
THEOREM 7 (Uniqueness of weak solutions). Assume H is C2 and sat-
isfies (19), and g satisfies (20). Then there exists at most one weak solution
of the initial-value problem (38).
Proof*. 1. Suppose that и and й are two weak solutions of (38) and write
w := и — u.
Observe now at any point (y, s) where both и and й are differentiable
and solve our PDE, we have
wt(y, s) = ut(y, s) - ut(y, s)
= —H(Du(y, s)) + H(Du(y, s))
f1 d
= — I —HlrDutyjSj + il — rjDufyjSyidr
Jo dr
= — I DH(rDu(y, s) + (1 — r)Du(y, s)) dr  (Du(y, s) — Du(y, s))
Jo
=: —b(y,s)  Dw(y,s).
Consequently
(39)	wt + b • Dw = 0 a.e.
*Omit on first reading.
3.3. INTRODUCTION TO HAMILTON-JACOBI EQUATIONS
133
2.	Write v fit™) > 0, where <f> : R —> [0, oo) is a smooth function to
be selected later. We multiply (39) by 0'(w) to discover
(40)	vt + b • Dv = 0 a.e.
3.	Now choose £ > 0 and define uE :=	* и, йЕ :=	* й, where т]Е is the
standard mollifier in the x and t variables. Then according to §C.4
(41)	l-D^I < Lip(u), |£)й£| < Lip(fi),
and
(42)	DuE —» Du, DuE —> Du a.e., as t-+0.
Furthermore inequality (c) in the definition of weak solution implies
(43)	D2ue, D2ue < C (1 + J) I
for an appropriate constant C and all £ > 0, у € Rn, s > 2£. Verification is
left as an exercise.
4.	Write
(44)	be(y,s):= [ DH(rDuE(y,s) + (1 — r)Du£(y,s)) dr.
Jo
Then (40) becomes
Vt + be • Dv = (be — b)  Dv a.e.;
hence
(45)	vt + div(vb£) = (divbe)z> + (be — b) • Dv a.e.
5.	Now
-1 n
divb£ = / 52 HPkPi(rDuE + C1 - r)D^)(^4;xfc + (1 - r)uEXlXk) dr
k,l=l
(46)	< C (1 + - )
\ s J
for some constant C, in view of (41), (43). Here we note that H convex
implies D2H > 0.
134
3. NONLINEAR FIRST-ORDER PDE
6.	Fix xq € Rn, to > 0, and set
(47)	R := max{|D77(p)| | |p| < max(Lip(u),Lip(ri))}.
Define also the cone
C := {(x, t) | 0 < t < to, - ®o| < R(to — t)}.
Next write
e(t) = I	v(x,t)dx
J B(x0,B(t0-t))
and compute for a.e. t > 0:
e(t) = I	Vfdx — R	vdS
J B(xo,R(to— t))	JdB(xo,R(to—t))
= /	— div(vbe) + (divbe)z> + (be — b) • Dv dx
-	R [	vdS by (45)
JdB(xo,R(to-ty)
= — [	v(be • v + .R) dS
J dB(xo,R(to—t')')
+ /	(div be)z> + (be — b) • Dv dx
JB(x0,R(t0-t))
<	I	(divbe)z> + (be — b) • Dvdx by (41), (44)
J B(xo,R(to—t))
<	C (1 +	) e(t) + i	(be — b) • Dv dx
X &/	J B(x0,R(t0-t))
by (46). The last term on the right hand side goes to zero as £ —> 0, for a.e.
to > 0, according to (41), (42) and the Dominated Convergence Theorem.
Thus
(48)	e(t) < C ^1 + e(t) for a.e. 0 < t < to-
7.	Fix 0 < £ < r < t and choose the function 0(z) to equal zero if
|z| < £[Lip(u) + Lip(ii)]
and to be positive otherwise. Since и = й on Rn x {t = 0},
v = (f>(w) = ф(и — й) = 0 at {t = e}.
3.3. INTRODUCTION TO HAMILTON-JACOBI EQUATIONS
135
Thus e(e) = 0. Consequently Gronwall’s inequality (see §B.2) and (48)
imply
e(r) < е(£)е^Гс(1+«)^ = 0.
Hence
|u — й| < e[Lip(u) + Lip(-il)] on B(xq, R(to — r)).
This inequality is valid for all e > 0, and so и = й in B(xo,R(to — r)).
Therefore, in particular, u(xo,to) — vfxQ,to).	□
5(?/)|
(49)
|y|| •
In light of Lemmas 3, 4 and Theorem 7, we have
THEOREM 8 (Hopf-Lax formula as weak solution). Suppose H is C2
and satisfies (19), and g satisfies (20). If either g is semiconcave or H is
uniformly convex, then
/x — y\
tL ------ 4-
\ t J
is the unique weak solution of the initial-value problem (38) for the Hamilton-
Jacobi equation.
Examples, (i) Consider the initial-value problem:
щ 11Du|2 = 0 in Rn x (0, oo)
и = |x| on Rn x {t = 0}.
Here H(p) = ||p|2 and so L(q) = ||g|2. The Hopf-Lax formula for the
unique, weak solution of (49) is
(50)	u(x, t) = min I ————h
v	v ’ j/eR" [ 2t
Assume |x| > t. Then
\ 2t J t |y|
and this expression equals zero if x = у + -^t, у = (|x| — t)^	0. Thus
u(x,t) = |x| — | if |x| > t. If |ж| < t, the minimum in (50) is attained at
у = 0. Consequently
136
3. NONLINEAR FIRST-ORDER PDE
Observe that the solution becomes semiconcave at times t > 0, even though
the initial function g(x) = |x| is not semiconcave. This accords with Lemma
4.
(ii) We next examine the problem with reversed initial conditions:
(51)
in Rn x (0, oo)
on IR” x {t = 0}.
Then
u(x,t) = min f --------|y|l.
v 7 yeR« ( 2t 1 J
Now
and this equals zero if x = у — fyt, у = (|x| + Thus
u(x,t) = —|x| - (x € Rn, t > 0).
The initial function g(x) = — |ж| is semiconcave, and the solution remains so
for times t > 0.	□
In Chapter 10 we will again study Hamilton-Jacobi PDE and discover
another notion of weak solution, which is applicable even if H is not convex.
3.4.	INTRODUCTION TO CONSERVATION LAWS
In this section we investigate the initial-value problem for scalar conservation
laws in one space dimension:
(1)
Щ + F(u)x = 0
и = g
in IR x (0, oo)
on IR x {t = 0}.
Here F : IR —> IR and g : IR —» IR are given and и : IR x [0, oo) —> IR is
the unknown, и = u(x,t). As noted in §3.2, the method of characteristics
demonstrates that there does not in general exist a smooth solution of (1),
existing for all times t > 0. By analogy with the developments in §3.3.5, we
therefore look for some sort of weak or generalized solution.
3.4. INTRODUCTION TO CONSERVATION LAWS
137
3.4.1.	Shocks, entropy condition.
a.	Distribution solutions; Rankine—Hugoniot condition.
We open our discussion by noting that since we cannot in general find a
smooth solution of (1), we must devise some way to interpret a less regular
function и as somehow “solving” this initial-value problem. But as it stands,
the PDE does not even make sense unless и is differentiable. However,
observe that if we temporarily assume и is smooth, we can as follows rewrite,
so that the resulting expression does not directly involve the derivatives of
u. The idea is to multiply the PDE in (1) by a smooth function v and then
to integrate by parts, thereby transferring the derivatives onto v.
More precisely, assume
( v : R x [0, oo) —> R is smooth, with
(2)	5
[ compact support.
We call v a test function. Now multiply the PDE Ut -I- F(u)x = 0 by v and
integrate by parts:
0 = i i (щ + F(u)x)v dxdt
JO J—oo
/•OO /*OO	/*OO
(3)	= - /	/ uvt dxdt— / uvdx\t=o
Jo J—oo	J—oo
roc roo
— I / F(u)vx dxdt.
Jo J—oo
In view of the initial condition и = g on R x {t = 0}, we thereby obtain the
identity
/•OO roo	yoo
(4)	/	/ uvt -I- F(u)vx dxdt + gv dx|t=o = 0.
J0 J—oo	J—oo
We derived this equality supposing и to be a smooth solution of (1), but
the resulting formula has meaning even if и is only bounded.
DEFINITION. We say that и € L°°(R x (0, oo)) is an integral solution
of (1), provided equality (4) holds for each test function v satisfying (2).
Suppose then that we have an integral solution of (1). What can we
deduce about this solution from the identities (4)?
We partially answer this question by looking at a situation for which u,
although not continuous, has a particularly simple structure. Let us in fact
138
3. NONLINEAR FIRST-ORDER PDE
suppose in some open region V C R X (0, oo) that и is smooth on either side
of a smooth curve C. Let Vi be that part of V on the left of the curve and
Vr that part on the right. We assume that и is an integral solution of (1),
and that и and its first derivatives are uniformly continuous in Ц and in Vr.
First of all, choose a test function v with compact support in VJ. Then
(4) becomes
/•OO /»ОО	roo 1*00
(5)	0= I / uvt + F(u)vx dxdt = — /	/ (ut + F(u)x]v dxdt,
JO J—oo	Jo J—oo
the integration by parts being justified since и is C1 in Ц and v vanishes
near the boundary of VJ. The identity (5) holds for all test functions v with
compact support in V), and so
(6)	щ + F(u)x = 0 in Vi.
Likewise,
(7)	ut + F(u)x = 0 in Vr.
Now select a test function v with compact support in V, but which does
not necessarily vanish along the curve C. Again employing (4), we deduce
1*00 roo
0= I I uvt + F(u)vxdxdt
JO J—oo
(8)	J[ uvt + F(u)vx dxdt
+ УУ uvt + F(u)vx dxdt.
Now since v has compact support within V, we have
// uvt + F(u)vxdxdt = — II \ut + F(u)x]v dxdt
J Jvt	J Jvi
(9)	-I- f (щи2 + F(ui)v1)vdl
Jc
= [ (щи2 + F(ui')i'1)v dl
Jc
in view of (6). Here и = (и1, и2) is the unit normal to the curve C, point-
ing from Vi into Vr, and the subscript “I” denotes the limit from the left.
Similarly, (7) implies
// uvt + F(u)vx dxdt — — / (uru2 + F(ur)u1)vdl,
JJvr	Jc
3.4. INTRODUCTION TO CONSERVATION LAWS
139
Rankine-Hugoniot condition
the subscript “r” denoting the limit from the right. Adding this identity to
(9) and recalling (8) gives us:
[ [(F(ui) - Е^игУ)!/1 + (ui - ur)p2]vdl = 0.
Jc
This equality holds for all test functions v as above, and so
(10)	(F(rq) — F(ur))p1 4-(tq — ur)p2 = 0 along C.
Now suppose C is represented parametrically as {(x, t) | x = s(t)} for
some smooth function s(-) : [0, oo) —» IR. We can then take v = (p1,!/2) =
(1 + s2)-1/2(l, — s). Consequently (10) implies
(11)	F(ui) - F(ur) = s(ui - Ur)
in V, along the curve C.
Notation.
{[[u]] = ui — ur = jump in и across the curve C
[[F(u)]] = F(ui) — F(ur) = jump in F(u)
a = s = speed of the curve C.
□
140
3. NONLINEAR FIRST-ORDER PDE
Let us then rewrite (11) as the identity
(12)
[№)]] = ф]]
along the discontinuity curve. This is the Rankine-Hugoniot condition. Ob-
serve that the speed a and the values щ, ur, F(ui), F(ur) will generally vary
along the curve C. The point is that even though these quantities may
change, the expressions [[^(u)]] = F(ui) — F(ur) and <r[[u]] = s(ui — ur)
must always exactly balance.
Example 1 (Shock waves). Let us consider the initial-value problem for
Burgers’ equation:
(13)
и = g
in IR x (0, oo)
on IR x {t = 0},
with the initial data
(14)
p(®) = <
1 — X
0
if x < 0
if 0 < x < 1
if x > 1.
According to the characteristic equations (cf. §3.2.5) any smooth solution
и of (13), (14), takes the constant value z° = g(x°) along the projected
characteristic
y(s) = (5(x°)s+ x°,s) (s > 0)
for each x° € IR. Thus
u(x,t) := <
1
l—x
1-t
0
if x<tyQ<t<l
if t <x <1, 0 < t < 1
if x > 1, 0 < t < 1.
Observe that for t > 1 this method breaks down, since the projected
characteristics then cross. So how should we define и for t > 1?
Let us set s(t) = lyl, and write
1 if x < s(t)
0 if s(t) < x
if t > 1. Now along the curve parameterized by s(-), ui = 1, we have ur = 0,
F(iq) = |(uz)2 = F(ur) = 0. Thus [[E(u)]j = | = <r[[u]], as required by
the Rankine-Hugoniot condition (12).	□
3.4. INTRODUCTION TO CONSERVATION LAWS
141
Formation of a shock
b.	Shocks, entropy condition.
We try now to solve a similar problem by the same techniques.
Example 2 (Rarefaction waves and nonphysical shocks). Again consider
the initial-value problem (13), for which now we take
0
1
(15)
if x
if x
< 0
> 0.
The method of characteristics this time does not lead to any ambiguity
in defining u, but does fail to provide any information within the wedge
{0 < x < t}. To illustrate this lack of knowledge, let us first set
ui(x,t)
JO
I 1
if x <
if x >
t
2
f
2
It is easy to check that the Rankine-Hugoniot condition holds and, indeed,
that и is an integral solution of (13), (15). However, we can create another
such solution by writing
U2(<r,t)
1	if x > t
| if 0 < x < t
0	if x < 0.
The function U2, called a rarefaction wave, is also a continuous integral
solution of (13), (15).	□
142
3. NONLINEAR FIRST-ORDER PDE
Rarefaction wave
Thus we see that integral solutions are not in general unique. Presum-
ably the class of integral solutions includes various “nonphysical” solutions,
which we want somehow to exclude. Can we find some further criterion
which ensures uniqueness?
Entropy condition. Let us recall from §3.2.5 that for the general scalar
conservation law of the form
ut + F(u)x = 0,
the solution u, whenever smooth, takes the constant value z° = g(x°) along
the projected characteristic
(16)	y(s) = (F'(g(x°))s + x°, s) (s > 0).
Now we know that typically we will encounter the crossing of characteristics,
and resultant discontinuities in the solution, if we move forward in time.
However, we can hope that if we start at some point in Rn x (0, oo) and
3.4. INTRODUCTION TO CONSERVATION LAWS
143
go backwards in time along a characteristic, we will not cross any others.
In other words, let us consider the class of, say, piecewise-smooth integral
solutions of (1) with the property that if we move backwards in t along any
characteristic, we will not encounter any lines of discontinuity for u.
So now suppose at some point on a curve C of discontinuities that и has
distinct left and right limits, ui and ur, and that a characteristic from the
left and a characteristic from the right hit C at this point. Then in view of
(16) we deduce
(17)
F\ui) ><r> F'(ur).
These inequalities are called the entropy condition (from a rough analogy
with the thermodynamic principle that physical entropy cannot decrease as
time goes forward). A curve of discontinuity for и is called a shock provided
both the Rankine-Hugoniot identity (12) and the entropy inequalities (17)
hold.
Let us further interpret the entropy condition under the additional as-
sumption that
(18)
F is uniformly convex.
This means F" > в > 0 for some constant 0. Thus in particular F' is strictly
increasing. Then (17) is equivalent to our requiring the inequality
(19)
ui > ur
along any shock curve.
□
Example 3. We again return to Burgers’ equation (13), now for the initial
function
0
1
0
p(®) =
(20)
if	x	<	0
if	0	<	x <	1
if	x	>	1.
For 0 < t < 2, we may combine the analysis in Examples 1 and 2 above
to find
{0	if	x	<	0
—	if	0	<	rr	<	/
1	if	t<x<l+-
X 11 L ''•s.	X i- 2
0	if	x	>	1	+	|
144
3. NONLINEAR FIRST-ORDER PDE
the entropy condition (19). We calculate the behavior of the shock curve by
applying the Rankine-Hugoniot jump condition (12). Now
.. .. S(t) ..	. \11	1 / ®(^)\	• / \
[[«]] =	[[F(«)]] = 9 (
C	£ \ v J
along the shock curve for t > 0. Thus (12) implies
s(f) = ^ (<>2).
AV
Additionally s(2) = 2, and so we can solve this ODE to find s(t) = (2t)1//2
(t > 2). Hence we may augment (21) by setting
0
X
t
0
u(x,t) = <
if x < 0
if 0 < x < (2*)1/2
if x > (2г)1/2
(t > 2).
See the illustration.
□
3.4.2. Lax—Oleinik formula.
We now try to obtain a formula for an appropriate weak solution of
the initial-value problem (1), assuming as above that the flux function F is
uniformly convex. With no loss of generality we may as well also take
(22)
F(0) = 0.
3.4. INTRODUCTION TO CONSERVATION LAWS
145
As motivation, suppose now g € L°°(R) and define
(23)	h(x) := [ g(y)dy (x € R).
Jo
Recall the Hopf-Lax formula from §3.3 and set
(24) w(x, t) := min < tL ( -—- ) + 7i(y) >	(ж € K, t > 0),
j/eR I	\ t J J
where
(25)
L = F*.
Thus w is the unique, weak solution of this initial-value problem for the
Hamilton-Jacobi equation:
(26)
( wt + F(wx) = 0
(	w = h
in R x (0, oo)
on R x {t = 0}.
For the moment assume w is smooth. We now differentiate the PDE
and its initial condition with respect to x, to deduce
H” E^'Wx'jx — 6
wx = g
in R x (0, oo)
on R x {t = 0}.
Hence if we set и = wx, we discover и solves problem (1).
The foregoing computation is only formal, as we know that w defined
by (24) is not in general smooth. But recall from §3.3 that w is in fact
differentiable a.e. Consequently
(27)	u(x,t)
min [tL f -—+ h(y) I
is defined for a.e. (x, t) and is presumably a leading candidate for some sort
of weak solution of the initial-value problem (1). Our intention henceforth
is to justify this expectation.
First, we will need to rewrite the expression (27) into a more useful form.
Notation. Since F is uniformly convex, F' is strictly increasing and onto.
Write
(28)
for the inverse of F'.
G:= (F')-1
□
146
3. NONLINEAR FIRST-ORDER PDE
THEOREM 1 (Lax-Oleinik formula). Assume F : R —> R is smooth,
uniformly convex, and g € L°°(R).
(i)	For each time t > 0, there exists for all but at most countably many
values of x € R a unique point y(x,t) such that
min [tL ( ——+ hfy) | — tL (-—+ h(y(x, t)).
I \ t / J \ t )
(ii)	The mapping x i-> y(x, t) is nondecreasing.
(iii)	For each time t > 0, the function и defined by (27) is
(29)
for a.e. x. In particular, the formula (29) holds for a.e. (x,t) €
Rn x (0,oo).
DEFINITION. We call equation (29) the Lax-Oleinik formula for the
solution (1), where h is defined by (23), L by (25).
Proof. 1. First, we note
L(g) = max (qp - F(pf) = qp* - F(p*fi
peR
where F'(p*) = q. But then p* = G(q) according to (28), and so
L(q) = qG(q)-F(G(q)) (q G R)
(cf. §3.3.1). In particular, L is C2. Furthermore
(30)	L'(q) = G(q) + qG'(q) - F' (G(q))G'(q) = G(q)
by (28); and L"(q) = G'(q) > 0. This and (22) imply L is nonnegative and
strictly convex.
2.	Fix t > 0, Xi < X2- As in §3.3 there exists at least one point y\ G R
such that
(31)	[tL	+/i(yi)| = min (tL	+h(y)
t X *	/ J I \ ^ /
We next claim
(32)	tL f^2	+ h(?/i) < tL f^2 y\+h(y) ify<yi.
\ L /	\ L j
3.4. INTRODUCTION TO CONSERVATION LAWS
147
To see this, we calculate X2 — yi = ^(xi — yi) + (1 — t\x2 — y) and xi — у =
(1 - t)(xi - yi) + t(x2 - y) for
n /	У1-У ,
0 < т := ----------- < 1.
X2 - X! + yi - у
Since L" > 0, we thus have
т(х2-уЛ	(xt - yA	(x2-y\
\ t J	\t)	\ t J
T fxl-y\ xr (x1-y1\	(x2-y\
\ t J \ t J \ t J
and hence
(зз)
\ t j \ t j \ t j \ t j
Now notice from (31) that
 г /Ж1 - У1А . , / \ / ,r (x\ - y\ , , .
tZ I —-— I + /i(yi) < tL I —-— I + h(y).
We multiply (33) by t, add 7i(yi) + Л(у) to both sides, and add the
resulting expression to the above inequality to obtain (32).
3.	In view of (31), in computing the minimum of tL + h(y) we
need only consider those у > yi, where yi satisfies (31). Now for each x € R
and t > 0, define the point y(x,t} to equal the smallest of those points у
giving the minimum of tL (^f^) + h(y). Then the mapping x i—► y(x,t) is
nondecreasing and is thus continuous for all but at most countably many x.
At a point x of continuity of y(-,t), y(x,t) is the unique value of у yielding
the minimum.
4.	According to the theory developed in §3.3 for each fixed t > 0, the
mapping
x i-> w(x, t) := min < tL ( -—- j + /i(y) >
yeR (	\ t ) J
is differentiable a.e. Furthermore the mapping x i—> y(x, t) is monotone and
consequently differentiable a.e. as well. Thus given t > 0, for a.e. x the
mappings x i—> Z(I~^1-^) and so also x i—> /i(y(x, t)) are differentiable as
well.
148
3. NONLINEAR FIRST-ORDER PDE
Consequently formula (27) becomes
= ~&c tL	+ h(y(x,t))
r/ f x — y(x,t)\	,	..	d,, ,
= L ------------- I (1 - yx(x,tfi) + — h(y(x,t)).
\ t J	ox
But since у i—> tL	+ h(y) has a minimum at у = y(x, t), the mapping
z и-> tL ( x y^z,t^ j + h(y(z,tf) has a minimum at z = x. Therefore
~L'	—У&, t))	+ ^h(y(x ty) = Q,
and hence
u(l,t) = i' f= G (x~^
X	У \ t
according to (30).
We now investigate the precise sense in which formula (29) provides us
with a solution of the initial-value problem (1).
THEOREM 2 (Lax-Oleinik formula as integral solution). Under the as-
sumptions of Theorem 1, the function и defined by (29) is an integral solution
of the initial-value problem (1).
Proof. As above, define
w(x, t) = min < tL
v ’ i/eRl
(x € R, t > 0).
Then Theorem 6 in §3.3.2 tells us w is Lipschitz continuous, is differentiable
for a.e. (x,t), and solves
(34)
f wt + F(wx) = 0
(	w = h
a.e. in К x (0, oo)
on® x {i = 0}.
Choose any test function v satisfying (2). Multiply the PDE wt +
F(wx) = 0 by vx and integrate over R x (0, oo):
(35)
/•OO /*OO
0 = /	/ [wt + F(wx)] vx dxdt.
J0 J—oo
3.4. INTRODUCTION TO CONSERVATION LAWS
149
Observe
These integrations by parts are valid since the mapping x i—> w(x, t) is
Lipschitz continuous, and thus absolutely continuous, for each time t > 0.
Likewise 1i—> w(x, t) is absolutely continuous for each x € R. Now w(x, 0) =
/i(x) = fa g(.y)dy, and so wx(x, 0) = g(x) for a.e. x. Consequently
wtvx dxdt =
Substitute this identity into (35) and recall и = wx a.e., to derive the integral
identity (4).	□
3.4.3. Weak solutions, uniqueness.
a. Entropy condition revisited.
We have already seen in §3.4.1 that integral solutions of (1) are not
generally unique. Since we believe the Lax-Oleinik formula does in fact
provide the “correct” solution of this initial-value problem, we must see if it
satisfies some appropriate form of the entropy condition discussed in §3.4.1.
This is not straightforward, however, since it is not usually the case that the
function и defined by the Lax-Oleinik formula is smooth, or even piecewise
smooth.
We identify now a kind of “one-sided” derivative estimate for the func-
tion и defined by the Lax-Oleinik formula (27). This estimate—which is
an analogue for conservation laws of the semiconcavity estimate from Lem-
mas 3, 4 in §3.3.3 for Hamilton-Jacobi equations—will turn out to be a
uniqueness criterion.
LEMMA (A one-sided jump estimate). Under the assumptions of The-
orem 1, there exists a constant C such that the function и defined by the
Lax-Oleinik formula (29) satisfies the inequality
(36)
u(x + z,t) — u(x,t) <
C
—z
for all t > 0 and x, z E R, z > 0.
150
3. NONLINEAR FIRST-ORDER PDE
DEFINITION. We call inequality (36) the entropy condition.
It follows from (36) that for t > 0 the function x i—> u(x,t) — yX is
nonincreasing, and consequently has left and right hand limits at each point.
Thus also x i-> u(x,t) has left and right hand limits at each point, with
ui(x,t) > ur(x,t). In particular, the original form of the entropy condition
(19) holds at any point of discontinuity.
Proof. We know from §3.3 that in computing the minimum in (29) we need
only consider those у such that < C for some constant C; verification
is left to the reader. Consequently we may assume, upon redefining G if
necessary off some bounded interval, that G is Lipschitz continuous.
2. As G = (F')~1 and y(-,t) are nondecreasing, we have
>g(L-«^ + ^\ for2>0
_ \ t J
f X + z - y(x + z,t)\ Lip(G)z
(j I ------------- I----------
_ \ t J t
.	4 Lip(G)2
= u(x + z,t)-------.
□
b. Weak solutions, uniqueness.
We now establish the important assertion that an integral solution which
satisfies the entropy condition is unique.
DEFINITION. We say that a function и € L°°(R x (0, oo)) is an entropy
solution of the initial-value problem
( ut + F(u)x = 0 in R x (0, oo)
[	и = g on Rn x {t = 0}
provided
/•oo ЛОО	/*OO
(i)	/	/ uvt + F(u)vx dxdt + / 5vdz|t=o = 0
•/0 J —oo	J —oo
for all test functions v : R x [0, oo) —> R with compact support, and
(ii)	u(x + z,t) — u(x, t) < C(1 -I-
for some constant C > 0 and a.e. x, z E R, t > 0, with г > 0.
3.4. INTRODUCTION TO CONSERVATION LAWS
151
THEOREM 3 (Uniqueness of entropy solutions). Assume F is convex
and smooth. Then there exists—up to a set of measure zero—at most one
entropy solution of (37).
Proof*. 1. Assume that и and й are two entropy solutions of (37), and
write w := и — й. Observe for any point (x,t) that
f1 d
F(u(x,t)) - F(u(x,t)) = / —F(ru(x,t) + (1 — r)u(x, t)) dr
Jo dr
= / F'(ru(x, t) + (1 — r)S(x, t)) dr (u(x, t) — u(x, t))
Jo
=: b(x,t)w(x,t').
Consequently if v is a test function as above,
0 = [ f (u — u)vt + [F(u) — F(u)]vxdxdt
/ OQ\	*0 OO
roo roc
= /	/ w[vt + bvx] dxdt.
Jo J—oo
2.	Now take e > 0 and define uE = т]£ * и, u£ = r/£ * u, where t]£ is the
standard mollifier in the x and t variables. Then according to §C.4
(39)	K||L~ < llulkoo, ||ue||/,oo <
(40)	uE —> и, йЕ —> u a.e., as e —> 0.
Furthermore the entropy inequality (ii) implies
(41)	uEx(x,t), u£x(x,t) < C (1 + 0
for an appropriate constant C and all £ > 0, x € R, t > 0.
3.	Write
b£(x, t) := f F\ruE(x,t) + (1 — r)uE(x,t)) dr.
Jo
Then (38) becomes
(42)	0 = [ i w[vt + b£vx] dxdt + [	[ w[b — b£]vx dxdt.
Jo J—oo	Jo J—oo
*Omit on first reading.
152
3. NONLINEAR FIRST-ORDER PDE
4.	Now select T > 0 and any smooth function ф : R x (0,T) —> R
with compact support. We choose v to be the solution of the following
terminal-value problem for a linear transport equation:
vf + b£vx = <ф in R x (0, T)
v = 0 on R x {t = T}.
Let us solve (43) by the method of characteristics. For this, fix x G R,
0 < t < T, and denote by a;e(-) the solution of the ODE
ie(s) = be(xe(s),s) (s > t)
x£(t) = x,
and set
(45)	v£(x, t) := — у ф(хе(з), s) ds (x e R, 0 < t < T).
Then v£ is smooth and is the unique solution of (43). Since |be| is
bounded and ф has compact support, v£ has compact support in R x [0, T).
5. We now claim that for each s > 0, there exists a constant Cs such
that
(44)
(46)	|v®| < onRx(s,T).
To prove this, first note that if 0 < s < t < T, then
be,x(x,t) = [ F"(ru£ + (1 - r)u£)(ru£x + (1 - r)uex) dr
(47)
V 7	C C
-7-7
by (41), since F is convex.
Next, differentiate the PDE in (43) with respect to x:
(48)	v£tx + bevxx + be,xvx = фх.
Now set a(a;,t) := extvx(x,t), for
(49)	A = f + !•
Then
o-t + b£ax = Xa + 7*
(50)	= Xa + ext[-b£,xvx + фх\ by (48)
= [A - b£,x]a + емфх.
3.4. INTRODUCTION TO CONSERVATION LAWS
153
Since v£ has compact support, a attains a nonnegative maximum over
R x [s, T] at some finite point (a?o, to)- If to = T, then vx = 0. If 0 < to < T,
then
at(®o,to) < 0, ax(xo,to) = 0.
Consequently equation (50) gives
(51)	[A - be,x]a + ext°ipx < 0 at (a?o,to)-
But since b£,x < j and A is given by (49), inequality (51) implies
a(xo,to) < ~eXt°ipx < eXT|\фх\\L™.
A similar argument shows
a(xi,ti) > —eA7"||-0x||b°=>
at any point (cci,ti) where a attains a nonpositive minimum. These two
estimates and the definition of a imply (46).
6.	We will need one more inequality, namely
/ОО
|i4(<r,t)|<fa: < D
-oo
for all 0 < t < r and some constant D, provided r is small enough.
To prove this, choose т > 0 so small that ф = 0 on R x (0, r). Then if
0 < t < r, we see from (45) that v is constant along the characteristic curve
a;e(-) (solving (44)) for t < s < r. Select any partition a?o < xi < • • • < x^.
Then t/o < У1 < • • • < Un, where уг := x^s) (г = 1,..., IV) for
( ie(s) = b£(x£(s),s) (t<s<r)
( X£(t) = Xi-
As ve is constant along each characteristic curve а?г(-), we have
N	N
-ve(xi-i,t)\ = ^\ve(yi,r) -U£(t/j-l,r)|
1=1	i=l
< var ve(-, r),
“var” denoting variation with respect to x. Taking the supremum over all
such partitions, we find
/ОО	roc
|u£(a;,t)| dx = varve(-,t) < varve(-,r) = I |и£(а;,т)| dx < C,
•oo	«/ —oo
154
3. NONLINEAR FIRST-ORDER PDE
since v£ has constant support and estimate (41) is valid for s = r.
7.	Now, at last, we complete the proof by setting v = v£ in (42) and
substituting, using (43):
Then in view of (40), (46), and the Dominated Convergence Theorem,
I£ —> 0 as e —> 0
for each r > 0. On the other hand, if 0 < т < T, we see
/ОО
\vex\dx<rC, by (52).
oo
Thus
wip dxdt = 0
for all smooth functions ip as above, and so w = и — й = 0 a.e.
□
3.4.4. Riemann’s problem.
The initial-value problem (1) with the piecewise-constant initial function
(53)
ui
if x < 0
if x > 0
is called Riemann’s problem for the scalar conservation law (1). Here щ, ur €
R are the left and right initial states, щ ф ur
We continue to assume F is uniformly convex and C2, and as before we
write G = (F')"1.
THEOREM 4 (Solution of Riemann’s problem).
(i) If щ > ur, the unique entropy solution of the Riemann problem (1),
(53) is
(54)
u(x,t) :=
ui
if I
if I
< cr
> cr
(x G R, t > 0),
3.4. INTRODUCTION TO CONSERVATION LAWS
155
where
Shock wave solving Riemann’s problem for ui>ur
(55)
_ F(uz) - F(ur)
щ — ur
(ii) If ui < ur, the unique entropy solution of the Riemann problem (1),
(53) is
Щ if f < F'(uj)
(56)
u(x,t):={ G(f) z/F'(uz)<f
< F'(ur)
(a; € R, t > 0).
ur if j > F'(ur)
Remarks, (i) In the first case the states щ and ur are separated by a shock
wave with constant speed a. In the second case the states щ and ur are
separated by a rarefaction wave.
(ii) We know from the theory set forth in §§3.4.2-3.4.3 that the Lax-
Oleinik formula must generate these solutions, and it is an interesting ex-
ercise to verify this directly. We will instead construct the functions (54),
(56) from first principles and verify they are in fact entropy solutions. By
uniqueness, then, they must agree with Lax-Oleinik formulas. This is a nice
illustration of the power of the uniqueness assertion, Theorem 3.	□
Proof. 1. Assume щ > ur. Clearly и defined by (54), (55) is then an inte-
gral solution of our PDE. In particular since a = [[F(u)]]/[[u]], the Rankine-
Hugoniot condition holds. Furthermore note
F'(ur) < a =	= Г F'(r)dr < F'(ui)
Ul Ur J ur
156
3. NONLINEAR FIRST-ORDER PDE
Rarefaction wave solving Riemann’s problem for U]<ur
in accordance with (17). Since щ > ur, the entropy condition holds as well.
Uniqueness follows from Theorem 3.
2. Assume now that ui < ur. We must first check that и defined by (56)
solves the conservation law in the region {F'(tq) < f < F'(ur)}. To verify
this, let us ask the general question as to when a function и of the form
u(x,t) = v(—j
\t '
solves (1). We compute
ut + F(u)x = ut + F'(u)ux
,(x\ x . .(x\ 1
= —v I — I -о + F (v)v I — I -
\t)t2 v ' \t) t
,{X\ 1 Г ,, .	£1
= <?);И>)-7].
Thus, assuming v' never vanishes, we find F' (v(|)) = y. Hence
.	.	fx\
u(x,t) = v — = G —
v	’	\t)
solves the conservation law. Now v(^) = щ provided у = F'(iq); and
similarly v(^) = ur if f = F'(ur).
As a consequence we see that the rarefaction wave и defined by (56) is
continuous in R x (0, oo), and is a solution of the PDE щ + F(u)x = 0 in
each of its regions of definition. It is easy to check that и is thus an integral
solution of (1), (53). Furthermore, since as noted in §3.4.3 we may as well
assume G is Lipschitz continuous, we have
+	_ fx\ Lip(G)z
----j — G (—) <---------
t J \ t f t
3.4. INTRODUCTION TO CONSERVATION LAWS
157
if F'(u/)t < x < x + z < F'(ur)t. This inequality implies that и also satisfies
the entropy condition. Uniqueness is once more a consequence of Theorem 3.
□
3.4.5. Long time behavior.
a. Decay in sup-norm.
We now employ the Lax-Oleinik formula (29) to study the behavior of
our entropy solution и of (1) as t —► oo. We assume below that F is smooth,
uniformly convex, F(0) = 0, and g is bounded and summable.
THEOREM 5 (Asymptotics in L°°-norm). There exists a constant C such
that
(57)	|u(z, t)| <
for all x G R, t > 0.
Proof. 1. Set
(58)	ст := F'(0);
then
(59)	G(ct) = 0,
and therefore
(60)	L(ct) = ctG(ct) - F(G(ct)) = 0, L'(ct) = 0.
2.	In view of (60) and the uniform convexity of L,
(x — y\ r (x — у — at \
-------- \ =tL[ ------------+ ст
t J \ t )
Г	/	.\	/	.\ 2"
, r i s ,Z/ s I x — у — at\	n (x — у — at\
(61)	> t L(a) + L'(ct) ------------ +0 ---------------
\ c /	\ c /
gk-y-CTti2
t
for some constant в > 0. Since h = fq gdy is bounded by M := we
see from (61) that
tL (izJl] + Л(„) >	- M.
\ t/ J
158
3. NONLINEAR FIRST-ORDER PDE
On the other hand,
r (x — (x — at) \	r
tL (------------ I + h(x — at) < M.
Thus at the minimizing point y(x,t) we have
|x -»(M) -otp 2M
t
and so
(62)
X - y(x,t)
t
for some constant C.
3.	But since G(a) = 0, for any x 6 R, t > 0 we have
according to (62).
□
Example 3 in §3.4.1 shows this t x/2 decay rate to be optimal.
b. Decay to N-wave.
Estimate (57) asserts that the £°°-norm of и goes to zero as t —> oo. On
the other hand we note from Example 3 in §3.4.1 that the T^-norm of и need
not go to zero; indeed, the integral of и over R is conserved (Problem 13).
We instead show here that и evolves in L1 into a simple shape, assuming
now that
g has compact support.
Given constants p, q, d, a, with p, q > 0, d > 0, we define the correspond-
ing N-wave to be the function
NT(	if -(pdt)1/2 < X - at < (qdt)1/2
(	0 otherwise.
3.4. INTRODUCTION TO CONSERVATION LAWS
159
TV-wave
The constant cr is the velocity of the TV-wave.
Now define cr by (58), set
(64)	d := F"(0) > 0,
and also write
/У	r<x>
qdx, o:=2max / gdx.
yeRjy
Note p, q > 0 and
(66)	G'(a) = 1
a
THEOREM 6 (Asymptotics in ZAnorm). Assume that p,q > 0. Then
there exists a constant C such that
/ОО	p
|u(-,t) -N(-,t)\dx <
for all t > 0.
Proof. 1. From estimate (62) in the proof of Theorem 5 we have
(68)
Now
(x — at) — y(x, t)
t
C
~ t№'
160
3. NONLINEAR FIRST-ORDER PDE
Consequently (59), (66) and (68) imply
(69)
1 (ж - at) - y(x, t)
u(z,t) - ------------------
U	V
2. Since g has compact support, we may assume for some constant R > 0
that g = 0 on R П {|ar| > R}. Therefore
( h- if x < —R
h(x) = <
l/i+ if x > R,
for constants h±. A calculation shows
(70)	min h = ~+h- = -| + h+.
M.	z	z
We next set
(71)	£ = £W:=^72 (t>0^
the constant A to be selected later.
3.	We now claim that if A is sufficiently large, then
(72)	u(x, t) = 0 for x — at <—R — (pd(l + е)^2
and
(73)	u(x, t) = 0 for x — at > R +	+ e)^)1/2.
In fact, since (64) implies
a
we deduce from (61) and (62) that
tL	+О(Г1/Л
\ t J d 2t	\	/
Hence
(74)	tL M = 11(1 -J*)-	+ hM + О (r VS) .
Assume
(75)	x — at < — R — (pd(l + e)^)1/2.
3.4. INTRODUCTION TO CONSERVATION LAWS
161
Then h(x — at) = h- and so
((x — (x — at))\	.
-----------j + h(x — at) = tL(a) + h- = h-.
Now if у < —R, then
(x — y\	, , ч
—I— )	Ml/) — >
c /
since L > 0. On the other hand if у > —R, we employ (74) and (70) to
estimate
tL M	-^h_+O (i-V2)
\ t J	a Zt	Z	\	/
2pj(^)t_p	_+ott-‘^ by(75)
Zat Z	x /
= ^4 + ^-+o(r1/2) by (71)
> h_,
provided A is large enough.
We conclude that (75) forces y(x, t) = x — at, and so u(x, t) = G(a) = 0.
This establishes assertion (72), and the proof of (73) is analogous.
4. Next we assert for A and t large enough that
(76)	y(x,t) >—R if x — at = R — (pd(l — e)t)x/2.
t
To see this, notice that y(x, t) < —R implies as above that
(x — y\ , , ,
------j + h(y) > h-.
Select now a point z such that h(z) = minh = —% + h- and \z\ < R. Then
we can as before invoke (74) to estimate
(x — z\ .	1 |(a; — at) — zl2 p ,	/ i/э\
—- + h(z) <	+ h_ + О (t-xl2\
t J	(L Zv	Z	X z
аМ1-е)»_Р+ь+0Л-1/г\
2dt 2	\	/
= -^4 + h- + О (i-1/2) < h-,
2ti	\	/
162
3. NONLINEAR FIRST-ORDER PDE
for A large enough. This proves (76) and a similar argument establishes
that
(77)	y(x, t) < R if x — at = — R + (<?d(l — e)^)1/2.
5.	Remember from the proof of Theorem 1 in §3.4.2 that the mapping
x и-> y(x,f) is nondecreasing. Hence (69), (76) and (77) imply for large t
that
(78)	f	f if
I R — (pd(l — e)^)1/2 < x — at < —R + (gd(l — e)^)1/2.
6.	Owing to Theorem 5, we have |u| = O(t“i) and by definition |1V| =
O(t~i). In addition (71) implies ((l±e)t)2 — ti = 0(1). Using these
bounds along with (72), (73) and (78), we estimate
[ |u(x, t) — N(x, t)| dx = О ft-1/2) ,
as desired.	□
Example 3 (continued). Observe we have p = 0, q = 2, a = 0, d = 1 in
Example 3 of §3.4.1. In this case
f f if 0 < x < (2t)V2
N(x,t) = P	V ’
( 0 otherwise,
and so in fact и = N for times t > 2.	□
We will study systems of conservation laws in Chapter 11.
3.5. PROBLEMS
1.	Prove
u(x, t,a,b) = a  x — tH(a) + b (a G Rn, b G R)
is a complete integral of the Hamilton-Jacobi equation
щ + H(Du) = 0.
2.
(a)	Write down the characteristic equations for the PDE
(*)	щ + b • Du = f in Rn x (0, oo),
3.5. PROBLEMS
163
where b G IR", f = f(x,t).
(b)	Use the characteristic ODE to solve (*) subject to the initial
condition
и — g on R" x {t = 0}.
Make sure your answer agrees with formula (5) in §2.1.2.
3.	Solve using characteristics:
(a)	xiuX1 + x2uX2 = 2u, u(a;1,1) =
(b)	uuX1 + uX2 = 1, u(<ri,<ri) = |a?i.
(c)	XiUX1 + 2x2uX2 = 3u, u(a?i, a?2,0) = g(xi, a?2).
4.	Verify that formula (61) in §3.2.5 provides an implicit solution of the
scalar conservation law.
5.	Write L = H*, if H : R" —> IR is convex.
(a)	Let H(p) = £ |p|r, for 1 < r < oo. Show
L(q) = -|g|s, where - + - = 1.
s	r s
(b)	Let H(p) = | aijPiPj + £4=1 biPi, where A	is &
symmetric, positive definite matrix, b G IR". Compute L(q).
6.	Let H : Rn —> R be convex. We say q belongs to the subdifferential of
H at p, written
<z e dH(p),
if
Я(г) > H(p) + q- (r — p) for all r G R".
Prove q G dH(p) if and only if p G dL(q) if and only if p • q =
H(p) + L(q), where L = H*.
7.	Prove that the Hopf-Lax formula reads
U(X,e> = {tL (~t^) +9^}
= min [tL (X + g(g)|
y&B(x,Rt) I \ t / J
for R = supRn \DH(Dg)\, H = L*. (This proves finite propagation
speed for a Hamilton-Jacobi PDE with convex Hamiltonian and Lip-
schitz continuous initial function g. Hint: Use the previous problem.)
8.	Let E be a closed subset of R". Show that if the Hopf-Lax formula
could be applied to the initial-value problem
' щ + |Du|2 = 0 in R" x (0,00)
<	(	0 r E
и = <	, „ on R" x {t = 0},
(+00 x E	L J
164
3. NONLINEAR FIRST-ORDER PDE
it would give the solution
9.	Fill in all details for the proof of Lemma 4 in §3.3.3.
10.	Assume u1, u2 are two weak solutions of the initial-value problems
( ult + Я(Ииг) = 0 in Rn x (0, oo)
[	г? = p* on Rn x {t = 0} (г = 1,2),
for H as in §3.3. Prove the L°°-contraction inequality
8ир|г?(-,г) - u2(-, t)| < sup Is1 — g2| (t > 0).
Show that
if 4a; + t2 > 0
0 if 4a; + t2 < 0
is an (unbounded) entropy solution of щ +	= 0.
12.	Assume u(x + z) — u(x) < Ez for all z > 0. Let ue =	* и, and show
< E.
13.	Assume F(0) = 0, и is a continuous integral solution of the conserva-
tion law
( щ + F(u)x = 0 in R x (0, oo)
(	и = g on R x {t = 0},
and и has compact support in R x [0, oo]. Prove
for all t > 0.
Compute explicitly the unique entropy solution of
'	/ 2 \
щ + ( ) =0 in R x (0, oo)
и = g onRx {t = 0},
0
2
0
S(a;) = *
if x < — 1
if — 1 < x < 0
if 0 < x < 1
if x > 1.
3.6. REFERENCES
165
Draw a picture documenting your answer, being sure to illustrate what
happens for all times t > 0.
3.6.	REFERENCES
Section 3.	1 A nice reference for this material is Courant-Hilbert [C-H,
Chapter 2].
Section 3.	2 This derivation of the characteristic differential equations is
found in Caratheodory [С]. The proof of Theorem 2 fol-
lows Garabedian [G, Chapter 2], John [J, Chapter 1], etc.
Chester [CH] and Sneddon [SN] are also good texts for more
on first-order PDE. Example 3 in §3.2.2 is from Zwillinger
[ZW],
Section 3.	3 See Lions [LI], Rund [RU] and Benton [BE] for more on
Hamilton-Jacobi PDE. The uniqueness proof, which is due
to A. Douglis, is from [BE].
Section 3.4 See Lax [LA] and Smoller [S, Chapters 15,16] (from which
I took the proof of Theorem 3, due to O. Oleinik). Theo-
rems 5 and 6 are from DiPerna [DP] and I am indebted to
M. Struwe for help with the proofs. A good overall reference
on nonlinear waves is Whitham [WH],
Chapter 4
OTHER WAYS
TO REPRESENT
SOLUTIONS
4.1	Separation of variables
4.2	Similarity solutions
4.3	Transform methods
4.4	Converting nonlinear into linear PDE
4.5	Asymptotics
4.6	Power series
4.7	Problems
4.8	References
This chapter collects together a wide variety of techniques that are some-
times useful for finding certain more-or-less explicit solutions to various par-
tial differential equations, or at least representation formulas for solutions.
4.1.	SEPARATION OF VARIABLES
The method of separation of variables tries to construct a solution и to a
given partial differential equation as some sort of combination of functions
of fewer variables. In other words, the idea is to guess that и can be written
as, say, a sum or product of as yet undetermined constituent functions, to
plug this guess into the PDE, and finally to choose the simpler functions to
ensure и really is a solution. This technique is best understood in examples.
167
168
4. OTHER WAYS TO REPRESENT SOLUTIONS
Example 1. Let U C Rn be a bounded, open set with smooth boundary.
We consider the initial/boundary-value problem for the heat equation
(1)
' щ — Au = 0 in U x (0, oo)
< u = 0 on dU x [0, oo)
и = g on U x {t = 0},
where g : U —> R. is given. We conjecture there exists a solution having the
multiplicative form
(2)
u(x,t) = u(t)w(jr) (x G U, t> 0);
that is, we look for a solution of (1) with the variables x = (a?i,..., xn) 6 U
“separated” from the variable t G [0,T].
Will this work? To find out, we compute
ut(x,t) = v'(t)w(x), Au(x,t) = v(t)Aw(x).
Hence
0 = щ(х,Р) — Nu(x,t) = v'fyw^x') — u(t)Aw(a?)
if and only if
v'(t) = Aw(x)
u(t) w(x)
for all x G U and t > 0 such that w(x), v(t) Ф 0. Now observe the left hand
side of (3) depends only on t and the right hand side depends only on x.
This is impossible unless each is constant, say
v (t) __ _ Aw(a;)
v(t)	w(x)
Then
(4)
v' = g.v,
(5)	Aw = pw.
We must solve these equations for the unknowns w, v and //.
Notice first that if p is known, the solution of (4) is v = deMf for an
arbitrary constant d. Consequently we need only investigate equation (5).
4.1. SEPARATION OF VARIABLES
169
We say that A is an eigenvalue of the operator —A on U (subject to zero
boundary conditions) provided there exists a function w, not identically
equal to zero, solving
Г —Aw = Aw in U
( w = 0 on dU.
The function w is a corresponding eigenfunction. (See Chapter 6 for the
theory of eigenvalues, eigenfunctions.)
If A is an eigenvalue and w is a related eigenfunction, we set p = —A
above, to find
(6)	и = de~xtw
solves
, ,	( ut~ &u = 0 in U x (0, oo)
[ и = 0 on dU x [0, oo),
with the initial condition u(-,0) = dw. Thus the function и defined by
(6) solves problem (1), provided g = dw. More generally, if Ai,..., Am are
eigenvalues, w\,... ,wm corresponding eigenfunctions, and d-[,...,dm are
constants, then
m
(8)	u = ^dke~Xktwk
fc=i
solves (7), with the initial condition u(-,0) = 'Y^k=1dkwk. If we can find
m, wi,..., etc. such that dkwk = g, we are done.
We can hope to generalize further by trying to find a countable sequence
Ai,... of eigenvalues with corresponding eigenfunctions wi,..., so that
OO
(9)	^2 dkWk = 9 in U
k=i
for appropriate constants di,.... Then presumably
OO
(10)	и = ^2 dke~Xktwk
k=i
will be the solution of the initial-value problem (1).
This is an attractive representation formula for the solution, but depends
upon (a) our being able to find eigenvalues, eigenfunctions and constants
satisfying (9), and (b) our verifying that the series in (10) converges in some
appropriate sense. We will discuss these matters further in Chapters 6, 7,
within the context of Galerkin approximations.	□
170
4. OTHER WAYS TO REPRESENT SOLUTIONS
Remark. Take note that only our solution (6) is determined by separation
of variables; the more complicated forms (8) and (10) depend upon the
linearity of the heat equation.	□
Example 2. Let us next apply the separation of variables technique to
discover a solution of the porous medium equation
(11)	ut — A(u7) = 0 in Rn x (0, oo),
where и > 0 and 7 > 1 is a constant. The expression (11) is a nonlinear
diffusion equation, in which the rate of diffusion of some density и depends
upon и itself. This PDE describes flow in porous media, thin-film lubrica-
tion, and a variety of other phenomena.
As in the previous example, we seek a solution of the form
(12)	u(x, t) = v(t)w(jr) (jeRn, t > 0).
Inserting into (11), we discover that
v'(t)	AuP(x)
v(t)7	w(x)
for some constant p and all x E R", t > 0, such that w(x), v(t) 0.
We solve the ODE for v and find
v = ((1 ~7)/zt + A)~,
for some constant Л, which we will take to be positive. To discover w, we
must then solve the PDE
(14)	A(w7) = pw.
Let us now guess that
w = |a;|Q,
for some constant a that must be determined. Then
(15)	pw — A(w7) = p|a;|a — 07(07 + n — 2) | a?|a7-2.
So in order that (14) hold in R", we should first require that о = 07 — 2,
and hence
Returning to (15), we see that we must further set
(17)	p = 07(07 + n — 2) > 0.
In summary then, for each Л > 0 the function
и = ((1 - 7)/zt + A)T^|rr|“
solves the porous medium equation (11), the parameters o, p defined by
(16), (17).	□
4.1. SEPARATION OF VARIABLES
171
Remark. Observe that since 7 > 1, this solution blows up for x Ф 0 as
t —»t*, for t* := (7Д)М- Physically, a huge amount of mass “diffuses in from
infinity” in finite time. See §4.2.2 for another, better behaved solution of the
porous medium equation, and see §9.4.1 for more on blow-up phenomena for
nonlinear diffusion equations.	□
In the previous example separation of variables worked owing to the
homogeneity of the nonlinearity, which is compatible with functions и having
the multiplicative form (12). In other circumstances it is profitable to look
for a solution in which the variables are separated additively:
Example 3. Let us turn once again to the Hamilton-Jacobi equation
(18)	щ + H(Du) = 0 in Rn x (0,00)
and look for a solution и having the form
u(x, t) = w(a?) + v(t) (x G R", t > 0).
Then
0 = t) + H(Du(x, t)) = v'(t) + H(Dw(x))
if and only if
H(Dw(xy) = p = —v'(t) (x G R",t > 0)
for some constant p. Consequently if
Я(£>ш) = p
for some p G R, then
u(x, t) = w(x) — pt + b
will for any constant b solve щ + H(Du) = 0. In particular, if we choose
w(x) — a - x for some a 6 Rn and set p — H(a), we discover the solution
и — a • x — H(a)t + b
already noted in §3.1.
□
172
4. OTHER WAYS TO REPRESENT SOLUTIONS
4.2.	SIMILARITY SOLUTIONS
When investigating partial differential equations it is often profitable to look
for specific solutions u, the form of which reflects various symmetries in the
structure of the PDE. We have already seen this idea in our derivation of
the fundamental solutions for Laplace’s and the heat equations in §2.2.1 and
§2.3.1, and our discovery of rarefaction waves for conservation laws in §3.4.4.
Following are some other applications of this important method.
4.2.1.	Plane and traveling waves, solitons.
Consider first a partial differential equation involving the two variables
x E R, t G R. A solution и of the form
(1)	u(x, t) = v(x — at) (x G R, t G R)
is called a traveling wave (with speed a and profile v). More generally, a
solution и of a PDE in the n + 1 variables x = (si,..., xn) G Rn, t G R
having the form
(2)	u(x, t) = v(y  x — at) (x G R", t G R)
is called a plane wave (with wavefront normal to у G Rn, velocity and
profile v).
a. Exponential solutions.
In view of the Fourier transform (discussed later, in §4.3.1), it is par-
ticularly enlightening when studying linear partial differential equations to
consider complex-valued plane wave solutions of the form
(3)	u(x,t) = е^'м\
where ш G C and у = (t/i,... ,yn) G Rn, ш being the frequency and {t/i}”=1
the wave numbers. We will next substitute trial solutions of the form (3) into
various linear PDE, paying particular attention to the relationship between
у and w forced by the structure of the equation.
(i)	Heat equation. If и is given by (3), we compute
щ — Au = (iw + |y|2)u = 0,
provided ш = г|у|2. Hence
u _ eiy-x-\y\2t
solves the heat equation for each у G R". Taking real and imaginary parts,
we discover further that 1 cos(y  x) and e_l^ 1 sin(y • x) are solutions as
4.2. SIMILARITY SOLUTIONS
173
well. Notice in this example that since ш is purely imaginary, there results
a real, negative exponential term e-^ 1 in the formulas, which corresponds
to dissipation.
(ii)	Wave equation. Upon our substituting (3) into the wave equation,
we discover
utt - Au = (-w2 + \y|2)u = 0,
provided w = ±|y|. Consequently
u = gi(?/-x±|y|t)
solves the wave equation, as do the pair of functions cos(y • x ± \y\t) and
sin(y • x ± |y|t). Since w is real, there are no dissipation effects in these
solutions.
(iii)	Dispersive equations. We now let n = 1 and substitute и =
ez(yx+wt) |П|.О ^iry’s equation
nt “I” Uxxx ~~ 0.
We calculate
Щ + uXxx = i(w - У3)и = 0,
whenever w = y3. Thus
u — ei(yx+y3t)
solves Airy’s equation, and once again as w is real there is no dissipation.
Notice however that the velocity of propagation is y2, which depends non-
linearly upon the frequency of the initial value eiyx. Thus waves of different
frequencies propagate at different velocities: the PDE creates dispersion.
Likewise, if n > 1 and we substitute и = ег(у'х+^*) into Schrodinger’s
equation
iut + Au = 0,
we compute
iut + Au = — (ш + |y|2)u = 0.
Consequently ш = — |y|2, and
u _ ei(yx-\y\2t)
Again, the solution displays dispersion.	□
174
4. OTHER WAYS TO REPRESENT SOLUTIONS
b. Solitons.
We consider next the Korteweg-de Vries (KdV) equation in the form
(4)	ut + 6uux + uxxx = 0 in R x (0, oo),
this nonlinear dispersive equation being a model for surface waves in water.
We seek a traveling wave solution having the structure
(5)	u(x,t) = v(x — at) (xER, t > 0).
Then и solves the KdV equation (4), provided v satisfies the ODE
(6)	— av' + fxvv' + v'" = 0	( ' = — | .
\	ds)
We integrate (6) by first noting
(7)	— av + 3v2 + v" = a,
a denoting some constant. Multiply this equality by v' to obtain
—avv' + Зг>2г/ + v"v' = av',
and so deduce
(г/)2 з a 2
(8)	= —v3 + -v2 +av + b
where b is another arbitrary constant.
We investigate (8) by looking now only for solutions v which satisfy
v,v',v" —>• 0 as s —> ±oo. (In which case the function и having the form
(5) is called a solitary wave.) Then (7), (8) imply a = b = 0. Equation (8)
thereupon simplifies to read
Hence v' = ±г>(<т — 2г)1/2.
We take the minus sign above for computational convenience, and obtain
then this implicit formula for v:
, ч	dz
(9)	S~ L г(а-2г)1/2+с’
4.2. SIMILARITY SOLUTIONS
175
for some constant c. Now substitute z — ^sech20. It follows that —
—a sech2 0 tanh в and z(cr — 2г)1/2 = sech2 0tanh0. Hence (9) becomes
/ X	2 „
(10)	s = —0 + c,
yer
where 0 is implicitly given by the relation
(11)	sech2 в = u(s).
We lastly combine (10) and (11), to compute
u(s) = sech2 (— c)^l (sGR).
Conversely, it is routine to check v so defined actually solves the ODE (6).
The upshot is that
u(x, t) = sech2 ((x — at — c)^l (x G R, t > 0)
is a solution of the KdV equation for each c G R, a > 0. A solution of this
form is called a soliton. Note the velocity of the soliton depends upon its
height.	□
Remark. The KdV equation is in fact utterly remarkable, in that it is
completely integrable, which means that in principle the exact solution can be
computed for essentially arbitrary initial data. The relevant techniques are
however beyond the scope of this book: see Drazin [D] for more information.
□
c. Traveling waves for a bistable equation.
Consider next the scalar reaction-diffusion equation
(12)	ut - uxx = f(u) in R x (0, oc),
where f : R —> R has a “cubic-like” shape.
Graph of the function f
176
4. OTHER WAYS TO REPRESENT SOLUTIONS
We assume, more precisely, f is smooth and verifies
(a)	/(0) = f(a) = /(1) = 0
(b)	f < 0 on (0, a), f > 0 on (a, 1)
(c)	/'(0) < 0, /'(1) < 0
(d)	fof(z)dz>°
for some point 0 < a < 1.
We look for a traveling wave solution of the form
(14)	u(x, t) = v(x — at),
the profile v and velocity a to be determined, such that
и —> 0 as x —> —oo,	и —> 1 as x —> +oo.
Now since f' < 0 at z = 0,1, the constants 0 and 1 are stable solutions of
the PDE (and since f > 0 at z = a, the constant a is an unstable solution).
So we want our traveling wave (14) to interpolate between the two stable
states z — 0,1 at x — =poo.
Plugging (14) into (12), we see v must satisfy the ordinary differential
equation
(15)	v" + av' + /(v) = 0
subject to the conditions
(16)	lim v(s) = 1, lim v(s) = 0, lim v\s) = 0.
s—>+oo	s—>—oo	s—>±oo
।' = - Y
\ ds J ’
We outline now (without complete proofs) a phase plane analysis of the
ODE problem (15), (16). We begin by setting
w := v'.
Then (15), (16) transform into the autonomous first-order system:
(17)
with
(18)
( v' = w
( w' — —aw — /(v),
lim (v, w) = (1,0), lim (v, w) = (0,0).
S—>OQ	S—> —OO
4.2. SIMILARITY SOLUTIONS
177
Now (0,0) and (1,0) are critical points for the system (17), and the eigen-
values of the corresponding linearizations are
,,m	-a±(^-4/'(0))V2	.
(iyl ло ~	2	’ Л1 ~	2
In view of (13)(c), Aj, A± are real, with differing sign, and thus (0,0)
and (1,0) are saddle points for the flow (17). Consequently an “unstable
curve” Wu leaves (0,0) and a “stable curve” Ws approaches (1,0), as drawn.
Furthermore, by calculating eigenvectors corresponding to (19) we see
(20)
Wu is tangent to the line w = A^v at (0,0)
Ws is tangent to the line w = A]“(v — 1) at (1,0).
Stable and unstable curves
Note that Aq ,A±,VV“ and Ws depend upon the parameter cr. Our in-
tention is to find cr < 0 so that
(21)	Wu = Ws in the region {v > 0, w > 0}.
Then we will have a solution of (17), (18), whose path in the phase plane is
a heteroclinic orbit connecting (0,0) to (1,0).
To establish (21), we fix now a small number e > 0 and let L denote the
vertical line through the point (a + e, 0). We claim
(22)
WSOL^0, WUOL^0
178
4. OTHER WAYS TO REPRESENT SOLUTIONS
if <t < 0. To check this assertion, define
E(v, w)
(y,w E. R)
and compute
-^E(v(t),w(t)) = w(t)w'(t) +/(v(t))v'(t)
etc
=	by (17).
As cr < 0, we see that E is nondecreasing along trajectories of the ODE
(17). Note also the level sets of E have the shapes illustrated.
Level curves of E
Consider next the region R, as drawn. The unstable curve enters R from
(0,0) and cannot exit through the bottom, top or left hand side. Using (17)
we deduce that Wu must exit R through the line L, at a point (a+s, wo(cr)).
Similarly we argue Ws must hit L at a point (a + s, wi(cr)). This verifies
claim (22).
We next observe
(23)
wo(O) < wi(0);
this follows since trajectories of (17) for a — 0 are contained in level sets of
E. We assert further that
(24)
W0(cr) > Wl(cr)
4.2. SIMILARITY SOLUTIONS
179
The region S
provided a < 0 and |cr| is large enough. To see this, fix (3 > 0 and consider
the region S, as drawn.
Now along the line segment T := {0 < v < a + e, w = (3v}, we have
w' _ —aw - /(v) _ _ _ /(u)
v'	w	fiv
180
4. OTHER WAYS TO REPRESENT SOLUTIONS
Since |	1 is bounded for 0 < v < a + e, we see
(25)	onT,
v	p
provided cr < 0 and |cr| is large enough.
The calculation (25) shows that Wu cannot exit S through the line
segment T, and so wo(cr) > (3(a + e) if cr = cr(/3) is sufficiently negative. On
the other hand, wi(cr) < wi(0) for all a < 0. Thus we see that (23) will
follow once we choose (3 large enough and then cr sufficiently negative.
Since wo and wi depend smoothly on a, we deduce from (22) and (23)
that there exists cr < 0 with
(26)	wo(<t) = wi(<t).
For this velocity a there consequently exists a solution of the ODE (17),
(18). Hence we have found for our reaction-diffusion PDE (12) a traveling
wave of the form (14).	□
Remark. A more refined analysis demonstrates that the velocity cr veri-
fying (26) is unique. Hence given the nonlinearity f satisfying hypotheses
(13), there exists a unique velocity for which there is a corresponding trav-
eling wave. Compare this assertion with Example 2 above, where we found
soliton traveling waves for each given velocity.	□
4.2.2.	Similarity under scaling.
We next illustrate the possibility of finding other types of “similarity”
solutions to PDE.
Example (A scaling invariant solution). Consider again the porous medium
equation
(27)	щ — A(u7) = 0 in R" x (0, oo),
where и > 0 and 7 > 1 is a constant.
As in our earlier derivation of the fundamental solution of the heat equa-
tion in §2.3.1, let us look for a solution и having the form
(28)	u(z,t) = —(жеГ, t>0),
where the constants a, (3 and the function v : R" —> R must be determined.
Remember that we come upon (28) if we seek a solution и of (27) invariant
under the dilation scaling
u(x,t) 1—► Xau(X@x, At);
4.2. SIMILARITY SOLUTIONS
181
so that
и(х,^ = Xau(X@x, Xt)
for all A > 0, x € R", t > 0. Setting A = t-1, we obtain (28) for v(y) :=
We insert (28) into (27), and discover
(29)	at~'a+V)v(y) + {3t~(a+Vy  Dv(y) +	= 0
for у = t~^x. In order to convert (29) into an expression involving the
variable у alone, let us require
(30)	a + 1 = cry + 2/3.
Then (29) reduces to
(31)	av + (3y • Dv + Д(гР') = 0.
At this point we have effected a reduction from n + 1 to n variables. We
simplify further by supposing v is radial; that is, v(y) — w(|y|) for some
w : R —> R. Then (31) becomes
TL — 1
(32)	aw + ffrw' + (w1)" 4	(w7)7 = 0,
r
where r = |y|,' = Now if we set
(33)	a = n/3,
(32) thereupon simplifies to read
+ /3(rnwY = 0.
Thus
гп-х(и;7)' + /3гпи; = а
for some constant a. Assuming lim^oo w, w' = 0, we conclude a = 0;
whence
(w7)z = —[3rw.
But then
(w7-1)' =	^/3r.
7
Consequently
w7-1 = b — dr2,
*1
182
4. OTHER WAYS TO REPRESENT SOLUTIONS
b a constant; and so
/	7 — 1	+ 7-1
(34)	w — I b-----—/3r2 I ,
\	27	/
where we took the positive part of the right hand side of (34) to ensure
w > 0. Recalling г?(у) = w(r) and (28), we obtain
(35)	(^Rn,t>0),
where, from (30), (33),
n	1
36	01 n(7 - 1) + 2 ’	n(7 - 1) + 2 '
The formulas (35), (36) are Barenblatt’s solution to the porous medium
equation.	□
Remarks. Observe that Barenblatt’s solution has compact support for each
time t > 0. This is a general feature for (appropriately defined) weak, non-
negative solutions of the porous medium equation with compactly supported
initial data. The nonlinear parabolic PDE (27) becomes degenerate wher-
ever и = 0, and so the set {u > 0} moves with finite propagation speed.
Consequently the porous medium equation (27) is often regarded as a bet-
ter model of diffusive spreading than the linear heat equation (which predicts
infinite propagation speed).	□
4.3.	TRANSFORM METHODS
In this section we develop some of the theory of Fourier and Laplace trans-
forms, which provides extremely powerful tools for converting certain linear
partial differential equations into either algebraic equations or else differen-
tial equations involving fewer variables.
4.3.1.	Fourier transform.
In this section all functions are complex-valued, and - denotes the
complex conjugate.
4.3. TRANSFORM METHODS
183
a. Definitions and elementary properties.
Definition of Fourier transform on L1. If и G L^R"), we define its
Fourier transform
(1)	“(»):=	dx fa e R")
and its inverse Fourier transform
(2)	йМ:=7т4л/' <=“’»«*: to SR”)-
Since \е±гх'у| = 1 and и G L^R”), these integrals converge for each у G R".
We intend now to extend definitions (1), (2) to functions и G L2(R").
THEOREM 1 (Plancherel’s Theorem). Assume и G L1(R") П L2(R").
Then й,й G L2(Rn) and
(3)	II&IIl2(R") = IH|L2(Rn) - ||u||L2(Rn).
Proof. 1. First we note that if v,w G L^R72), then v,w G L°°(R"). Also
(4)
/ u(z)w(z) dx =
Rn
/ v(y)w(y)dy,
Rn
since both expressions equal ^„/2 Jr" Jr^ e lx'yv(x)w(y) dxdy. Further-
more, as we will explicitly compute below in Example 1 of §4.3.2,
eix-y-t\x\2 dx =
7T\"/2	\y\2
— e 4t
t)
(t > 0).
। 12	— —
Consequently if e > 0 and ve(x) := e~£>x> , we have v£(y) =	• Thus
(4)	implies for each e > 0 that
(5)
e £\y\2 dy =
1 Г < а
———/ w(x)e 4s dx
(2e)"/2JRn v
2.	Now take и G Z1(R") П Z2(R") and set u(x) := u(—x). Let w :=
и * v G L^R") П C(R") and check (cf. Theorem 2 below) that
w = (27r)"/2uu G L°°(R").
184
4. OTHER WAYS TO REPRESENT SOLUTIONS
But
= /9 Vn/2 [ е~гхуй{—х) dx = u(y\,
(2тг)П/2 jRn
and so w = (2тг)"у/2|й|2.
Now w is continuous and thus
b W2 [ w(x)e~^dx= (27r)n/2w(0),
e—*0 (2s)"/2 JRn
where we employed the lemma from §2.3.1. Since w = (2тг)"/2|й|2 > 0, we
deduce upon sending e —> 0+ in (5) that w is summable, with
[ w(y)dy = (2tt)"/2w(0).
/ж»
Hence
/ |u|2d?/ = w(0) = I u(x)v{—x)dx = / |u|2dx.
JR"	JR"	Jr"
The proof for й is similar.	□
Definition of Fourier transform on L2. In view of the equality (3)
we can define the Fourier transforms of a function и 6 L2(Rn) as follows.
Choose a sequence	C Z1(Rn) П L2(Rn) with
uk —> и in Z2(R").
According to (3), ||ufe - uj||L2(Kn) - ll^fc - uj||L2(Rn) = 11^ - uj||L2(Rra), and
thus	is a Cauchy sequence in L2(R"). This sequence consequently
converges to a limit, which we define to be u:
uk^u in L2(R").
The definition of u does not depend upon the choice of approximating
sequence	We similarly define u.
Next we record some useful formulas.
THEOREM 2 (Properties of Fourier transform). Assume u,v E L2(R”).
Then
(i)	uvdx = jyp uvdy,
(ii)	Dau = (iy)au for each multiindex a such that Dau € T2(R"),
(iii)	(u * v\= (2тг)п/2ш),
(iv)	и = (й)'.
4.3. TRANSFORM METHODS
185
Proof. 1. Let u,v G L2(R") and a G C. Then
||u + ov||^2(Rn) = ||й +
Expanding, we deduce
/ |u|2 + |cw|2 + u(av) + u(av) dx = / |й|2 + |m)|2 + u(av) + u(av) dy;
JR"	JR"
and so according to Theorem 1,
I auv + auv dx = I auv + auv dy.
JR"	JR"
Take а — 1,г and combine the resulting equalities to deduce
/ uvdx = uv dy.
JR"	JR"
This proves (i).
2.	If и is smooth and has compact support, we calculate
D^u(y) =	[ e~ix yDau(x) dx
(27T)n/J JR"
=	/ /9 / D^e~iX ^u(x)dx
(27^/2 jRn	'
= /9 W2 f e~lxy(iy)au{x)dx = (iy)au(y).
(27T)"/z Tgn
By approximation the same formula is true if Dau G L2(R”).
3.	We compute for u,v G L^R") П L2(R") and у G R" that
(u*v)(y) =	1 I е~гх'у [ u(z)v(x - z) dzdx
(27r)n/J	J^n
= f e~iz-yu(z) ( [ e^x-*>yv(x - z}dx\ dz
(2тт)п^ jRn	yjR»	)
= [ e~tz yu(z)dz v(y) = (2тг)"/2й(?/)?)(?/).
JR"
4.	Fix z G R", £ > 0 and write ve(x) := elx'z ^l2. Then
V-{y) =
f e-ix.(y-z)-£\xfd =
fR"	(2e)”/2
186
4. OTHER WAYS TO REPRESENT SOLUTIONS
where we followed computations from the proof of Theorem 1. Utilizing
formula (4), we deduce for и G Zx(Rn) Г) Z1 2(R") that
eiz-y~£\y\2 dy =
1 f	|x-z|2
---;—7- / u(x)e dz
(2sH2JRn k >
The expression on the right converges to (27r)"/2u(z) as e —> 0+, for each
Lebesgue point of u. Thus
zo <Z/2 / u(y)elz ydy = u(z) for a.e. z.
This proves (iv).
b. Applications.
The Fourier transform is an especially powerful technique for studying
linear, constant-coefficient partial differential equations.
Example 1 (Bessel potentials). We investigate first the PDE
(6)
—Ди + и = f in R",
where f G L2(R”). To find an explicit formula for u, we take the Fourier
transform, recalling Theorem 2, (ii) to obtain
(7)
(l + |y|2)u(y) = /(y) Ь^п)-
The effect of the Fourier transform has been to convert the PDE (6) into
the algebraic equation (7), the solution of which is trivial:
1 + |y|2 ’
Thus
(8)
and so the only real problem is to rewrite the right hand side of (8) into a
more explicit form.
Invoking Theorem 2,(iii), we see
(9)
f*B
(27t)"/2 ’
4.3. TRANSFORM METHODS
187
where
(10)
i + Ы2’
We solve for В as follows. Since 1
nfeF = r^‘(1+l’|2)«-Thus
tadt for each a > 0, we have
Z11\ D (	1 V 1
(11) B = \ T+"M2) = (2^
/ e^y-^2 dy] dt.
R"	/
Now if a, b G R, b > 0, and we set z = b1/,2x — we find
, 2	e-a?/4b r
гах-Ьх2 j _ ________ /
M/2 /г
Г denoting the contour |lm(z) = — 5^72 in the complex plane. Deforming
Г into the real axis, we compute Jr e~z2dz = e~x2 dx = 7г1/2; and hence
(12)
eiax~bx2__ e~a2/4b /^X1/2
Thus
11 l*OQ
e^-y-^dy = П / e“'
j=i
e 4t
by (12). Consequently, we conclude from (11), (13) that
1	e-*-^
(14)	B(x) =	/ ------^—dt (x e R").
v	’	2"/2 Jo t"/2
В is called a Bessel potential. Employing (9), we derive then the formula
1 /-oo r -t- !*~у|2
<15)	“«=0^72/,	<*eR")
for the solution of (6).	□
188
4. OTHER WAYS TO REPRESENT SOLUTIONS
Example 2 (Fundamental solution of heat equation). Consider again the
initial-value problem for the heat equation
(16)
ut — Ди = 0
и = g
in Rn x (0, oo)
on Rn x {t — 0}.
We establish a new method for solving (16) by computing u, the Fourier
transform of и in the spatial variables x only. Thus
( ut + |y|2u = 0
I й = g
for t > 0
for t = 0;
whence
u = e~^2g.
Consequently и =	, and therefore
(17)
_ g*F
~ (27t)"/2’
where F = e But then
F= (e~^2
by (13). Invoking (17), we compute
1 f	|X-vl2
(18) u(z,t) = ——^/ e 4* g(y)dy (ж G R", t > 0),
(47rt)"/2 jRn
in agreement with §2.3.1. The Fourier transform has provided us with a new
derivation of the fundamental solution of the heat equation.	□
Example 3 (Fundamental solution of Schrodinger’s equation). Let us next
look at the initial-value problem for Schrodinger’s equation
(19)
iut -I- Ди — 0 in Rn x (0, oo)
и = g on R" x {t = 0}.
Here и and g are complex-valued.
If we formally replace t by it on the right hand side of (18), we obtain
the formula
1 f ilx —
(20)	u(a;,t) = -	/ e 4« g(y)dy (x G Rn, t > 0),
(47T2i)"/2
4.3. TRANSFORM METHODS
189
where we interpret i5 as e“. This expression clearly makes sense for all
times t > 0, provided g G L^R"). Furthermore if |y|2<7 G L1(R"), we can
check by a direct calculation that и solves iut + Au = 0 in Rn x (0, oo). (We
will not discuss here the sense in which u(', t) —► g as t —► 0+, but see §4.5.3
below and Problem 5.)
Let us next rewrite formula (20) as
Ц(М) = Fr"7'W2 f e^e^9(y)dy.
(4mt)n/2 Jr™
г|з:|2	i|^|2	.
Since |e 4t « | = 1, we can check as in Theorem 1 that if g G L1(R") П
L2(Rn), then
(21)	IH-,*)IIl2(R") = I|5||l2(R") (t > 0).
Hence the mapping g >—> u(-,t) preserves the L2-norm. Therefore we can
extend formula (20) to functions g G L2(Rn), in the same way that we
extended the definition of Fourier transform.	□
Remark. We call
1 i|xl2
(22)	Ф(ж, t) :=	e » (x	0)
(47TZt)"/2
the fundamental solution of Schrodinger’s equation. Note that formula (20),
и = g * Ф, makes sense for all times t 0, even t < 0. Thus we in fact have
solved
, .	( iut + Au = 0 in R" x (—oo, oo)
( и = g on R" x {t = 0}.
In particular, Schrodinger’s equation is reversible in time, whereas the heat
equation is not (in spite of Theorem 11 in §2.3.4).	□
Example 4 (Wave equation). We next analyze the initial-value problem
for the wave equation
(24)
( utt — Au = 0 in R" x (0, oo)
( и = g, щ = 0 on R" x {t = 0},
where for simplicity we suppose the initial velocity to be zero. Take as before
u to be the Fourier transform of u in the variable x G R". Then
, .	( utt + Ы2й = 0 for t > 0
[ u = g, Ut = 0 for t = 0.
190
4. OTHER WAYS TO REPRESENT SOLUTIONS
This is an ODE for each fixed у G R". We look for a solution having the
form й = /3et7 (/3,7 G C). Plugging into (25) gives 72 + |y|2 = 0 and so
7 = ±г|у|. Remembering the initial conditions from (25), we deduce
u= ^(e^l Te’^l).
2
Inverting, we find
u(x,t) = |(erfM+e-^l) ;
and consequently
(26) u(x, t) =	(e‘(-»+‘l»l> +	dy
(27t)"/2 jRn 2
for x G R", t > 0. We will further analyze this formula in certain asymptotic
limits later, in §4.5.3. See also Example l(ii) in §4.2.1.	□
Example 5 (Telegraph equation). The initial-value problem for the one-
dimensional telegraph equation is
( utt + 2dut -uxx = 0 in R x (0, oc)
' '	( и = g, Ut — h on R x {t = 0},
for d > 0, the term “2dtq” representing a physical damping of wave propa-
gation. As before
Г utt + 2dut + |y|2u = 0	for t > 0
( й = g, щ = h	for t = 0.
We again seek a solution of the form й = /Зе*7 (/3,7 G C). Plugging into (28),
we deduce 72 + 2d7 +|y|2 = 0; whence 7 = —d± (d2 — I?/!2)1/2. Consequently
_ Г е-л(/31(2/)е7^4+ /32(2/)е-7^) if |y| < d
и(уЛ) - I	+^2(y)e-i«(y)t) if |y| > d
for 7(y) := (d2 - |y|2)1/2(|?/| < d), <5(y) := (|y|2 - d2)1/2 (|y| > d), where
/31 (y) and /32(1/) are selected so that
s(y) = A(y) + /Ш)
~ Г Z?i(3/)(7(3/) — d)-I-^2(з/)(—7(3/) — rf) if |y| < d
У t/3i(y)(h5(i/) — d) +/32(1/)(—г<5(?/) — d) if |y| > d.
We thereby obtain the representation formula:
“(*' (> =	dy
(2tt)"/2 J{\y\<d}
+ 7T^72 /	+ 02(y^y-s^ dy.
(27Г)п/2
Notice the terms e~dt, which correspond to damping as t —> 00.	□
4.3. TRANSFORM METHODS
191
4.3.2. Laplace transform.
Remember that we write R+ = (0, oo).
DEFINITION. IfuE LX(R+), we define its Laplace transform to be
(29)	u*(s) := i e~stu(fi)dt (s > 0).
Jo
Whereas the Fourier transform is most appropriate for functions defined
on all of R (or R"), the Laplace transform is useful for functions defined
only on R_|_. In practice this means that for a partial differential equation
involving time, it may be useful to perform a Laplace transform in t, holding
the space variables x fixed. (This is the opposite of the technique from
Examples 2-5 above.)
Example 1 (Resolvents and Laplace transform). Consider again the heat
equation
J vt - Av = 0 in U x (0, oc)
' '	( v = f on U x {t — 0},
and perform a Laplace transform with respect to time:
v#(z, s) = f e~stv(x,t)dt (s > 0).
Jo
What PDE does v# satisfy? We compute
Av#(z,s) = i e~stAv(x,t')dt = i e~stvt(x,t') dt
Jo	Jo
= s [ e~stv(x, t) dt + e-s*v|*=^° = sv#(x, s) — f(x).
Jo
Think now of s > 0 being fixed, and write u(x) := v#(x, s). Then
(31)	—Au + su = f in U.
Thus the solution of the resolvent equation (31) with right hand side f is
the Laplace transform of the solution of the heat equation (30) with initial
data f. (If U = R" and s = 1, we could now represent v in terms of the
fundamental solution, to rederive formula (15).)	□
The connection between the resolvent equation and the Laplace trans-
form will be made clearer by the discussion in §7.4 of semigroup theory.
192
4. OTHER WAYS TO REPRESENT SOLUTIONS
Example 2 (Wave equation from the heat equation). Next we employ some
Laplace transform ideas to provide a new derivation of the solution for the
wave equation (cf. §2.4.1), based—surprisingly—upon the heat equation.
Suppose и is a bounded, smooth solution of the initial-value problem:
(32)
( utt —	= 0 in R" x (0, oo)
[ и = g, щ = 0 on R" x {t = 0},
where n is odd and g is smooth, with compact support. We extend и to
negative times by writing
(33)	u(x, t) = u(x, —t) if x G R", t < 0.
Then
utt — Au = 0 in R" x R.
Next define
(34) v(x,t) :=
1	r°°
------;-7- /	e~s2/4tu(z, s) ds (x G R", t > 0).
(47rt)1/2 J-oo k V
Hence
limv = g uniformly on R".
In addition
1	r00
Лг,(ЖЛ) ~ (47rt)V2
x, s) ds
| e s2/4tuss(x, s) ds
—oo
/•oo
I —e~s2/itus{x,s}ds
—OO
/•oo / o2 1 \
Consequently v solves this initial-value problem for the heat equation:
f vt — Дг> = 0 in R" x (0, oo)
( v = g on R” x {t = 0}.
As v is bounded, we deduce from §2.3 that
1 f i*-»i2
(35)	v(s,t) =	, / e « g(y)dy.
(4irt)n'z J^n.
4.3. TRANSFORM METHODS
193
We equate (34) with (35), recall (33), and set A = , thereby obtaining the
identity
✓ \ n~ 1
f u(x, s)e~As2ds = ( - ) f e~x\x~yfg(y)dy.
Jo	4 \ 7Г / J^n
Thus
f°° / x —As2 ,	па(п) /АА 2 Г°° _Ar2 п_1„, x >
(36)	/ u(x, s)e A ds = —-— I — ) / e r 1G(x;r) dr,
Jo	2	\7Г/ Jo
for all A > 0, where
(37)	G(x;r) = + g(y)dS(y).
J 8B(x,r)
We will solve (36), (37) for u. To do so, we write n = 2k + 1 and note
where we integrated by parts к times for the last equality.
Owing to (36) (with r replacing s in the expression on the left), we
deduce
Upon substituting r = r2 we see that each side above, taken as a function
of A, is a Laplace transform. As two Laplace transforms agree only if the
original functions were identical, we deduce
, x na(n) J1 d \k,	..
(38)	u(x,t) = ^2^+1 4 ytQtJ (4	G(x,t)).
Now n = 2k + 1 and a(n) =	= г(з’м) • Since Г (|) = 7г1/2 and
Г(х + 1) = хГ(ж) for x > 0 (cf. [RD, Chapter 8]), we can compute
na(n)	П7Г1/2	1	1
irk2k+1 = 2k+M (I + 1) = (n-2)(n-4)---5-3 =
194
4. OTHER WAYS TO REPRESENT SOLUTIONS
We insert this deduction into (38) and simplify:
n —3 /	\
1 Z") X1 Z") \ 2 l Г	\
(39)	u(x,t) = — - (--)	r~2-/	gdS]	(reRn, t>0).
'yndtytotj у	J QB(x,t) у
This is formula (31) in §2.4.1 (for h = 0).	□
4.4. CONVERTING NONLINEAR INTO LINEAR PDE
In this section we describe several techniques which are sometime useful for
converting certain nonlinear equations into linear equations.
4.4.1.	Hopf—Cole transformation.
a.	A parabolic PDE with quadratic nonlinearity.
We consider first of all an initial-value problem for a quasilinear parabolic
equation:
, .	( щ — aAu + b|Du|2 = 0 in Rn x (0, oo)
[	и = g on Rn x {t = 0},
where a > 0. This sort of nonlinear PDE arises in stochastic optimal control
theory.
Assuming for the moment и is a smooth solution of (1), we set
w ф(и),
where ф : R —> R is a smooth function, as yet unspecified. We will try to
choose ф so that w solves a linear equation. We have
Wt = ф'(и)щ, Aw = ф'(и)Аи + <£'z(tz)|.D-u|2;
and consequently (1) implies
Wt = ф'(и)щ - ф'(и)\аДи — b|Du|2]
= aAw — [a$z,(u) + &</>z(u)]|Du|2
= aAw,
provided we choose ф to satisfy аф" + Ьф' = 0. We solve this differential
equation by setting ф = e~. Thus we see that if и solves (1), then
(2)
— bu
w = e °-
4.4. CONVERTING NONLINEAR INTO LINEAR PDE
195
solves this initial-value problem for the heat equation (with conductivity a):
( Wt — aAw = 0 in Rn x (0, oo)
(3)	| w = on Rn x {t = 0}.
Formula (2) is the Hopf-Cole transformation.
Now the unique bounded solution of (3) is
w(x, t) = ----	[ e e~9^dy (x e Rn, t > 0);
(47rat)n/2 jRn
and, since (2) implies
и = — - logw,
b
we obtain thereby the explicit formula
(4)	u(x, t) = log (	1 , [ e~^~~-9Mdy\	(a; £ Rn, t > 0)
b \(Virat)n'z J^n	/
for a solution of quasilinear initial-value problem (1).
b.	Burgers’ equation with viscosity.
As a further application, we examine now for n = 1 the initial-value
problem for the viscous Burgers’ equation:
щ — auxx + uux = 0 in R x (0, oo)
и = g on R x (0, oo).
(5)
If we set
(6)
w
I u(y,f)dy
—OO
and
‘X
(7)
•oo
(8)
(cf. §3.4), we have
wt — awxx +	=0 in R x (0, oo)
w = h on R x {t = 0}.
This is an equation of the form (1) for n = 1, b = and so (4) provides the
formula
-|a:-y|2 h(y) \
6 4at 2a dy j .
'R	/
But then since и = wx, we find upon differentiating (9) that
roo x—y ~1*-у|2 _ Mjd ,
—T^e 4at 2a dy
u(x, t) =	----- (x e R, t > 0)
J-oc e iat 2“
is a solution of problem (5), where h is defined by (7). We will scrutinize
this formula further in §4.5.2.
(9)
(10)
w(x,t) = —2a log
196
4. OTHER WAYS TO REPRESENT SOLUTIONS
4.4.2.	Potential functions.
Another technique is to utilize a potential function to convert a nonlinear
system of PDE into a single linear PDE. We consider as an example Euler’s
equations for inviscid, incompressible fluid flow:
(11)
' (a)	Uj	+ u • Du	= —Dp + f	in R3 x (0, oo)
< (b)	divu	= 0	in R3 x (0,oo)
(c)	u	=	g	on R3 x {t = 0}.
Here the unknowns are the velocity field u = (u1, u2, u3) and the scalar
pressure p; the external force f = (У1,/2,/3) an(l initial velocity g =
(51,52,53) are given. Here D as usual denotes the gradient in the spatial
variables x = (xi,X2,xs). The vector equation 11(a) means
з
*4 + 52“^; = ~Pxi + f (* = 1,2,3).
1=1
We will assume
(12)	divg = 0.
If furthermore there exists a scalar function h : R3 x (0, oo) —> R such that
(13)	f = Dh,
we say that the external force is derived from the potential h.
We will try to find a solution (u,p) of (11) for which the velocity field
u is also derived from a potential, say
(14)	u = Dv.
Our flow will then be irrotational, as curlu = 0. Now equation (11) (b) says
(15)	0 = divu = Av
and so v must be harmonic as a function of x, for each time t > 0. Thus
if we can find a smooth function v satisfying (15) and Du(-,0) = g, we can
then recover u from v by (14).
How do we compute the pressure p? Let us observe that if u = Dv,
then u  Du = |D(|Du|2). Consequently (ll)(a) reads D (vt + ||Du|2) —
D(—p + h), in view of (13). Therefore we may take
(16)	vt + ^\Dv\2 + p = h.
This is Bernoulli’s law. But now we can employ (16) to compute p, since v
and h are already known.
4.4. CONVERTING NONLINEAR INTO LINEAR PDE
197
4.4.3. Hodograph and Legendre transforms.
a. Hodograph transform.
The hodograph transform is a technique for converting certain quasilinear
systems of PDE into linear systems, by reversing the roles of the dependent
and independent variables. As this method is most easily understood by
an example, we investigate here the equations of steady, two-dimensional,
irrotational fluid flow:
(17)
' (a) (cr2(u) - (u1)2)^ - u1u2(u^2 +u2X1)
*	+(°'2(u) - (u2)2)ux2 = 0
. (b) игХ2 - u2X1 = 0
in R2. The unknown is the velocity field u = (it1, it2), and the function
cr(-) : R2 —> R, the local sound speed, is given.
The system (17) is quasilinear. Let us now, however, no longer regard
u1 and u2 as functions of Xi and x2:
(18)
u1 = U1(xi,X2), U2 = U2(xi,X2),
but rather regard x1 and x2 as functions of щ and U2'.
(19)
X1 = ^(Ul, U2), X2 = X2(U\,U2}-
We have exchanged sub- and superscripts in the notation to emphasize the
interchange between independent and dependent variables.
According to the Inverse Function Theorem (§C.5) we can, locally at
least, invert equations (18) to yield (19), provided
(20)
5(1?, U2)	12	1	2
9(xi,x2) “	°
in some region of R2. Assuming now (20) holds, we calculate
w22 =	-w2! = -Jx2ui
ux2 = ~Jxl2> ux! = Jx2U2.
We insert (21) into (17), to discover
' (a) (cr2(u) - “i>22 + U!U2(x^2 +x2J + (cr2(u) - г
. (b) xl2 - x2± = 0.
(21)
(22)
^=0
This is a linear system for x = (a;1, x2), as a function of и = (гц, u2).
198
4. OTHER WAYS TO REPRESENT SOLUTIONS
Remark. We can utilize the method of potential functions (§4.4.2) to sim-
plify (22) further. Indeed, equation (22) (b) suggests that we look for a single
function z = z(uj such that
a;1 = zU1
9
X£ = zU2.
Then (22) (a) transforms into the linear, second-order PDE
(23)	(cr (u)	U-^^Zu2U2 "h 2'U1'U2ZuiU2 T (^ (^)	^2)%uiui — 9.
div
b. Legendre transform.
A technique closely related to the hodograph transform is the classical
Legendre transform, a version of which we have already encountered before,
in §3.3. The idea is to regard the components of the gradient of a solution
as new independent variables.
Once again an example is instructive. We investigate the minimal sur-
face equation (cf. Example 4 in §8.1.2)
Du
(l + lDul2)1^
which for n = 2 may be rewritten as
(24)	(1 + UX2)uX1Xl 2'UI1'Ux2'llx1x2 + (1 + 'Uxi)V‘X2X2 = 0-
Let us now assume that at least in some region of R2 , we can invert the
relations
(25)	p1 = uX1(xi,x2), p2 = uX2(xi,x2),
to solve for
(26)	x1 = x1(p1,p2), x2 = x2(pi,p2)-
The Inverse Function Theorem assures us we can do so in a neighborhood
of any point where
(27)	J = det D2u 7^ 0.
4.5. ASYMPTOTICS
199
Now define
(28)
n(p) :=x(p)-p-u(x(p)),
where x = (ж1, a;2) is given by (26), p = (р1,рг)- We discover after some
calculations that
(29)
UXiXl — JVp2P2
*	~	*^^P1P2
' 11X2X2 ~ JVpiPl-
Upon substituting the identities (29) into (24), we derive for v the linear
equation
(30)	(1 + p%)vP2P2 + 2p1p2vP1P2 + (1 + pj)vpipi = 0.
Remark. The hodograph and Legendre transform techniques for obtaining
linear out of nonlinear PDE are in practice tricky to use, as it is usually not
possible to transform given boundary conditions very easily.	□
4.5. ASYMPTOTICS
It is often the case that even when explicit representation formulas can be
had for solutions of partial differential equations, these are too complicated
to be of much immediate use. In such circumstances it sometimes becomes
profitable to study the formulas in various asymptotic limits, whereupon
simplifications often appear.
Following are several rather complicated examples, illustrating typical
issues involved in asymptotics for PDE. The results in this section are ex-
plained only heuristically, mostly without formal proofs.
4.5.1. Singular perturbations.
A singular perturbation is a modification of a given PDE by adding a
small multiple г times a higher order term. In accordance with the informal
principle that the behavior of solutions is governed primarily by the high-
est order terms, a solution ue of the perturbed problem will often behave
analytically quite differently from a solution и of the original equation.
Example 1 (Transport and small diffusion). We illustrate this idea by
studying formally the effects of small diffusion upon the transport of dye
within a moving fluid in R2.
200
4. OTHER WAYS TO REPRESENT SOLUTIONS
Flow of dye without diffusion
Suppose we are given a smooth vector field b : R2 —> R2, b = (ft1, ft2),
representing the steady fluid velocity. Assume dye has been continuously
injected at unit rate into the fluid at the origin, and let u(x) represent the
density of dye at the point x € R2, x = (xi,X2). Then, formally at least as
we shall see,
(1)	div(ub) = <50 in R2,
where <5o is the Dirac measure on R2 giving unit mass to the point 0. This
PDE implies that the dye density is transported with the fluid motion at
points x 0.
Consider now for e > 0 the singular perturbation:
(2)	—sAu£ + div(u£b) = <50 in R2.
The new term “sA” represents a small, isotropic diffusion of the dye within
the background fluid motion. We are interested in understanding in an
approximate way the structure of the solution u£ of (2) and, in particular,
describing if and how u£ approximates и for small s > 0.
a.	Analysis of problem (1).
We turn our attention first to the unperturbed PDE (1). Consider the
characteristic ODE
, .	f x(t) = b(x(t)) (t > 0)
1 }	( x(0) = 0,
the solution x(t) = (a;1(t), r2(t)) of which we assume to trace out a curve
C, as drawn.
Given a point x E R2 near C, we write
(4)
x = x(0 + уИх(*))>
4.5. ASYMPTOTICS
201
where 1/ = (z/1,//2) is the (upward pointing) unit normal to С, у e R, and
t is the time required for the solution of the ODE (3) to reach the point
x(t) along C closest to x. We hereafter regard (t,y) as providing a new
coordinate system near the curve C; so that x = (x\y,t),x2(y,t)).
Using (3) and (4), we compute
d(x\x2}
d(t, y)
= det
dx1
dt
dx2
~3F
dx1
dy
dx2
~3y
= det
b1 + yiA
b2 + yv2
Let us write a = |b|, v = (—b2,b^/a and i> -- —акт — —кЬ (where a =
speed, к = curvature, r =	— unit tangent). We then simplify, to obtain
(5)
д(х\х2)
d(t, y)
= cr(l - «у).
Return now to the PDE (1), which we rewrite to read
(6)	b  Du + (div b)u = bo in R2.
As in §3.2 we see и = 0 off the curve C. Let us next guess и has the form
(7)	u(x) = p(t)b(y)
in the (t, y)-coordinates, 6 denoting the Dirac measure on R giving unit mass
to the origin.
What is p(t)? To compute it, take R to be a small, smooth region in
the (a?i, a?2)-plane, with boundary intersecting the curve C at the points
x(ti) and x(t2), 0 < ti < t2- Let R' denote the corresponding region in the
(t, y)-plane. Then using (5) we calculate
[ udx= f p(t)b(y)cr(t)(l — ку) dydt = f p(f)a(t)dt.
Jr Jr'	Jh
Now fR и dx represents the total amount of dye within the region R, which
is to say, the total amount released between times ti and t2- This is simply
t2 — ti. Thus
/ p(t)a(t)dt = t2 - ti-
Jh
This identity holds for all 0 < ti < t2, and so p(t) = a(t)~r. Hence (7) says
(8)	u(x,t) = S(y)/a(t)
202
4. OTHER WAYS TO REPRESENT SOLUTIONS
is a solution of (1), for <r(t) := |b(x(t))|, t > 0. In other words, и represents
the density along the curve C of the dye, whose concentration varies inversely
with the speed of the fluid.
We can further confirm this formula as follows. Let v G C^°(R2). Then
using (5), we compute
[ Dv -budx = [ Dv • b^j^-cr(t)(l — ny) dydt
JR2	JR2 a(t)
= f Dv(x(t))  b(x(t)) dt
Jo
Г°° d
= Jo dtv^t^dt= ~V^'
Hence we may indeed interpret и defined by (8) as a weak solution of the
unperturbed PDE (1).
b.	Analysis of problem (2) for 0 < e << 1.
We look now at the perturbed problem (2). We expect that at time
t > 0, the diffusing dye will fill a ball of radius approximately O((et)1//2)
about the point x(t). The dye will thus be mostly concentrated in a plume
as drawn, about the curve C.
We wish to understand the structure of the solution u£ of (2) within this
plume, as e —> 0.
Since the width is presumably of order ©(s1/2) for times 0 < ti < t < t2
and the total mass of dye corresponding to the same time interval is <2 — <1,
we expect u£ to be of order O(s-1/2) along C. This suggests that for us to
understand the asymptotics as s —> 0, we should turn our attention to the
rescaled variables
(9)
z := £ 1/2y, v£ := E1/2!!6,
the powers of s selected so that z,v£ = 0(1).
4.5. ASYMPTOTICS
203
We must therefore rewrite the PDE (2) in terms of the new variables t, z
and v£. For this, we need first to study the structure of the velocity field b
along the curve C. Let us therefore write
(10)	b = cr(t)r + {a(f)r + &(t)u}y + O(y2),
where, as noted before, т = Ъ/а is a unit tangent vector to C. Now for any
smooth function w:
and	dx2 Wt = wxi-&- +Wx2~fa = WZ1<7(1 - «у)7-1 + wX2a(l - ку)т2 dx1	dx2 Wy = wxi~d^- +Wx2~g^ = wxivl +wX2v2.
Thus	
(И)	I	wtu2 — Wya(l — ку)т2 1 WX1 ~	<7(1 - Ky) I	_ -WtU1 + Wy<7(l - ку)?-1 1 WX2 — I	<7(1 - ку)
Therefore using (10), (11) (for w = u£), we can compute:
b • Du£ — [ат + (ат + /Зи)у + O(y2)]  Du£
=тгЧ+-tFh++°to2i D“e D-
(1 - ку) cr(l - ку) y
Since v£ = £l/‘2u£, z = s-1/2y, we can rewrite the foregoing as
(12)	b • Dv£ = vf + /3zv£ + ©(e1/2).
Similarly, we calculate using (10) and (11) (for w = Ьг,Ь2) that
divb = ^4^j^1'2 “ 62|/I1 +17'6» “
+ [/(ат2 + (3v2) - т2(атг + ^z/1)] + O(y)
=	+ &} + O(y).
\(T	/
Here we used the identity т = аки. It follows that
(13)	(divb)ue = (— + /3^ v£ + (^(e1/2).
204
4. OTHER WAYS TO REPRESENT SOLUTIONS
In addition, a similar heuristic argument, the details of which we omit,
suggests that
(14)
eAu£ = vzz + O(e1/2).
Combining now (12)-(14) and recalling (2), we at last deduce v£ satisfies
(15)
vt ~ v£zz + (J3zv£)z + -v£ = O(e1/2).
<7
We suppose now that as e —> 0, the functions v£ converge in some sense
to a limit:
(16)	v£ —> v in R2.
Then presumably from (15) we will have
(17)	vt — vzz + (J3zv)z + -v = 0 in R x (0, oo).
<7
We therefore expect
(18)	u£ = e-1/V = e-1/2(v + o(1)),
with v solving (17). The PDE (17) is consequently a parabolic approximation
(in the variables t, z = E-1/2y) to our elliptic equation (2). The proper initial
condition should be
(19)	v = on R x {t = 0}.
cr(O)
We will see in Problem 6 that an explicit solution of (18), (19) can be
found, in terms of the solution of an ODE involving (3-
4.5.2. Laplace’s method.
Laplace’s method concerns the asymptotics as e —> 0 of integrals involv-
ing expressions of the form e“;/£, I denoting some given function.
Example 2 (Vanishing viscosity method for Burgers’ equation). We next
investigate the limit as e —> 0 of the solution u£ of the initial-value problem
for the viscous Burgers’ equation
( uf + u£u£ — eu£ = 0 in R x (0, oo)
[	u£ = g on R x {t = 0}.
4.5. ASYMPTOTICS
205
Remembering formula (10) from §4.4.1, we note
(21)
u£(x,t) =
roo x—V
-oo t e
~K(x,y,t)
2e dy
— K(x,ytt)
в 2e
dy
for
(22) K^yjty.= ^-^L + h(y) (z,yeR, t>0),
where h is an antiderivative of g.
What happens to u£ as £ —> 0? Mathematically the term U£uxx” in (20)
makes the partial differential equation act somewhat like the heat equation,
in that the solution u£ is infinitely differentiable in Rn x (0, oo), in spite of
the nonlinearity. This follows from the explicit formula (21). On the other
hand, an obvious guess is that the solutions u£ should converge as £ —» 0 to
a solution и of the conservation law
(23)
in R x (0, oo)
on R x {t — 0}.
Physically, we regard the term U£u£xx” as imposing an “artificial viscosity”
effect, which we are now sending to zero. We expect that this vanishing
viscosity technique should allow us to recover the correct entropy solution и
of (23), which may have discontinuities across shock waves, as the limit of
the solutions u£ of (20), which are smooth.
We must understand the limiting behavior of the expression on the right
hand side of (21), as £ —> 0.
LEMMA (Asymptotics). Suppose that k, I : R —> R are continuous func-
tions, that I grows at most linearly and that k grows at least quadratically.
Assume also there exists a unique point yo E R such that
fc(yo) = minfc(y).
Then
(24)
firn
= *(Уо)-
206
4. OTHER WAYS TO REPRESENT SOLUTIONS
Proof. Write ко = к(уо). Then the function
fcp-^(y)
в e
u£(y) :=----------. ,, .—	(y £ R)
^evy7 poo ko~kM ,	77	7
I e * dz
J —oo
satisfies
(25)
Me > 0,
p£(y) —> 0 exponentially fast for у yo, as e —> 0.
Consequently
Г° l(y)e e dy f°°
Ит	----- = lin?> / l(y^e(.y) dy = Ы-
e^o f^e-^dy
□
Return now to (21), (22). We observe K(x, y, t) = tL (^-^) + h(y),
where L = F* for F(z) = According to the analysis in §3.4, for each
time t > 0 the mapping у !-»• K(x,y,t) attains its minimum at a unique
point у = y(x, t) for all but at most countably many points x. But then the
lemma implies
(26) lim u'(x, t) =	= G f	= U(I, ()
e—>0	t	\ t )
The final equality in (26) is the Lax-Oleinik formula for the unique en-
tropy solution of the initial-value problem (23). It is a powerful endorsement
of the methods from §3.4 that this formula has reappeared in the context of
vanishing viscosity. (See also Problem 3.)	□
We will later discuss the vanishing viscosity method for symmetric hy-
perbolic systems in §7.3.2, for Hamilton-Jacobi equations in §10.1, and for
systems of conservation laws in §11.4.
4.5.3. Geometric optics, stationary phase.
This section investigates the behavior of certain highly oscillatory so-
lutions of the wave equation. We begin with some crude, but instructive,
calculations.
4.5. ASYMPTOTICS
207
a. Geometric optics.
Example 3 (Oscillating solutions). Let us once more turn our attention to
the wave equation
(27)	ин — Ди = 0 in Rn x (0, oo),
and we now regard the solution и as taking complex values. We fix £ > 0
and seek a solution и = u£ of (27) having the form
(28)	u£(x, t) =	'a£(x,t) (xeRn,t>0),
the real-valued function p£ representing the phase, and the real-valued func-
tion a£ representing the amplitude. The proposed form (28) for the solution
is called the geometric optics ansatz*. The idea is that highly oscillatory
solutions of the wave equation can be understood by studying a PDE for the
phase function in the limit as e —> 0. Following is a formal demonstration.
Substituting (28) into (27), we find after some computations that
0 = u£tt - Ди£ = e^£ г—^£- a£ + 2^ +
_	(^ae _	. 2iDP£ ' Da£ . Дае
С-	I	<л>	q U> |	|
\ £ £2 £
We cancel the term etp£'£ and take the real part of the resulting expression,
to find
(29)	a£((pf)2 -\Dp£\^ = e\a£tt- Д^)-
Now if as £ —> 0
(30)	p£ —> p, a£ —> a	0
in some sense, then presumably from (29) it follows that
(31)	pt ± \Dp\ = 0 in R” x (0, oo).
We may informally regard the straight line characteristics of these Hamilton-
Jacobi PDE as rays along which the solution (28) concentrates in the high-
frequency limit as £ —> 0.
More generally, let us consider the second-order hyperbolic PDE
(32)	utt ~ akl(x)uXkXl =0 in Rn x (0, oo)
k,l=l
* ansatz = formulation (German).
208
4. OTHER WAYS TO REPRESENT SOLUTIONS
with akl = alk (k, I = 1,..., n). We again look for a complex-valued solution
и = u£ of the form (28), and calculate
n
0 = uft - V akluxkx,
k,l=l
/ . e	/ e\2
= e^/e(^Lae_ (Pl\ ae +
\ E	\E )
-е^/4^ак1С^а£-
4,z=i e
2ipfaf F \
-^+a4
PxkPx,£ , ^Px^xt , e
-2 a	c- ‘t'“®
We once again cancel elp£//£ and take real parts to find
«e [ (pD2 - о^РхьРч ] = e2 [ a£tt - 52 aklaxkxt ] 
у	к,1=1	у	у	к,1=1	у
Hence if (30) holds in some sense, we may then expect
/	\ 1/2
(33)	pt ± I 5? aklpXkpXl j =0 in Rn x (0, oo).
\k,z=i	У
See below and also §4.6.1, §7.2.4 for further elaboration of these ideas. □
b. Stationary phase.
The foregoing example suggests that the Hamilton-Jacobi PDE (31)
somehow “controls the high frequency asymptotics for the wave equation”.
However the range of validity of the geometric optics ansatz is highly un-
certain in the preceding strictly formal computations. To understand more
clearly the behavior of the solution, we employ next the method of station-
ary phase, which is a variant of Laplace’s method had by replacing the —1
in the exponent (cf. §4.5.2) with i.
Example 4 (Stationary phase for the wave equation). Look again at the
initial-value problem for the wave equation
( u£tt - Nu£ = 0 in Rn x (0, oo)
| u£ = g£, u£t=Q on Rn x {t = 0},
where we hereafter assume g£ to have the rapidly oscillating structure
(35)	9£(x) = a(x)e (x G Rn).
4.5. ASYMPTOTICS
209
Here e > 0, a,p e C^°(Rn), and we suppose
(36)	Dp ф 0 on the support of a.
Utilizing formula (26) from §4.3.2, we can write
U£(z,t) = , * /9 [	+ei^y-t\y\)>) dy G Rn, t > 0)
(2тг)п/2 jRn 2
Invoking (35), we see
(37)	u£(x, t) = ^(7£ (x,t) + I£_(x, tf),
where
7|(x, t) =	[	[ a(2)e<(—dydz.
(2?r)n Jr* JR"
Changing variables gives
(38)	= f f a(2)ef*‘(«'»’«-‘> dydz,
{2тге)п J^n Jjgn
for
(39)	ф±(х, у, z, t):=(x- z)-y± t\y\ + p(z).
We want to study the asymptotics of as e —> 0. Let us pause in this
example and develop some general machinery, which we will later apply to
(38), (39).	□
Example 4 motivates our considering general integral expressions of the
form
(40)	Ie := [ e^a(y)dy (x e Rn),
JRn
where а, ф are smooth functions, a has compact support, and e > 0. We
wish to understand the limiting behavior of I£ as e —> 0.
We first examine the special case that ф is linear in y:
LEMMA 1 (Asymptotics for linear terms). Let a G C^°(R”) and p 6 Kn,
p 0. Then for m = 1,2,...
[ еёр уа(у) dy = O(sm) as £ —> 0.
JR"
210
4. OTHER WAYS TO REPRESENT SOLUTIONS
Proof. Without loss we may assume p = (pi,... ,рп), Pi / 0. Then for
m = 1,2,...
г i	/ £ \m f dm i
/ e'™a(y) dy=(~)	/ —(e7^)a(y) dy
Rn	\гР1 /
□
Next we suppose ф is quadratic in y:
LEMMA 2 (Asymptotics for quadratic terms). Let a G C^°(Rn) and sup-
pose A is a real, nonsingular, symmetric matrix. Then
1 f i л	gizsgnX
(41>	s»“^=n^W0)+o<£))
Here sgn A, the signature of A, denotes the number of positive eigenvalues
of A minus the number of negative eigenvalues.
Proof. 1. First we claim for each ф G Cf?°(Rn) that
lim [ f eix Ax-6\x\2~ix^y)dxdy
,лп. 8^0+J^J^n.
(42)	n/2	r .	,
= йЛ л .-172sgn A / e~^'A~ dy-
I det Al1/-!
To confirm this, we start by assuming A is diagonal:
(43)	A = diag(Ab..., An) (Afe / 0, к = 1,... ,n).
Now for fixed y, A G R and ё > 0, we have
f ei^2-8x2-ixydx = e^^ [ e(iX~S>)(x-2?ЙЬ)) dx
Jr	Jr
_j2._
g 4(гХ —6)
= (ё - гА)1/2
where Г = {z = (6—гА)1/2^— 2(Ixs‘)') I x and we take Re(6 —гА)1/2 >
0. Thus Г is a line in the complex plane, which intersects the x-axis at an
angle less than . We consequently may deform the integral over Г into the
4.5. ASYMPTOTICS
211
integral along the real axis: see Problem 7. Hence Jre 2,2 dz = JRe x2dx =
7Г1/2, and thus
У2 ,
g 4(iA—6)
Since A has the diagonal form (43), we consequently deduce
Js(y)-= [ eixAx-6\x\2-ix-ydx
n
Ук
ei(iXk-6)
П
= »/2 TT ------------
k=l
2. Now let ф e C~(Rn). Then
[ <KyWy)dy = 7Tn/2
JR"
n _________yl____
" e4(iAfc-«)
£1 (4 - iAt)V2 iy-
Applying the Dominated Convergence Theorem, we deduce
iy?
П	~ АЛ
(44) lim [ ф(у)Т6(у^у = тгп<
6^0 JR"
Recall we are supposing Re(—гАд,)1/2 > 0. Thus if Ад > 0, (—гАд)1/2 =
|Afc^2e~. If instead Ад < 0, then (—гАд)1/2 = |Адр/2е*г. Therefore
JJ(-^Afc)1/2 = | det Ap/2e-1r sgn A,
fc=i
and so (44) gives (42), provided A is diagonal.
If A is not diagonal, we rotate to new coordinates to diagonalize A, and
again verify (42).
3. Let us now write a£(y) := e^y'Ay. Then if a £ C£°(Rn),
/ a(x)ae(x)dx = / a(y)a£(-y) dy.
JR"	JRn
According to (42) (with ^A replacing A):
^.n/2
^.n/2
=й^е,?дал<1+0(£|л)'
212
4. OTHER WAYS TO REPRESENT SOLUTIONS
a£ interpreted as in (42). Consequently
1 Г	ет88пЛ Г
a(x)a£(x)dx =	/ a(y)(l + O(e|y|2)) dy.
JR"	I det Al1/2 J^n.
But a(y) dy = (2тг) 2a(0) and a(y)\y\2dy < oo. Formula (41) follows.
□
For a general phase function ф, we will employ the following result to
change variables and thereby convert locally to one of the earlier cases.
LEMMA 3 (Changing coordinates). Assume ф : Rn —> R is smooth.
(i) Suppose that
£>ф(0) 0.
Then there exists a smooth function Ф : Rn —> Rn such that
Г Ф(0) = 0, БФ(0) = I, and
t ф(Ф(х)) = $(0) + Вф(0) • x for |a;| small.
(ii) (Morse Lemma) Suppose instead that
£>ф(0) = 0, detD2<£(0) ± 0.
Then there exists a smooth function Ф : Rn —> R” such that
( Ф(0) = 0, БФ(0) = I, and
( ) t ф(Ф(х)) = ф(0) + ^x -Е2ф(0)х for |x| small.
In other words, we can change variables near 0 to make ф affine in case (i),
quadratic in case (ii).
Proof. 1. Assume rn := D<^(0)	0. Then there exist vectors ri,... , rn_i
so that {rfc}£=1 is an orthogonal basis of Rn. Define f : R" x Rn —> Rn by
f (x, y) := (n  (y - x),..., rn_i  (у - x), ф(у) - <£(0) - Оф(0)  x).
Therefore
(ri \
Гп/
the {rfc}^=1 regarded as row vectors, and so detD2/f(0,0)	0. The Implicit
Function Theorem (§C.6) implies we can find Ф : Rn —> Rn such that
Ф(0) = 0 and
f(x, Ф(х)) = 0 for |x| small.
4.5. ASYMPTOTICS
213
In particular,
Г <^(Ф(а:)) = </>(0) + D</>(0)  x
t г*, • (Ф(ж) — x) = 0 (fc = 1,..., n — 1).
Differentiating with respect to x, we deduce as well
(РФ(0) - L)rfe = 0 (k = l,...,n)
and so £>Ф(0) = I. This proves assertion (i).
2.	Fix x G R". Then ^(t) := </>(tz) satisfies
V>(1) = ^(0) + V>'(0) + [ (1 - t)/(t) dt.
Jo
Thus if D</>(0) = 0, we have
(48)	</>(«) = </>(0) + ~x • А(ж)ж
for the symmetric matrix
A(z) := 2 / (1 — t)D2^>(tz) dt.
Jo
Observe A(0) = В2ф(0). Let us hereafter suppose D2</>(0) is nonsingular,
and so the same is true for A(x), provided |z| is small. Furthermore, we
may assume upon rotating to new coordinates if necessary that
A(0) = D2</>(0) is diagonal.
3. We now claim that there exists for each m G {0,1,..., n} a smooth
mapping Фт : R" —> R", such that
(49)
' Фт(0) = 0, ИФт(0) = I, and
, ^(Фт(ж)) — <^(0) + I 52г=1 ФхгХг (0)x2 + 5 52iJ=m+lata(^)XjXj,
for |ж| small, where Am = ((am)) is smooth and symmetric.
Observe in particular that (49) implies
(50)	a^(0) = ^^(0) (i,j = m + l,...,n),
and thus a™+1’m+1(z)	0 for |ж| sufficiently small.
214
4. OTHER WAYS TO REPRESENT SOLUTIONS
4. Assertion (49) for m = 0 is (48) with Aq = A and Фо the iden-
tity mapping. So next assume by induction that (49) holds for some m G
{0,..., n — 1} and write
фт(х) := </.(Фт(х)).
Then
1 m	j n
(51) фт(х) = </>(0) + -\2</>зд(0)хг2 + - У? a%(x)xiXj for |ж| small.
i=l	i,j=m+l
Define a mapping IIm+i : R" —> R", Пт+1(у) = x, by writing
Пт+1(у) :—
(dm (,У)О-
------------75J2
<Pxm+1xm+1 ('J J
n m+lj'z \
V- yjdm ’7(y)1
' / j m+l,m+l/ \ J ’ ’ ’ ’ ’ Уп
j=m+2am (У)
for small |y|. It follows then from (51) that
<Ш = </>(0) + T 22 </>зд(0)ж2 + 22 bm+l(y)xixL
г=1	t,j=m+2
where
IJ z \	Orn+1’*(j/)Orn+1’J({/) • ’	I о
flm(w -	7 I, 3 = m + 2, . . . , n
am \У)
0	otherwise.
:=
Since D2</>(0) is diagonal, (50) implies
Пт+1(0) = 0, Dnm+i(0) = I.
Consequently we can define for small |ж| the inverse mapping Sm+i :=
у = Sm+i(z). Therefore
i=l	t,j=m+2
for Am+i := Bm+i о Sm+i. This is statement (49), with m + 1 replacing m
and with Фт+1 := Фт о Sm+1.
The case m = n is assertion (ii) of the theorem.	□
4.5. ASYMPTOTICS
215
The stationary phase method. We can at last combine the information
gleaned in Lemmas 1-3 to explain informally the stationary phase technique
for deriving the asymptotics of
e e a(y)dy
as £ —> 0. We will assume
(52)
D</> vanishes within the support of a
only at the points yi,.   ,yN,
and furthermore
(53)	D20(yk) is nonsingular (к = 1,... ,N).
Fix 6 > 0 so small the balls {B(yk, ^)}fcLi are disjoint. Then for m = 1,...,
e e
-u£iW)
a(y)dy ^O(em)
as £ —> 0.
This follows since we can employ Lemma 3,(i) to change variables near any
point у (Jill -^(1/, d) to make ф affine, with nonvanishing gradient. Thus
Lemma 1 and a partition of unity argument gives the stated estimate.
On the other hand if 5 > 0 is small enough, we can employ Lemma 3,(ii)
to compute
( ei^a(y')dy= [	е^(ф(1))а(Ф(х))| det £>Ф(х)| dx
В(ук,6)	4ф-ЦВ(ук,6))
J*-\B(yk,6y)
| det ЛФ(х)| dx
= e^ld	+°W).
IdetD^yfe)!1/^
according to Lemma 2. We thereby obtain the asymptotic formula
TV	i<t>(.vk')
(54)	4 = (2,r£)”/2 Y. Mtnw Ц1/ 4n(D1*ta>,(o(») + OW).
“ | det Dtytyk)]1/2
as e
0.
216
4. OTHER WAYS TO REPRESENT SOLUTIONS
Example 4 (Stationary phase for the wave equation, continued). We can
now apply the foregoing theory to (38), which states
!> = test f f <№
(2тге) Jjjn Jjgn
This is of the form (40), with (z, t) replacing x and (y, z) replacing y.
Define for fixed x € Rn, t > 0, the set where the mapping (y, z) e->
Ф+(х, у, z, t) is stationary:
S+ := {(y, 2) | DytZ</)+(x, y, z, t) = 0}.
Recall from (39) that ф+(х, y,z,t) = (x — z)  у + t|y| + p(z); and so
Г Оуф+ = (ж - z) + (y / 0),
t Егф+ = -y + Dp(z).
Consequently
(55)	S+ = ^(y,z))x = z~^^, y = Dp(z)j,
and we here and henceforth assume that у = Dp(z) 0 if (y, z) e S+.
Now if у 0,
2	%.ф+\	=/^P(s) -M
’^+	\D^	-I &p)'
for P(y) := I — We have у = Dp(z) on the stationary set S+, and so
det(D2^+) - (-1)"det (l - -l-D2pP(Dp)\.
\ |Dp|	)
Now the symmetric matrix S(z) := ^^^D2pP(£>p) has n real eigen-
values Ai(z),..., An(z). Since E(z)Dp = 0, we may take An(z) = 0. The
other eigenvalues Ai(z),..., An-i(z) turn out to be the principal curvatures
Kq(z),..., Kn-i(z) of the level surface qf p passing through z. Since P2 = P,
the nonzero eigenvalues of y^D2pP(Dp) are the nonzero principal curva-
tures. Thus on S+,
n— 1
(56)	det(D2^+) = (-l)n Ip1 -
i=l
4.5. ASYMPTOTICS
217
We apply the stationary phase estimates for xq g Rn and small to > 0.
If to is small enough, we can invoke the Implicit Function Theorem to solve
uniquely the expressions
xq = z
. Dp(z)
°\Dp(zW
у = Dp(z)
for yo = y(xo,to), zo = 2(zo,to)- Thus the asymptotic formula (54) (with
2n replacing n) implies
(57)
(Wo) = , *	[ [ a(z)e^+^°’y'z’t°'1 dydz
(27Г£)П jRn jRn
Д sgn(L>220+)
= 1Л + Г)2 Л 11/2 e'£0+	+ °(£)1 as £ —> 0,
| det D2^1/2
ф^. and Оугф+ evaluated at (хо,Уо> zo.to). Recall further that (56) gives us
an explicit function for det(Dy^+). A similar asymptotic formula holds for
IL(xo,to')- Since u£(xo,to) = j(7^(xo,to) + -fS(^o,io))> we derive detailed
information concerning the limits as £ —> 0, at least for small times to > 0.
□
Remark (Optics and stationary phase). It remains to discuss briefly the
connections between the formal geometric optics and the stationary phase
approaches. Recall that the former brought us to the two Hamilton-Jacobi
equations
(58)	pt ± |Dp| = 0
for the phase function of ue = aEe'^~ = (a + o(l))e p+^( \ Now the charac-
teristic equations for the PDE pt — \Dp\ = 0 are
(59)
, P(s) = 0,
as previously discussed in §3.2.2. In particular given a point x G Rn, t > 0,
where t is small, the projected characteristic x(-) is a straight line, starting
at the unique point z satisfying
z = x +1
Dp(z)
|Dp(2)|
But this relation is precisely what determines the stationary set S+ above.
Likewise, the characteristics of the partial differential equation pt + |Dp| = 0
determine the stationary set S~ for ф-.	□
218
4. OTHER WAYS TO REPRESENT SOLUTIONS
4.5.4. Homogenization.
Homogenization theory studies the effects of high-frequency oscillations
in the coefficients upon solutions of PDE. In the simplest setting we are given
a partial differential equation with two natural length scales, a macroscopic
scale of order 1 and a microscopic scale of order e, the latter measuring the
period of the oscillations. For fixed, but small, e > 0 the solution uE of the
PDE will in general be complicated, having different behaviors on the two
length scales.
Homogenization theory studies the limiting behavior u£ —> и as e —> 0.
The idea is that in this limit the high-frequency effects will “average out”,
and there will be a simpler, effective limiting PDE that и solves. One of
the difficulties is even to guess the form of the limiting partial differential
equation, and for this formal, multiscale expansions in £ may be useful.
Example 5 (Periodic homogenization of an elliptic equation). This example
assumes some familiarity with the theory of divergence-structure, second-
order elliptic PDE, as developed later in Chapter 6.
Let U denote an open, bounded subset of R", with smooth boundary
dU, and consider this boundary-value problem for a divergence structure
PDE:
(60)	Jint'
гг=0 in dU.
Here f : U —> R is given, as are the coefficients au (i,j = 1,... ,ri). We will
assume the uniform ellipticity condition
Y. > Ж12
M=1
for some constant 0 > 0 and all G R". We suppose also
(61)	the mapping у av(y) is Q-periodic (y G Rn),
Q denoting the unit cube in Rn. Thus the coefficients аг] (|) in (60) are
rapidly oscillating in x for small e > 0, and we inquire as to the effect this
has upon the solution us. (In applications u£ represents, say, the electric
field within a nonisotropic body having small-scale, periodic structure.)
In the following heuristic discussion let us assume
(62)	u£ —> и as £ —> 0
4.5. ASYMPTOTICS
219
in some suitable sense and try to determine an equation which и satisfies.
The trick is to suppose u£ admits the following two-scale expansion:
(63)	u£(z) = uq(z, x/e) + £Ui(z, x/e) + £2u2(x,x/e) + ...,
where u, : U x Q —> R (г = 0,1,...), щ = Ui(x, y). We are thus thinking
of the terms щ as being both functions of the macroscopic variable x and
periodic functions of the microscopic variable у = f • The plan is to plug (63)
into (60), and to determine thereby Uq, Ui, etc. We are primarily interested
in и — uq.
Now if u(z) = w(x,x/e) for some function w = w(x,y), then =
(&1 +	• • • ’n- Thus’ writin§
n
Lv = -
i,j=l
we have
(64)
T 1 r
L =	4—1/2 + 1/3,
£2 £
where
(65)
'(a) Lxw := ~^=1(аЧ(у)и>у^,
< (b) L2w := -	(y)wXi)Vj + (al^y)wyi)Xj
, (c) L3w :=
Next plug the expansion (63) into the PDE Lu£ = f, and utilize the decom-
position (64), (65) to find
1	1
-z-LiUq-I—(Tiui + L2U0) + (TiU2 + Ь2щ + Ьз^о)
£z	£
+ {terms involving £,£2,... } = f.
Equating like powers of £, we deduce
(66)
(a)	L1U0 = 0,
(b)	L\U\ + L2uq = 0,
(c)	Liu2 + L2ui + L3uo = /, etc.
We examine these PDE to deduce information concerning uo,ui,U2-
Now in view of (65)(a), (66)(a) for each fixed x, uo(x,y) solves Lyuo = 0
220
4. OTHER WAYS TO REPRESENT SOLUTIONS
and is Q-periodic. It turns out that the only such solutions are constant in
y. Thus in fact
(67)
Uq = u(x) depends only on x.
Next employ (67), (65) (b), (66) (b) to discover
(68)	Liui =	аг3(у)у^.
ij=l
We can as follows separate variables to represent щ more simply. For г =
1,..., n, let хг = x’(y) solve
, _OA	J £i ^ = - a” (y)yj in Q
(69)	<
l Хг Q-periodic.
As the right hand side of the PDE in (69) has integral zero over Q, this
problem has a solution уг (unique up to an additive constant). Here we are
applying the Fredholm alternative: see Chapter 6.
Using (69) we obtain
(70)	U!(x,y) = - ^x\y)uXi{x) + Й1(ж),
i=i
til denoting an arbitrary function of x alone.
Finally let us recall (66) (c):
(71)
L\U2 = f —	— L3U0.
In view of (65)(c) this PDE will have a Q-periodic solution (in the variable
y) only if the integral of the right hand side over Q is zero. Thus we require
(72)
/ £2ui + £3uody= / fdy = f(x).
JQ	JQ
Owing to (65) (b) and (70),
4.6. POWER SERIES
221
Since и = uo, this calculation and (72) imply
“52 (/ aV(y)-52a'’fe(y)X^(y)^pa:i^(^)=/(^)-
i,j=l \JQ	fc=l	/
That is,
(73)	f - E"j=i ^uXiXj =f in U
(	(	и = 0 on dU,
where
(74) агз := / av(y) - ajfc(y)y^ (у) dy (г, j = 1,... ,ri)
JQ
are the homogenized coefficients, and уг solves the corrector problem (69)
(г = 1,... ,n). Thus we expect u£ —> и as £ —> 0, and и to solve the limit
problem (73).	□
This example clearly illustrates the power of the multiscale expansion
method. It is not at all readily apparent that the high-frequency oscillations
in the coefficients of (60) lead to a constant coefficient PDE of the precise
form (73), (74).
4.6. POWER SERIES
We discuss in this final section solving boundary-value problems for partial
differential equations by looking for solutions expressed as power series.
4.6.1. Noncharacteristic surfaces.
We begin with some fairly general comments concerning the solvability
of the /ct/l-order quasilinear PDE
(1) У2 aa(Dk xu,... ,u,x)Dau + ao(Dk 1u,... ,u, x) = 0
|=k
in some open region U C R". Let us assume that Г is a smooth, (n — 1)-
dimensional hypersurface in U, the unit normal to which at any point z° e Г
is i/(a;0) = v = (i/b ... ,pn).
Notation. The jth normal derivative of и at x° E. Г is
d3u
—— := > D и I/
va
dx}...d^ 1 " n
ai+--+an=j	Xn
222
4. OTHER WAYS TO REPRESENT SOLUTIONS
Now let go,..., gk-i : Г —> R be k given functions. The Cauchy problem
is then to find a function и solving the PDE (1), subject to the boundary
conditions
(2)
du	dk ru
U = go, ^-=gl,---, q bi = gfe-1 onr.
du	du* 1
We say that the equations (2) prescribe the Cauchy data go,. . . , gk-i on Г.
We now pose a basic question:
{Assuming и is a smooth solution of the PDE (1),
do conditions (2) allow us to compute all the partial
derivatives of и along Г?
This must certainly be so, if we are ever going to be able to calculate the
terms of a power series representation formula for u.
a. Flat boundaries.
We examine first the special circumstance that U = R" and Г is the
plane {xn = 0}. In this situation we can take и = en, and so the Cauchy
conditions (2) read
,4.	dk~lu	,
(4) u = g0, -x—= gi,  • •, —fc_x = gfe-i on {xn - 0}.
OXn	(jxn
Which further partial derivatives of и can we compute along the plane
Г = {xn = 0}? First, notice that since и = go on all of Г we can differentiate
tangentially, that is, with respect to Xi {i = 1,..., n — 1), to find
du
dxi
9go ал i \
— o = l,.
Since we also know from (4) that
du
a— = gi,
uXp
we can determine the full gradient Du along Г = {xn = 0}. Similarly, we
have
d2u
дХп dx^

. , 71 — 1)
d2u _
9^-92,
4.6. POWER SERIES
223
and hence we can compute D2u on Г. Next, we see
' d39o if
dxi дх dxm
д3и	Л I1	if
____________ _ OXiUXj
дхгдх3	Qq2	• c
dXl	11
, 93	if
i, j, m = 1,..., n — 1
i, j = 1,... ,n — 1; m = n
i = 1,..., n — 1; j = m = n
i = j = m = n
along Г, and so we can compute D3u there. Continuing, it is straightfor-
ward to check that employing the Cauchy conditions (4) we can compute
u, Du,..., Dk~xu on Г.
Difficulties will arise, however, when we try to calculate Dku. In this
circumstance it is not hard to verify that we can determine each partial
derivative of и of order к along Г = {xn = 0} from the Cauchy data (4),
except for the /ci/l-order normal derivative
dku
dxk'
Here, at last, we turn to PDE (1) for help. We observe from (1) that if
the coefficient a(o,...,o,fc) nonzero, we can then solve for
(5)
дки	1
dxk	®(o,...,o,fc)
aaDau + ao
 |a|=fc
a/(0,...,0,fe)
with the coefficients aa(|a| = k) and ao evaluated at (Dfe-1u,... ,u,x) along
Г. Now in view of the remarks above, everything on the right hand side
of equality (5) can be calculated in terms of the Cauchy data along the
plane Г, and thus we have a formula for the missing kth partial derivative.
Consequently we can in fact compute all of Dku on Г, provided
(6)
a(o,...,o,fc) 0 0-
We say that the plane Г = {жп = 0} is noncharacteristic for the PDE
(1), if the function a(0,.is nonzero for all values of its arguments.
Can we calculate still higher partial derivatives? Assuming the nonchar-
acteristic condition (6), we observe that we can now augment our list (4) of
Cauchy data with the new equality
(7)
dku
dxk
= 9k on Г = {жп - 0},
224
4. OTHER WAYS TO REPRESENT SOLUTIONS
gk denoting the right hand side of (5). But then we can, as before, compute
all of Dk+1 и along Г, except for the term
dk+1u
dxk+1 ’
Again we employ the PDE (1). We differentiate (1) with respect to xn,
evaluate the resulting expression on the plane Г, and rearrange to find
dk+1u _	1	,
dxk+1 a(o.....o,fc)
the dots denoting the sum of various expressions, each of which can be
computed along Г in terms of <?o, • • , <7fe- Consequently we can ascertain all
of Dk+1 и on Г, and an induction verifies that in fact we can compute all the
partial derivatives of и on the plane Г.
b. General surfaces.
We now propose to generalize the results obtained above to the general
case that Г is a smooth hypersurface with normal vector field к
DEFINITION. We say the surface Г is noncharacteristic for the partial
differential equation (1) provided
(8)	aava 0 on Г,
|a|=fc
for all values of the arguments of the coefficients aa (|a| = k).
THEOREM 1 (Cauchy data and noncharacteristic surfaces). Assume that
Г is noncharacteristic for the PDE (1). Then if и is a smooth solution of
(1) and и satisfies the Cauchy conditions (2), we can uniquely compute all
the partial derivatives of и along Г in terms ofT, the functions go,. ,gk-i>
and the coefficients aa (|a| = kfiao-
Proof. We will reduce to the special case considered above.
For this, let us choose any point ж0 € Г and recall §C.l to find smooth
maps Ф, Ф : R" —> R", so that Ф = Ф-1 and
Ф(ГП5(Лг))с{Уп = 0}
for some r > 0. Define
v(y) := и(Ф(у));
4.6. POWER SERIES
225
so that
(9)	u(x) = и(Ф(ж)).
It is relatively easy to check now v satisfies a quasilinear partial differ-
ential equation having the form
(10)	52 baDav + i»0 = 0.
|a| =k
2.	We claim
(П)	\o,...,o,k)	0 on {yn = 0}.
Indeed from (9) we see that for any multiindex a with |a| = k, we have
dkv	(	dkv 1
Dau = —j-(D^n)a + < terms not involving —г >.
дУп	I	дУп J
Thus from (1) it follows that
0 = 52 ciaDau + ao
|a|=fe
= / , аа(£>Фп) + <! terms not involving —;
|a|=fe	°Уп	°Vn J
and so
b(o,...,o,fc) = £ аа(Т>ФТ-
|а |=k
But РФ" is parallel to i/ on Г. Consequently b(o,...,fe) *s a nonzero multiple
of the term
52 aa^a / 0.
|a | =k
This verifies the claim (11).
3.	Let us now define the functions ho, hi,..., hk-i : R"-1 —> R by
dv	dk-1v
(12)	v = ho, — = hi,...,	, = hk-i on {yn = 0}.
dyn	dy*
Thus we can compute ho,   , hk-i near у — 0 in terms of Ф and the functions
go, • • • ,9k-i- But then, using (11) and the special case discussed above, we
see that we can calculate all of the partial derivatives of v on {yn — 0} near
У - 0.
And finally, upon recalling (9), we at last observe we can compute all
the partial derivatives of и on Г near x°.	□
226
4. OTHER WAYS TO REPRESENT SOLUTIONS
Remark. It is sometimes convenient to recast the noncharacteristic condi-
tion (8) into a somewhat different form, by representing Г as the zero set of
a function w : R" —> R. So assume that we are given a function w with
Г = {w = 0}
and Dw 0 on Г. Then v = on Г , and so the noncharacteristic
condition (8) becomes
(13)	52 a<*(Dw)a ± 0 on Г.
|a|=fe
□
4.6.2. Real analytic functions.
We review in this section the representation of real-valued functions by
power series.
DEFINITION. A function f : R" —> R is called (real) analytic near xo if
there exist r > 0 and constants {/a} such that
f(x) = ^f^-xOr (|x-X0|<r),
a
the sum taken over all multiindices a.
Remarks, (i) Remember that we write xa = ж"1 • • • x"n, for the multiindex
a — (ai,... ,a„).
(ii)	If f is analytic near zo, then f is C°° near zq. Furthermore the
constants fa are computed as fa = °	, where a! = aila^l • • • anl. Thus
f equals its Taylor expansion about xq:
f(x) = ^^^f(xo)(x~xo)a	< r).
a
To simplify, we hereafter take zo = 0.	□
Example. If r > 0, set
f{x) :=----------------r for Ы < r/y/n.
r — (®i + --- + xn) 11 /v
Then
4.6. POWER SERIES
227
We employed the Multinomial Theorem for the third equality above and
recalled that	This power series is absolutely convergent for
|ж| < г/y/n. Indeed,
У#7|х«|=У'ГМ±^±ЫУ<оо,
< J 7* I I (V	' * \	T	/
a ‘	fc=O 4	7
since |®i| + • • • + |z„| < jx^y/n < r.	□
We will see momentarily that the simple power series illustrated in this
example is rather important, since we can use it to majorize, and so confirm
the convergence of, other power series.
DEFINITION. Let
f = '^foixa, g = 52^®“
a	a
be two power series. We say g majorizes f, written
g^ f,
provided
да > |/a| for all multiindices a.
LEMMA (Majorants).
(i) If g 3> f and g converges for |x| < r, then f also converges for
|ж| < r.
(ii) If f = faxa converges for |ж| < r and 0 < Sy/n < r, then f has
a majorant for |z| < Sy/n.
Proof. 1. To verify assertion (i), we check
52\f<*xa\	   \Xn\an < °° if и <r-
a	a
2. Let 0 < Sy/n < r and set у := s(l,..., 1). Then |y| = Sy/n < r and
so 52a faya converges. Thus there exists a constant C such that
|/аУа| < C for each multiindex a.
In particular,
< c = £_< r
-№№ sl“l - A!’
But then
g(x) :=--------—---------; = С?
s-(ziH-------|-jn)	#1а!
majorizes f for |ж| < Sy/n.	□
228
4. OTHER WAYS TO REPRESENT SOLUTIONS
Remark. We will later need to extend our notation to vector-valued se-
ries. So given power series {fk}™=1, {<7fe}fcLi, we set f =
g = (g1,..., gm), and write
g » f
to mean
?»/ (k = l,...,m).
4.6.3.	Cauchy-Kovalevskaya Theorem.
We turn now to our primary task of building a power series solution
for the kth-order quasilinear partial differential equation (1), with analytic
Cauchy data (2) specified on an analytic, noncharacteristic hypersurface Г.
a.	Reduction to a first-order system.
We intend to construct a solution и as a power series, but must first
transform the boundary-value problem (1), (2) into a more convenient form.
First of all, upon flattening out the boundary by an analytic mapping (as
in §4.6.1), we can reduce to the situation that Г C {xn = 0}. Additionally,
by subtracting off appropriate analytic functions, we may assume the Cauchy
data are identically zero. Consequently we may assume without loss that
our problem reads:
(14)
' Y,\a\=kaa(Dk 1U,...,U,x)DaU
<	+ao(Dk~1u,... ,u,x) = 0
for |ж| < r
for |a/| < r, xn = 0,
r > 0 to be found. As usual we write x' = (a?i,..., a?n-i)-
Finally we transform to a first-order system. To do so we introduce the
function
___du du d2u dk~1u\
the components of which are all the partial derivatives of и of order less
than k. Let m hereafter denote the number of components of u by m, so
that u : Rn —> Rm, u = (u1,..., um). Observe from the boundary condition
in (14) that u = 0 for |a/| < r, xn = 0.
Now for к 6 {1,... ,m — 1}, we can compute ukn in terms of
Furthermore in view of the noncharacteristic condition n(o,...,o,fc) / 0 near
4.6. POWER SERIES
229
0, we can utilize the PDE in (14) also to solve for u™n in terms of u and
(15) I
Employing these relations, we can consequently transform (14) into a
boundary-value problem for a first-order system for u, the coefficients of
which are analytic functions. This system is of the general form:
= Ej=i B>(u’ж/)и^+c(u’ж/) forkl<r
u = 0 for |a/| < r, xn — 0,
where we are given the analytic functions Bj : Rm x Rn-1 —> Mmxm (j =
1,... ,n — 1) and c : Rm x Rn-1 —> Rm. We will write Bj = ((Ьк1У) and
c = (c1,..., cm). Carefully note that we have assumed {Bj}"z/ and c do
not depend on xn. We can always reduce to this situation by introducing if
necessary a new component um+1 of the unknown u, with um+1 = xn.
In particular, the components of the system of partial differential equa-
tions in (15) read
n—1 m
(16)	ukXn = ^^bkl(u,x'fulXj +ck(u,x') (fc = l,...,m).
j=iz=i
b.	Power series for solutions.
Having reduced to the special form (15), we can now expand u into a
power series, and, more importantly, verify that this series converges near
0.
THEOREM 2 (Cauchy-Kovalevskaya Theorem). Assume	and
c are real analytic functions. Then there exist r > 0 and a real analytic
function
(17)	u = UgXa
a
solving the boundary-value problem (15).
Proof. 1. We must compute the coefficients
(18)	=
a!
in terms of	and c, and then show that the power series (18) so
obtained in fact converges if |rr| < r and r > 0 is small enough.
230
4. OTHER WAYS TO REPRESENT SOLUTIONS
2.	As the functions {В7}"=^ and c are analytic, we can write
(19)	Bj(z,x') =	(j = 1,..., n — 1)
7,6
and
(20)	c(z, x') =
7,5
these power series convergent if \z\ + |a/| < s for some small s > 0. Thus
E?L>^Bj(0,0)	_ D?Dsxc(fi,0)
(7 + 6)!	,C7’6-	(7 + 6)!
for J = 1,..., n — 1 and all multiindices 7,6.
3.	Since u = 0 on {xn =0}, we have
Dau(0)
(22)	uQ =-----j— = 0 for all multiindices a with an = 0.
Now fix i G	1} and differentiate (16) with respect to 37:
n—1 m ✓	m	\	m
uk — V V I bklul	+ bkl и1 -I- V* bkl up и1 I -I- ck 4- Y^ ck up
UXnXi	/ a / I aXiXj	aXj	UJ,Zp aXi aXj I ' СХг ' / CZp aXi •
1=1	P=1	'	P=1
In view of (22), we conclude икпХ. (0) = ck. (0,0).
If a is a multiindex having the form a = (04,..., an-i, 1) = (a/, 1), we
likewise prove by induction that
DQ-ufe(0) = DQcfc(0,0).
Next suppose a = (a', 2). Then
(n—1 m	\
7Шь^ + ск) by <16)
J=1 f=l	У Xn
(n—1 mm	m
j=l 1=1	P=1	P=1
Thus
4.6. POWER SERIES
231
The expression on the right hand side can be worked out to be a polynomial
with nonnegative coefficients involving various derivatives of	and c,
and the derivatives D@u, where < 1.
More generally, for each multiindex a and each к € {1,we
compute
= p‘(...	.....D^u,... )|I=u=o,
where p& denotes some polynomial with nonnegative coefficients.
Recalling (18)—(21), we deduce for each a, к that
(23)	ua = qa(..., Bj^^,..., c^/, • • •, U/3, • • • )j
where
(24)	is a polynomial with nonnegative coefficients
and
(25)	(3n < an — 1 for each multiindex /3 on the right hand side of (23).
4.	Thus far we have merely demonstrated that if there is a smooth
solution of (15), then we can compute all of its derivatives at 0 in terms of
known quantities. This of course we already know from the discussion in
§4.6.1, since the plane {xn = 0} is noncharacteristic.
We now intend to employ (22)-(25) and the method of majorants to show
the power series (17) actually converges if |ж| < r and r is small. For this,
let us first suppose
(26)	B* > Bj (j = l,...,n-l)
and
(27)	c* » c,
where
BJ:=EBjW7a;6 o = i,...,n-i)
7,5
and
<= >=
7,5
232
4. OTHER WAYS TO REPRESENT SOLUTIONS
these power series convergent for \z\ + |a/| < s. Then
(28)	0 < |BJ)7>6| <	0 < |c7ij| < c7j15
for all j, 7, 6.
We consider next the new boundary-value problem
fu* = B^u^a/Ju*• +c*(u*,a/) for |ar| < r
1	u* = 0 for |a/1 < r, xn = 0,
and, as above, look for a solution having the form
(30)	u*=£u*z“,
a
where
5. We claim
0 < l«4l < -S'
for each multiindex a.
The proof is by induction. The general step follows since
hal = |9aC • • > Bj,7,6, • • •,c7;£,...,Up,... )| by (23)
< 9a(- • • > |Bj,71«|,  • •, |c7,i|,.. •, |up|,...) by (24)
< <£(..., B*7 6,..., c*i6,..., u£,...) by (24), (28) and induction
Thus
(32)
u* » u,
and so it suffices to prove that the power series (30) converges near zero.
6. As demonstrated in the proof of assertion (ii) of the lemma in §4.6.2,
if we choose
b;
__________________Cr__________________
r - («1 4-----1- 2?n-i) - (zi +--1- zm)
4.7. PROBLEMS
233
for j = 1,..., n — 1, and
Cr
c* :=----7-----------------7------------;(1,   •, 1),
r - (a?i + • • • + xn_i) - (21 + • • • + zm)
then (26), (27) will hold if C is large enough, r > 0 is small enough, and
k'| + |z| < r.
Hence the problem (29) reads
f u* =____________________QjL________________ (y' „/* । i
I Xn r-(zi + --- + zn_i)-(u1* + --- + u™*) k^’Z J
|	for |ж| < r
I	u* = 0 for |a/| < r, xn = 0.
However, this problem has an explicit solution, namely
(33)	u* =v*(l,...,l),
for
(34)	>’W=^(r-(« + +*»-l)
- [(r - (a?i 4--h Zn-1))2 - 2mnCrxn]1^2').
This expression is analytic for |ж| < r, provided r > 0 is sufficiently small.
Thus u* defined by (33) necessarily has the form (30), (31), the power series
(30) converging for |ж| < r. As u* » u, the power series (17) converges as
well for |rr| < r.
This defines the analytic function u near 0. Since the Taylor expansions
of the analytic functions uIn and Bj(u> rr)uxJ + c(u, x) agree at 0,
they agree as well throughout the region |ж| < r.	□
Remark. The Cauchy-Kovalevskaya Theorem is valid also for fully nonlin-
ear, analytic PDE: see Folland [Fl]	□
4.7.	PROBLEMS
1.	Use separation of variables to find a nontrivial solution и of the PDE
uxiuxixi + 2uX1 uX2иХ1Х2 + uX2uX2X2 = 0 in IR .
2.	Consider Laplace’s equation Au = 0 in IR2, taken with the Cauchy
data
и = 0,	= £ sin(nxi) on {x2 = 0}.
234
4. OTHER WAYS TO REPRESENT SOLUTIONS
Employ separation of variables to derive the solution
и = sin(mri) sinh(nx2)-
What happens to и as n —> oo? Is the Cauchy problem for Laplace’s
equation well-posed? (This example is due to Hadamard.)
3.	Consider the viscous conservation law
(*)	ut + F(u)x — auxx — 0 in R x (0, oo),
where a > 0 and F is uniformly convex.
(i)	Show и solves (*) if u(x, t) = v(x—at) and v is defined implicitly
by the formula
/•U«) a
s =	.--------dz (s€R),
Л F(z) — az + b k h
where b and c are constants.
(ii)	Demonstrate that we can find a traveling wave satisfying
lim v(s) = ui, lim v(s) = ur
s—> —OO	s—>OO
for ui > ur, if and only if
F(ut) - F(ur)
a =-----------------.
ui — ur
(iii	) Let u£ denote the above traveling wave solution of (*) for a = e,
with us(0,0) = Vi+Ur_ Compute lime^o^£ and explain your
answer.
4.	Prove that if и is the solution of problem (23) for Schrodinger’s equa-
tion in §4.3 given by formula (20), then
IH,i)llL~(R") < (47r|^n/2l|g|lLi(R")
for each t 0.
5.	Utilize Lemma 2 in §4.5.3 to discuss the sense in which и defined by
formula (20) in §4.3.1 converges to the initial data g as t —> 0+.
6.	Show that we can construct an explicit solution of the initial-value
problem (18), (19) from Example 1 in §4.5.1, having the form
v(z,t) = *	* /Qe~*2/7(t) (z G R, t > 0),
4.8. REFERENCES
235
the function 7(f) to be found. Substitute into the PDE and determine
an ODE 7 should satisfy. What is the initial condition for this ODE?
7.	Justify in the proof of Lemma 2 in §4.5.3 the transformation of the
integral of e~z over the line Г to the integral over the real axis.
8.	Let n = 1 and suppose that u£ solves the problem
-(a(f M)x = f in (0,1)
us(0) = «e(l) = 0,
where a is a smooth, positive function, which is 1-periodic. Assume
also that f € L2(0,1).
(i)	Show that u£ и weakly in Hq(0, 1), where и solves
-auxx = f in (0,1)
u(0) = u(l) = 0,
for a := (£ a(y) 1 dy) \
(ii)	Check that this answer agrees with the conclusions (73), (74) in
§4.5.4.
(This problem requires knowledge of energy estimates, Sobolev spaces,
etc. from Chapters 5, 6.)
9.	Show that the line {t = 0} is characteristic for the heat equation
ut = uxx. Show there does not exist an analytic solution и of the
heat equation inR x R, with и = on {t = 0}. (Hint: Assume
there is an analytic solution, compute its coefficients, and show that
the resulting power series diverges except at (0,0). This example is
due to Kovalevskaya.)
4.8.	REFERENCES
Section 4.	1 See for instance Pinsky [P], Strauss [ST], Thoe-Zachman-
oglou [T-Z], Weinberger [WE], etc. for more on separation
of variables.
Section 4.	2 C. Jones provided the discussion of traveling waves for the
bistable equation, and J.-L. Vazquez showed me the deriva-
tion of Barenblatt’s solution. P. Giver’s book [O] explains
much more about symmetry methods for PDE.
Section 4.3 Stein-Weiss [S-W], Stein [SE], Hormander [Hl], Rauch [R],
Treves [T], etc. provide much more information concerning
Fourier transform techniques. M. Weinstein helped me with
Schrodinger’s equation. Example 5 is from Pinsky [P], and
236
4. OTHER WAYS TO REPRESENT SOLUTIONS
Section 4.	4
Section 4.	5
Section 4.	6
Section 4.	7
the solution of the wave equation in §4.3.2 is from Pinsky-
Taylor [Р-Т].
See Courant-Hilbert [C-H] for more on the hodograph and
Legendre transforms.
J. Neu contributed §4.5.1. Section 4.5.3 is based upon some
classroom lectures of J. Ralston, following Hormander [Н2].
The discussion of homogenization in §4.5.4 follows Bensous-
san-Lions-Papanicolaou [B-L-P],
See Folland [Fl, Chapter 1], John [J, Chapter 3], DiBene-
detto [DB, Chapter 1].
Problem 1 is due to Aronsson, and Problem 9 is from Mik-
hailov [М].
Chapter 5
SOBOLEV SPACES
5.1	Holder spaces
5.2	Sobolev spaces
5.3	Approximation
5.4	Extensions
5.5	Traces
5.6	Sobolev inequalities
5.7	Compactness
5.8	Additional topics
5.9	Other spaces of functions
5.10	Problems
5.11	References
This chapter mostly develops the theory of Sobolev spaces, which turn
out often to be the proper setting in which to apply ideas of functional
analysis to glean information concerning partial differential equations. The
following material is often subtle, and will seem largely unmotivated, but
ultimately will prove extremely useful.
Since we have in mind eventual applications to rather wide classes of
partial differential equations, it is worth sketching out here our overall point
of view. Our intention, broadly put, will be later to take various specific
PDE and to recast them abstractly as operators acting on appropriate linear
spaces. We can symbolically write this as
A : X Y,
239
240
5. SOBOLEV SPACES
where the operator A encodes the structure of the partial differential equa-
tions, including possibly boundary conditions, etc., and X, Y are spaces of
functions. The great advantage is that once our PDE problem has been
suitably interpreted in this form, we can often employ the general and ele-
gant principles of functional analysis (Appendix D) to study the solvability
of various equations involving A. We will later see that the really hard work
is not so much the invocation of functional analysis, but rather finding the
“right” spaces X, Y and the “right” abstract operators A. Sobolev spaces
are designed precisely to make all this work out properly, and so these are
usually the proper choices for X, Y.
We will utilize Sobolev spaces for studying linear elliptic, parabolic and
hyperbolic PDE in Chapters 6-7, and for studying nonlinear elliptic and
parabolic equations in Chapters 8-9 .
The reader may wish to look over some of the terminology for functional
analysis in Appendix D before going further.
5.1.	HOLDER SPACES
Before turning to Sobolev spaces, we first discuss the simpler Holder spaces.
Assume U C Rn is open and 0 < 7 < 1. We have previously considered
the class of Lipschitz continuous functions и : U —> R, which by definition
satisfy the estimate
(1)	|u(x) - u(y)I < С|я: - 2/1 (ж, у G U)
for some constant C. Now (1) of course implies и is continuous, and more
importantly provides a uniform modulus of continuity. It turns out to be
useful to consider also functions и satisfying a variant of (1), namely
(2)	|u(x) - u(y)I < C\x - yp (x, yeU)
for some constant C. Such a function is said to be Holder continuous with
exponent 'y.
DEFINITIONS, (i) If и : U —> R is bounded and continuous, we write
Ihllctt/) := sup l«(®)l-
xEU
(ii)	The 7t/l-Hdlder seminorm of и : U —> R is
r ,	( 1Ф) — u(y)I
%АУ
5.2. SOBOLEV SPACES
241
and the yth -Holder norm is
ll'ullc0^(C7) := ll'ullc(C7) +
DEFINITION. The Holder space
ск’\й)
consists of all functions и E Ck(U) for which the norm
(3)	Il'ullc*>->'(t7) := 57 IPaullc(t/) + 57
|a|<fc	|a|=fc
is finite.
So the space Ck,~!(U) consists of those functions и that are fc-times con-
tinuously differentiable and whose fc//l-partial derivatives are Holder contin-
uous with exponent 7. Such functions are well-behaved, and furthermore
the space Ck,y (f7) itself possesses a good mathematical structure:
THEOREM 1 (Holder spaces as function spaces). The space of functions
Ck'y(U) is a Banach space.
The proof is left as an exercise (Problem 1), but let us pause here to
make clear what is being asserted. Recall from §D.l that if X denotes a real
linear space, then a mapping || || : X —> [0,00) is called a norm provided
(i)	||u + v|| < ||u|| + ||v|| for all u, v G X,
(ii)	||Au|| = |A|||u|| for all и G X, A G R,
(iii)	||u|| = 0 if and only if и = 0.
A norm provides us with a notion of convergence: we say a sequence {и*,
С X converges to и G X, written Uk —> u, if lim^oo ||ufc—u|| = 0. A Banach
space is then a normed linear space which is complete, that is, within which
each Cauchy sequence converges.
So in Theorem 1 we are stating that if we take on the linear space
C'fe,7(tZ) the norm || • || = || •	defined by (3), then ||  || verifies
properties (i)-(iii) above, and in addition each Cauchy sequence converges.
5.2.	SOBOLEV SPACES
The Holder spaces introduced in §5.1 are unfortunately not often suitable
settings for elementary PDE theory, as we usually cannot make good enough
analytic estimates to demonstrate that the solutions we construct actually
242
5. SOBOLEV SPACES
belong to such spaces. What are needed rather are some other kinds of
spaces, containing less smooth functions. In practice we must strike a bal-
ance, by designing spaces comprising functions which have some, but not
too great, smoothness properties.
5.2.1.	Weak derivatives.
We start off by substantially weakening the notion of partial derivatives.
Notation. Let C'^°(I7) denote the space of infinitely differentiable functions
ф : U —>• R, with compact support in U. We will call a function ф belonging
to C£°(U) a test function.	□
Motivation for definition of weak derivative. Assume we are given a
function и 6 C'1(17). Then if ф e С£°(17)> we see from the integration by
parts formula that
(1)	I ифх^х = —1 ux$dx (i = 1,... ,n).
Ju	Ju
There are no boundary terms, since ф has compact support in U and thus
vanishes near dU. More generally now, if к is a positive integer, и E Ck(U),
and a = (ai,..., an) is a multiindex of order |a| = ai H-1- an = k, then
(2)	[ uDa ф dx = (—1)^ [ D^dx.
Ju	Ju
This equality holds since
Qai Qan
D ф=д^'"д^ф
and we can apply formula (1) |a| times.
We next examine formula (2), valid for и G Ck{U}, and ask whether
some variant of it might still be true even if и is not к times continuously
differentiable. Now the left hand side of (2) makes sense if и is only locally
summable: the problem is rather that if и is not Ck, then the expression
on the right hand side of (2) has no obvious meaning. We resolve this
difficulty by asking if there exists a locally summable function v for which
formula (2) is valid, with v replacing Daw.
DEFINITION. Suppose u,v E Lloc(fJ), and a is a multiindex. We say
that v is the ath-weak partial derivative of u, written
Dau = v,
5.2. SOBOLEV SPACES
243
provided
(3)	[ uDacl>dx =	[ офdx
Ju	Ju
for all test functions ф € C£°(U).
In other words, if we are given и and if there happens to exist a function
v which verifies (3) for all ф, we say that Dau = v in the weak sense. If there
does not exist such a function v, then и does not possess a weak o^-partial
derivative.
LEMMA (Uniqueness of weak derivatives). A weak atfl-partial derivative
of u. if it exists, is uniquely defined up to a set of measure zero.
Proof. Assume that v, v e L}OC(U) satisfy
f uD°^dx = (—I)'"' [ vфdx = (—[ i^dx
Ju	Ju	Ju
for all Ф e	Then
(4)	I (v — v^dx = 0
Ju
for all ф 6	whence v — v = 0 a.e.	□
Example 1. Let n = 1, U = (0,2), and
Define
{x	if	0	<	x <	1
1	if	1	<	x <	2.
fl	if	0	<	x <	1
v(x) = <
I 0 if 1 < X < 2.
Let us show u' = v in the weak sense. To see this, choose any ф E C^°(U).
We must demonstrate
/ иф' dx = — I иф dx.
Jo	Jo
But we easily calculate
I иф' dx= хф' dx + I ф' dx
Jo Jo Ji
= —[ фdx + </>(1) — </>(1) = — [ vфdx,
Jo	Jo
as required.
□
244
5. SOBOLEV SPACES
Example 2. Let n = 1, U = (0,2), and
( x if 0 < x < 1
u(x) — <
v ' I 2 if 1 < x < 2.
We assert u' does not exist in the weak sense. To check this, we must show
there does not exist any function v E L|OC(U) satisfying
(5)	/ иф' dx = — / уфйх
Jo	Jo
for all ф 6	Suppose, to the contrary, (5) were valid for some v and
all ф. Then
— I ьфИх = иф' dx= хф' dx + 2 / ф' dx
(6)	Jo	Jo	Jo	Ji
= — [ фdx — ф(Х).
Jo
Choose a sequence {фт}т=1 °f smooth functions satisfying
0 фт < 1,	H for all X 1.
Replacing ф by фт in (6) and sending m —> oo, we discover
1 = lim </>ш(1) = lim
m—►oo	т—>оэ
/ ифт dx- фт dx
.Jo Jo
= 0,
a contradiction.
□
More sophisticated examples appear in the next section.
5.2.2. Definition of Sobolev spaces.
Fix 1 < p < oo and let A: be a nonnegative integer. We define now
certain function spaces, whose members have weak derivatives of various
orders lying in various IF spaces.
DEFINITION. The Sobolev space
Wk’P(U)
consists of all locally summable functions и : U —> R such that for each
multiindex a with |a| < k, Dau exists in the weak sense and belongs to
5.2. SOBOLEV SPACES
245
Remarks, (i) If p = 2, we usually write
Hk(U) = Wk’2(W) (fc = 0,1,...).
The letter H is used, since—as we will see—Hk(U) is a Hilbert space. Note
that Я°(Я) = Z2(t7).
(ii)	We henceforth identify functions in Wk,p(U) which agree a.e. □
DEFINITION. If и 6 Wk’p(U), we define its norm to be

(E\a\<k dx)
L|a|<k eSS SW l-D^I
(1 < p < oo)
(? = oo).
DEFINITIONS, (i) Let {um}m=i> и 6 Wk'p(U). We say um converges
to и in Wk’p(U"), written
umи in Wk'p(U),
provided
lim ||um - u||ivfc,p([n = 0.
m—>oo	v 7
(ii)	We write
umи inW^(U),
to mean
um^u in Wk’p(V)
for each V CC U.
DEFINITION. We denote by
the closure of C?(U) in Wk'p(U).
Thus и 6 Wq ,p(U) if and only if there exist functions um 6 C^°(U) such that
um —► и in Wk,p(U}. We interpret Wn’p(U) as comprising those functions
и e Wk'p(U) such that
“Dau = 0 on dU" for all |a| < к — 1.
This will all be made clearer with the discussion of traces in §5.5.
246
5. SOBOLEV SPACES
Notation. It is customary to write
Яо>) = W^(U).
□
We will see in the exercises that if n = 1 and U is an open interval in
R1, then и E W1,P(U) if and only if и equals a.e. an absolutely continuous
function whose ordinary derivative (which exists a.e.) belongs to
Such a simple characterization is however only available for n = 1. In
general a function can belong to a Sobolev space, and yet be discontinuous
and/or unbounded.
Example 3. Take U = B°(0,1), the open unit ball in Rn, and
u(s) = |z|-a (x EU, x / 0).
For which values of a > 0, n,p does и belong to W1,P(I7)? To answer, note
first и is smooth away from 0, with
=	(^0),
and so
= jjpk (*/»)
Let ф 6 C£°(U) and fix £ > 0. Then
I ифхгЛх = — I иХ{ф<1х-\- ифС dS,
JU-B(p,e)	JU-B(0,e)	JdB(0,e)
v = (i/1,... ,i^n) denoting the inward pointing normal on 9B(0,e). Now if
a + 1 < n, |Du(x)| 6 L1(I7). In this case
[ v^i/dS < II^Hloo [ e~a dS <Cen~x~a 0.
SB(0,e)	JdB{0,e)
Thus
/ ифх^х~- / ux^dx
Ju	Ju
for all ф E	provided 0 < a < n — 1. Furthermore |Du(x)| = щ!ггт E
Lp(t7) if and only if (a + l)p < n. Consequently и 6 Wi,p(U) if and only if
a <	I*1 particular и W1,P(U) for each p > n.	□
5.2. SOBOLEV SPACES
247
Example 4. Let	be a countable, dense subset of U = B°(0,1).
Write
OO -
u^ = T^2^\x~rk\~a (xeU).
Then и 6	if and only if a < If 0 < a < we see that и
belongs to W1,P(U) and yet is unbounded on each open subset of U. □
This last example illustrates a fundamental fact of life, that although
a function и belonging to a Sobolev space possesses certain smoothness
properties, it can still be rather badly behaved in other ways.
5.2.3. Elementary properties.
Next we verify certain properties of weak derivatives. Note very carefully
that whereas these various rules are obviously true for smooth functions,
functions in Sobolev space are not necessarily smooth: we must always rely
solely upon the definition of weak derivatives.
THEOREM 1 (Properties of weak derivatives). Assume u,v 6 Wk’p(U'),
|a| < к. Then
(i)	Dau e	and Dp{Dau} = Da(D?u) = Da+f3u for all
multiindices a,0 with |a| + \/3\ < к.
(ii)	For each A,/z G R, Au + pv E Wk,p(U) and Da{Au + /w) = ADau +
pDav, |a| < k.
(iii)	If V is an open subset ofU, then и E Wk'p(V).
(iv)	IfC E C™(U), then Си E Wk'p(U} and
(7)	Da(Cu) =	(a jD^CDa~^u (Leibniz’ formula),
(3<a W'
where (^) =	
Proof. 1. To prove (i), first fix ф E	Then Б^ф E	and so
[ DauD^dx = (-1)|q| [ uDa+^dx
Ju	Ju
= (-1)|QI(-l)l“+^l f Da+/3^dx
Ju
= (-1)^1 [ Da+?^dx.
Ju
248
5. SOBOLEV SPACES
Thus D@(Dau) = Da+/3u in the weak sense.
2.	Assertions (ii) and (iii) are easy, and the proofs are omitted.
3.	We prove (7) by induction on |a|. Suppose first |a| = 1. Choose any
ф G Cc°°(f7). Then
[ (uD^dx = I uD°JC^) - u(DaW>dx
Ju	Ju
= - [ (CDau + uDa^dx.
Ju
Thus D°JC,u) = £Dau + uDa£, as required.
Next assume I < к and formula (7) is valid for all |a| < I and all functions
C Choose a multiindex a with |a| = I +1. Then a = /3 + 7 for some j/3] = I,
|7| = 1. Then for ф as above,
[ C,uD^dx= [ (uD^D^dx
Ju	Ju
= (-l)l/?l [ ^(^D^D^uD^dx
(by the induction assumption)
= (_1)1/?1+Ы f ^2 (^D^D^D^u^dx
<7<P 'VJ
(by the induction assumption again)
= (-l)H [ ^(^[DpCDa~pu + Da<;Da-au^ dx
<7<0
(where p = a + 7)
DaCDa-°u) ф dx,
since
Not only do many of the usual rules of calculus apply to weak derivatives,
but the Sobolev spaces themselves have a good mathematical structure:
5.2. SOBOLEV SPACES
249
THEOREM 2 (Sobolev spaces as function spaces). For each к = 1,...
and 1 < p < oo, the Sobolev space Wk,p(U) is a Banach space.
Proof. 1. Let us first of all check that ||u||wk’P(U) *s a norm- (See the
discussion at the end of §5.1, or refer to §D.l, for definitions.) Clearly
|| Au|| wk’P(W) = I'M IMI Wfc’P({7) >
and
||'u||iv*,p([7) = 0 if and only if и = 0 a.e.
Next assume u, v G Wk’p(U). Then if 1 < p < oo, Minkowski’s inequality
(§B.2) implies
/	\ i/p
\a|<fc	'
/	\ ^Р
<	( £ (||D%||lp(p) + ||L»Qyl|LP(C/))P J
'4|a|<fc	'
/	\ 1/P	/	\ 1/P
<	E iib““IIl.(u) + E )
'4|a:|<fc	'	''|a:|<k
= ||,u||lV*,P(L7) + |M| Wk’P(Uy
2.	It remains to show that Wk,p(U) is complete. So assume {um}^=1
is a Cauchy sequence in Wk-p(U). Then for each |a| < k, {£>aum}~=1 is a
Cauchy sequence in Since 2/(17) is complete, there exist functions
ua G 2/(17) such that
Daum —> ua in LP(17)
for each |a| < k. In particular,
Um «(0,...,0) =' u in LP(U).
3.	We now claim
(8)	и G Wk’p(U), Dau = ua (|a| < k).
To verify this assertion, fix ф E C£°(U). Then
I uD°^dx = lim / umD°^dx
Ju
= lim (-I)1"1/" Dau„^dx
Ju
= (_l)|a|/' иаф(],х.
Ju
Thus (8) is valid. Since therefore Daum —» Dau in 2/(17) for all |ог| < к, we
see that um —> и in Wk,p(U), as required.	□
250
5. SOBOLEV SPACES
5.3.	APPROXIMATION
5.3.1.	Interior approximation by smooth functions.
It is awkward to return continually to the definition of weak derivatives.
In order to study the deeper properties of Sobolev spaces, we therefore need
to develop some systematic procedures for approximating a function in a
Sobolev space by smooth functions. The method of mollifiers, developed in
§C.4, provides the tool.
Fix a positive integer k and 1 < p < oo. Remember that U£ = {x € U I
dist(®,atZ) > e}.
THEOREM 1 (Local approximation by smooth functions). Assume и E
Wk,p(U) for some 1 < p < oo, and set
u£ = y£ * и in U£.
Then
(i)	u£ e C°°(C7S) for each e > 0,
(ii)	u£ —> и in	as £ —> 0.
Proof. 1. Assertion (i) is proved in §C.4.
2.	We next claim that if |a| < к, then
(1)	Dau£ = rj£ * Dau in
that is, the ordinary ath-partial derivative of the smooth function us is the
£-mollification of the atfl-weak partial derivative of u. To confirm this, we
compute for x E U£
Dau£(x) = Da [ T]£(x - y)u(y) dy
Ju
= / D“t]£(x- y)u(y)dy
Ju
= (-l)|a| [ D“T]£(x - y)u(y) dy.
Ju
Now for fixed x E U£ the function ф(у) := y£(x — y) belongs to
Consequently the definition of the at/l-weak partial derivative implies:
[ DyT]e(x - y)u(y) dy = (-1)W [ y£(x — y)Dau(y) dy.
Ju	Ju
5.3. APPROXIMATION
251
Thus
Dau£(x) = (-1)I“I+W [ r]£(x - y)Dau(y) dy
Ju
= [T]s * Dau] (ж).
This establishes (1).
3.	Now choose an open set V CC U. In view of (1) and §C.4, Dau£ —>
Dau in 17 (V) as e —► 0, for each |a| < k. Consequently
И - <*IIU»(V) = E 11°““' -	- 0
|a|<fc
as e —> 0. This proves assertion (ii).	□
5.3.2.	Approximation by smooth functions.
Next we show that we can find smooth functions which approximate in
Wk'p(U), and not just in (I/). Notice in the following that we make no
assumptions about the smoothness of dU.
THEOREM 2 (Global approximation by smooth functions). Assume U
is bounded, and suppose as well that и 6 Wk'p(U) for some 1 < p < oo.
Then there exist functions um G C°°(U) Г) Wk,p(U') such that
umи in Wk'p(U).
Remark. Note carefully that we do not assert um G C°°(t/) (but see The-
orem 3 below).	□
Proof. 1. We have U = IJi^i Ui, where
Ui := {ж G U | dist(x, dU) > \/i} (i — 1,2,...).
Write Vi := Ui+S - Ui+1.
Choose also any open set Vo CC U so that U = (JSo Now let {£i)£S0
be a smooth partition of unity subordinate to the open sets {V)}?S0; that is,
suppose
U	= l ont/.
Next, choose any function и G Wk'p(U). According to Theorem l(iv) in
§5.2, Qu G Wk’p(U) and spt(Ciu) С V.
252
5. SOBOLEV SPACES
2.	Fix 6 > 0. Choose then Ег > 0 so small that u1 := r]£i * (Cu) satisfies
f ||ul -	₽([/)< 25TT = 0,1,...)
t spti? C Wi	(i = 1,...),
for Wi := Ui+4 - Ui D Vi (г = 1,...).
3.	Write v := Y^oul- This function belongs to C°°(U), since for each
open set V CC U there are at most finitely many nonzero terms in the sum.
Since и = C,iui we have for each V CC U
OO
I|u — u||vy*,p(y) < У7 — Ci^llWk^(U)
i=0
OO -
М3)
i=0
= <5.
Take the supremum over sets V CC U, to conclude ||v — u||wk’P(U) —
□
5.3.3.	Global approximation by smooth functions.
We now ask when it is possible to approximate a given function и G
Wk,p(U) by functions belonging to C°°(U), and not just C’oo(t7). Such an
approximation requires some condition to exclude dU being wild geometri-
cally.
THEOREM 3 (Global approximation by functions smooth up to the
boundary). Assume U is bounded and dU is C1. Suppose и G Wk’p(U)
for some 1 < p < oo. Then there exist functions um E C°°(U) such that
um—* и in Wk’p(U).
Proof. 1. Fix any point x° G dU. As dU is C1, there exist, according to
§C.l, a radius r > 0 and a C1 function 7 : Rn-1 —► R such that—upon
relabeling the coordinate axes if necessary—we have
U D B(s°,r) = {x E B(x°,r) I xn > 7(3:1,... ,xn_i)}.
Set V :=UOB(x°,r/2).
2. Define the shifted point
x£ := x + Xeen (x E V, e > 0),
5.3. APPROXIMATION
253
and observe that for some fixed, sufficiently large number A > 0 the ball
B(x£,e) lies in U Г) B(z°,r) for all x G V and all small e > 0.
Now define u£(x) := u(x£) (x G V); this is the function и translated a
distance Ae in the en direction. Next write v£ = r]£*u£. The idea is that we
have moved up enough so that “there is room to mollify within C7”. Clearly
v£ G C°°(V).
3. We now claim
(4)	v£ -+ и in Wfc’p(V).
To confirm this, take a to be any multiindex with |a| < k. Then
\\Dav£ - Dau\\LP(V) < \\Dav£ - Dau£\\LP(V) + \\Dau£ - Dau\\LP(v).
The second term on the right hand side goes to zero with e, since translation
is continuous in the ZAnorm; and the first term also vanishes in the limit,
by reasoning similar to that in the proof of Theorem 1.
4.	Select 6 > 0. Since dU is compact, we can find finitely many points
x® G dU, radii n > 0, corresponding sets Vi = U Г) B(x®, %), and functions
Vi e C°°(Vi) (г = 1,... ,2V) such that dU C Ut=i B°(x°, %), and
(5)	||vj — «||w*,p(yi) < <5-
Take an open set Vb CC U such that U C UiXo an<^ select, using Theo-
rem 1, a function i?o € С°°(Й)) satisfying
(6)	||vo — «||w*,p(y0) < <5-
5.	Now let {Ci}£o be a smooth partition of unity subordinate to the
open sets {Vi}^0 in U • Define v :=	Then clearly v G	In
254
5. SOBOLEV SPACES
addition, since и = Qu, we see using Theorem 1 in §5.2.3 that for each
|a| < fc:
N
\\Dav - Dau\\LP(U) < £ ||DWi) - Da(Qu)\\LP{Vi}
i=0
N
< C 52 llvi ~ 'ulliv*’P(vi) = CN6,
i=0
according to (5) and (6).
□
5.4. EXTENSIONS
Our goal next is to extend functions in the Sobolev space W1,P(U) to become
functions in the Sobolev space W1,p(Rn). This can be subtle. Observe for
instance that our extending и G W1,p([7) to be zero in Rn — U will not in
general work, as we may thereby create such a bad discontinuity along dU
that the extended function no longer has a weak first partial derivative. We
must instead invent a way to extend и which “preserves the weak derivatives
across dU".
Suppose 1 < p < oo.
THEOREM 1 (Extension Theorem). Assume U is bounded and dU is Cl.
Select a bounded open set V such that U CC V. Then there exists a bounded
linear operator
(1)
E :	W^R”)
such that for each и G W1,P(U):
(i)	Ей = и a.e. in U,
(ii)	Eu has support within V,
and
(iii)
the constant C depending only on p, U, and V.
DEFINITION. We call Eu an extension of и to Rn.
Proof. 1. Fix x° G dU and suppose first
(2)	dU is flat near a?0, lying in the plane {xn = 0}.
5.4. EXTENSIONS
255
A half-ball at the boundary
Then we may assume there exists an open ball B, with center x° and radius
r, such that
Г B+ := В П {xn > 0} C U
(В- :=ВП {xn < 0} C Rn - U.
2.	Temporarily suppose also и € C'oo(t7). We define then
(3)
( u(x)	if x G B+
u(x) := <	_
I -3u(xi,... ,xn-i, -xn) + 4u(xi,... ,xn-i,-^) ifx£B~.
This is called a higher-order reflection of и from B+ to B~.
3.	We claim
(4)	й e c\b).
To check this, let us write u~ := й|в-, u+ ;= й|в+. We demonstrate first
(5)	u~n = u+n on {a;n = 0}.
Indeed according to (3),
£ • • • •
and so
l{zn=0} UXn l{zn=0} 
This confirms (5). Now since u+ = u~ on {xn = 0}, we see as well that
(6)	uzJ{zn=0} = и;г,|{:г„=0}
for i = 1,..., n — 1. But then (5) and (6) together imply
r>%-|{ln=0} = T»%+|{ln=o}
256
5. SOBOLEV SPACES
for each |a| < 1, and so (4) follows.
4.	Using these calculations we readily check as well
(7)
llullw1-p(B) < C|N
for some constant C which does not depend on u.
5.	Let us next consider the situation that dU is not necessarily flat near
x°. Then utilizing the notation and terminology from §C.l, we can find a
C1 mapping Ф, with inverse Ф, such that Ф “straightens out dU near x°”.
coordinates
x-coordinates
Straightening out the boundary
We write у = Ф(ж), x = Ф(у), г?(у) := п(Ф(у)). Choose a small ball В
as drawn before. Then utilizing steps 1-3 above we extend u' from B+ to a
function й' defined on all of B, such that u' is C1 and we have the estimate
< С|Ы1и'1-р(в+)-
Let W := Ф(-В). Then converting back to the ж-variables, we obtain an
extension й of и to W, with
(8)
llullwi.pfw) < С'||п||Ил1,р([/).
6.	Since dU is compact, there exist finitely many points x(- € dU, open
sets Wi, and extensions щ of и to Wi (i = 1,..., N), as above, such that
Г C lXi wi- Take CC U so that U C lXo wh and let {&}гС0 be an
associated partition of unity. Write й :=	where Uq = u. Then
utilizing estimate (8) (with щ in place of и, щ in place of u) we obtain the
bound
(9)	||«||Wi,P(R«) < C||n||iv!-p(B)
for some constant C, depending on U,p, n, etc., but not on u. Furthermore
we can arrange for the support of й to lie within V DD U.
5.5. TRACES
257
7.	We henceforth write Eu := u, and observe that the mapping и i-> Eu
is linear.
Recall that the construction so far assumed и € СОО(С7). Suppose now
и € W1,p(t/), and choose um € C°°(U) converging to и in W1,P(W). Esti-
mate (9) and the linearity of E imply
||Eum - EuJ|iyi,P(Rn) < C||um - u/||wi,₽([/).
Thus {Eum}^=1 is a Cauchy sequence and so converges to й =: Eu. This
extension, which does not depend on the particular choice of the approxi-
mating sequence {um}^=1, satisfies the conclusions of the theorem. □
Remarks, (i) Assume now that dU is C2. Then the extension operator
E constructed above is also a bounded linear operator from W2,p(t/) to
W2,p(Rn). To see this, note first in steps 3, 4 of the proof that although й
is not in general C2, it does belong to W2p(B'). We also have the bound
||«||lV2,p(B) < С'||и||Ил2,р(В+),
which follows from the definition (3). As before, we consequently derive the
estimate
(10)
IIEuIIh^p^) < СЦиЦууг,?([/),
provided dU is C2, the constants C depending only only on U, V,n and p.
We will need these observations later.
(ii) The above construction does not provide us with an extension for
the Sobolev spaces Wk,p(U), if к > 2. This requires a more complicated
higher-order reflection technique.	□
5.5. TRACES
Next we discuss the possibility of assigning “boundary values” along dU to
a function и € W1,P(U}, assuming that dU is C1. Now if и € C(U}, then
clearly и has values on dU in the usual sense. The problem is that a typical
function и € W1,P(U) is not in general continuous and, even worse, is only
defined a.e. in U. Since dU has n-dimensional Lebesgue measure zero, there
is no direct meaning we can give to the expression “u restricted to dU”.
The notion of a trace operator resolves this problem.
For this section we take 1 < p < oo.
258
5. SOBOLEV SPACES
THEOREM 1 (Trace Theorem). Assume U is bounded and dU is C1.
Then there exists a bounded linear operator
T : PV1,₽(i7) Lp(dlT)
such that
(i) Tu = u\9U if и E W^U) П C(U)
and
(ii)
II^IIlp(w) < CMw1’ ₽([/),
for each и G W1,P(C/), with the constant C depending only on p and U.
DEFINITION. We call Tu the trace of и on dU.
Proof. 1. Assume first и E C'1(B). As in the first part of the proof of
Theorem 1 in §5.4 let us also initially suppose x° E dU and dU is flat near
x°, lying in the plane {xn = 0}. Choose an open ball В as in the previous
proof and let В denote the concentric ball with radius r/2.
Select C € C£°(B), with ( > 0 in B, ( = 1 on B. Denote by Г that
portion of dU within B. Set x' = (xi,... ,xn-i) G Rn-1 = {xn = 0}.
Then
[ \u\pdx' < [ CMW = - [ (СИ)хп dx
Jr	J{xn=0}	Jb+
(1)	= - [ hlPCxn +p|u|p_1(sgnu)uInCdx
Jb+
<C	[ |u|p + \Du\p dx,
Jb+
where we employed Young’s inequality, from §B.2.
2.	If xQ E dU, but dU is not flat near r°, we as usual straighten out the
boundary near x° to obtain the setting above. Applying estimate (1) and
changing variables, we obtain the bound
[ \u\pdS<C [ \u\p + \Du\p dx,
Jr	Ju
where Г is some open subset of dU containing x°.
3.	Since dU is compact, there exist finitely many points x® E dU and
open subsets Tj C dU (г = 1,..., TV) such that dU = Uili Гг and
||«||ьр(п) < c||«||G = i, • • •>№).
5.5. TRACES
259
Consequently, if we write
Tu :=
then
(2)	ЦТиЦ^р^) < С'||и||Ил1,Р([/)
for some appropriate constant C, which does not depend on u.
4.	Inequality (2) holds for и € C1(t7). Assume now и € W1,P(U'). Then
there exist functions um G C°°(U) converging to и in W1,P(U). According
to (2) we have
(3)	||Tnm -	< C||um — UilllVl,₽([/)!
so that {Tum}m=1 is a Cauchy sequence in Lp(dU). We define
Tu := lim Tum,
m->oc
the limit taken in Lp(dU). According to (3) this definition does not depend
on the particular choice of smooth functions approximating u.
Finally if и € W1,P(U') П C(U), we note that the functions um G C'OO(C/)
constructed in the proof of Theorem 3 in §5.3.3 converge uniformly to и on
U. Hence Tu = «lau-	□
We next examine more closely what it means for a function to have zero
trace.
THEOREM 2 (Trace-zero functions in W1,p). Assume U is bounded and
dU is C1. Suppose furthermore that и € Wi,p(U). Then
(4)	и E	if and only if Tu = 0 on dU.
Proof*. 1. Suppose first и E Wq’p(U). Then by definition there exist
functions um E such that
—► u	in W1,P(U).
As Tum = 0 on dU (m = 1,...) and T : W1,P(U) —» LP^dU) is a bounded
linear operator, we deduce Tu = 0 on dU.
2.	The converse statement is more difficult. Let us assume that
(5)	Tu = 0 on dU.
*Omit on first reading.
260
5. SOBOLEV SPACES
Using partitions of unity and flattening out dU as usual, we may as well
assume
, .	f и G W1,P(R”), и has compact support in R”,
W	( Tu = 0 on Ж” = Rn~1.
Then since Tu = 0 on Rre-1, there exist functions um G C^R”) such that
(7)	um и in W1,P(R")
and
(8)	Tum = um|R„_i ^0 in Lp(Rn-1).
Now if x' G Rn-1, xn > 0, we have
rxn
|’Um(-£ i ®n)| —	> 0) | + I	|nmixn (x , t) | dt.
JO
Thus
[ \um(x',xn)\pdx'< C ( [	|um(a/,0)|pdx'
Jr«-i	VR*-1
+xp-1 [ f \Dum(x', t)|p dx'dt ) .
Jo JR"-1	/
Letting m —> oo and recalling (7), (8), we deduce:
(9)	/	\u(x', xn)\p dx' < Схр~г f f \Du\p dx'dt
Jr™-1	Jo JR"-1
for a.e. xn > 0.
3. Next let £ G C°°(R) satisfy
C = 1 on [0,1], c = о on R - [0,2], 0 < C < 1,
and write
( Cm(x) := C(mxn) (xGR")
1.	•— u(®)(l Cm)-
Then
( wm,xn =	— Cm) — mtiC
1.	Dx/Wm = DX'U{J- Cm)-
Consequently
[ \Dwm - Du\p dx <C [ |Cm|p|T>«|p dx
-/Rif.
(10)	[Vm r
+ Cmp / \u\p dx'dt
Jo JR"-1
=: A + B.
5.6. SOBOLEV INEQUALITIES
261
Now
(И)
A —> 0 as m —> oo,
since 0 only if 0 < xn < 2/m. To estimate the term B, we utilize
inequality (9):
(fl/m	\ / p2/m /•
/ t^dt /	/	|£>u|p dx'dxn
Jo	J \Jo JR"-1
/•2/m f
< C / |Z>u|p dx'dxn —> 0 as m —> 0.
Jo JR"-1
Employing (10)-(12), we deduce Dwm —► Du in //(R”). Since clearly
wm —► и in //(R”), we conclude
wm и in W1,P(R”).
But wm = 0 if 0 < xn < 1/m. We can therefore mollify the wm to produce
functions umeC“(R") such that um —► « in W1,P(R”). Hence uG WO1,P(R”).
□
5.6. SOBOLEV INEQUALITIES
Our goal in this section is to discover embeddings of various Sobolev spaces
into others. The crucial analytic tools here will be certain so-called “Sobolev-
type inequalities”, which we will prove below for smooth functions. These
will then establish the estimates for arbitrary functions in the various rele-
vant Sobolev spaces, since—as we saw in §5.2—smooth functions are dense.
To clarify the presentation we will consider first only the Sobolev space
W1,p(f7) and ask the following basic question: If a function и belongs to
W1,P(U}; does и automatically belong to certain other spaces! The answer
will be “yes”, but which other spaces depends upon whether
(1)
(2)
(3)
1 <P < n,
p = n,
n < p < oo.
We study case (1) in §5.6.1, case (3) in §5.6.2, and the borderline case
(2) only later in §5.8.1.
262
5. SOBOLEV SPACES
5.6.1. Gagliardo—Nirenberg—Sobolev inequality.
For this section let us assume
(4)
1 < p < n
and first ask whether we can establish an estimate of the form
(5)	||uIIl«(R") — С11 Du 11 lp (Rn),
for certain constants C > 0, 1 < q < oo and all functions и E C(?°(Rn).
The point is that the constants C and q should not depend on u.
Motivation. Let us first demonstrate that if any inequality of the form (5)
holds, then the number q cannot be arbitrary, but must in fact have a very
specific form. For this, choose a function и E C^°(Rn), и 0, and define for
A > 0 the rescaled function
ua(x) := u(Xx) (x E Rn).
Applying (5) to u\, we find:
(6)	II^aIIl?(R") < ^II^aIIlp(R")-
Now
[ \ux\qdx= [ \u(Xx)\qdx =	[ \u(y)\qdy,
JR"	JR"	л JR"
and
[ \Dux\pdx = Xp { \Du(Xx)\pdx = ^ [ \Du(y)\pdy.
JR"	JR"	Xn
Inserting these equalities into (6), we discover
-^IMIl<*(R") < C'-^I|£Mlp(R"),
and so
(7)	|IuIIl9(R") — CX ₽ + « ||Pu||lp(R").
But then if 1 —	7^ 0, we can upon sending A to either 0 or oo in (7)
obtain a contradiction. Thus if in fact the desired inequality (5) holds, we
must necessarily have	1 — 5 + ^	= 0;	so that 4	= 1	— 1 q	= -22-.	□
J	p q ’	q p n ’ * n—p
This observation motivates the following
5.6. SOBOLEV INEQUALITIES
263
DEFINITION. If 1 < p < n, the Sobolev conjugate of p is
nP
p :=------------------------.
n — p
(8)
Note that
(9)
1	1	1	.
— =--------, p > p-
p*	p	n
The foregoing scaling analysis shows the estimate (5) can only possibly
be true for q = p*. Next we prove this inequality is in fact valid.
THEOREM 1 (Gagliardo-Nirenberg-Sobolev inequality). Assume 1 < p
< n. There exists a constant C, depending only onp and n, such that
(io)	IMIlp*(R’1) ~ c||-CMlp(R")>
for all и E C^(Rn).
Now we really do need и to have compact support for (10) to hold, as the
example и = 1 shows. But remarkably the constant here does not depend
at all upon the size of the support of u.
Proof. 1. First assume p = 1.
Since и has compact support, for each i = 1,..., n and x E Rre we have
• . . , Xi—i, yi, Xi-f-i, . . . , Xn) dyi
oo
and so

Consequently
n
(И)
.,yi,...,xn)\dyi
dxi
i=l ''-‘’о
Integrate this inequality with respect to aq:
(12)
dxi
|Du| dxidyi
264
5. SOBOLEV SPACES
the last inequality resulting from the general Holder inequality (§B.2).
Now integrate (12) with respect to x?:
•00
i=i
for
oo
= 3,... ,n).
Applying once more the extended Holder inequality, we find
fOO roo
n
i=3
We continue by integrating with respect to xs,..., xn, eventually to find
.. dx.
(13)
i=i
This is estimate (10) for p = 1.
2. Consider now the case that 1 < p < n. We apply estimate (13) to
v := |u|7, where 7 > 1 is to be selected. Then
= 7
(14)
We choose 7 so that = (7 — l)^i • That is, we set
p(n - 1)
n — p
5.6. SOBOLEV INEQUALITIES
265
in which case = (7 —	= 77^ = p*  Thus, in view of (5), estimate
(14) becomes
( [ |u|p daA	< C ( f |.Du|pdaA .
\dRn	/	VR"	/
□
THEOREM 2 (Estimates for W1,p, 1 < p < n). Let U be a bounded, open
subset ofW1, and suppose dU is C1. Assume 1 <p < n, and и € W1,p(lT).
Then и E IP* (U}, with the estimate
(15)	IMlp’(i/) < C'll«llw1-p(i7),
the constant C depending only on p, n, and U.
Proof. Since dU is C1, there exists according to Theorem 1 in §5.4 an
extension Ей = й £ W1,p(Rn), such that
( й = и in U, й has compact support, and
I ll^llw1'P(Rn) - С|Ы1 •
Because й has compact support, we know from Theorem 1 in §5.3 that there
exist functions um G C(?°(Rn) (m = 1,2,...) such that
(17)	um й in W1,p(Rn).
Now according to Theorem 1, ||um — tq||£p*(Rn) < C||Dum — PuJ|/,p(Rn) for
all l,m > 1. Thus
(18)	um —> й in IP (Rn)
as well. Since Theorem 1 also implies ||Um||LP*(Rn) < C||T>izm||£P(Rn), asser-
tions (17) and (18) yield the bound
II“IIlp*(R«) - C'll-C>“||Lp(Rn).
This inequality and (16) complete the proof.	□
THEOREM 3 (Estimates for wj’p, 1 < p < ri). Assume U is a bounded,
open subset of Rn. Suppose и G Wj’p(t7) for some 1 < p < n. Then we
have the estimate
IMItv(cz) < C||-EMlp(cz)
for each q G [l,p*], the constant C depending only on p,q,n and U.
Remark. This estimate is sometimes called Poincare’s inequality. The
difference with Theorem 2 is that only the gradient of и appears on the
righthand side of the inequality. (Other Poincare-type inequalities will be
established later, in §5.8.1.)	□
266
5. SOBOLEV SPACES
Proof. Since и 6 Wj’₽(l7), there exist functions um € Cf^(U) (m =
1,2,...) converging to и in W1,P(U). We extend each function um to be
0 on Rn — U and apply Theorem 1 to discover ||u||£₽*([/) —	||ь₽(С/)• As
\U\ < oo, we furthermore have ||u||l«(i/) < C||^Hlp*([/) if 1 — Q — P*-	□
Remarks, (i) In view of Theorem 3, on W01,p(Lr) the norm | |Z>u| |ь₽(С7) is
equivalent to | |u| |wi.p([/), if U is bounded.
(ii) We next consider the case
p — n.
Owing to Theorem 2 and the fact that p* —	> +oo as p —> n, we might
expect и €	provided	и E Wl,n(lT). This is however false if n > 1:
for example, if U = B°(0,1) the function и = log log (1 + belongs to
W1,n(B), but not to L°°({7). We will return to this borderline situation in
§5.8.1 below.	□
5.6.2. Morrey’s inequality.
Now let us suppose
(19)	n < p < oo.
We will show that if и E W1,P(U), then и is in fact Holder continuous, after
possibly being redefined on a set of measure zero.
THEOREM 4 (Morrey’s inequality). Assume n < p < oo. Then there
exists a constant C, depending only on p and n, such that
(20)	ll«l|c°^(Rn) — W1’P(Rn)
for all и E C1(Rn), where
7 := 1 — n/p .
Proof. 1. First choose any ball В(ж,г) C Rn.
We claim there exists a constant C, depending only on n, such that
(21)	/	\u(y) - u(x)\dy <C f , dy-
J B(x,r)	\У "И
5.6. SOBOLEV INEQUALITIES
267
To prove this, fix any point w € dB(0,1). Then if 0 < s < r,
d
—u(x + tw) dt
x + tw) • w dt
x + tw)\ dt.
s
Hence
— U
<	\Du(x+ tw)\dSdt
JO JdB(0,l)
= [ [	| Ди(ж ~Ь fw) | 2_i dSdt.
Jo .ЛЭВ(0,1)	tn
Let у = x + tw, so that t = |a: — y\. Then converting from polar coordinates,
we have
[ \u(x + sw)-u(x)\dS < f	dy
JdB(0,l)	J B(x,s) Iх - У\
f |L>n(y)|
“ /в(х,г) к - у1п-1
Multiply by sn-1 and integrate from 0 to r with respect to s:
f । i \	/ м , Гп r |Z>u(y)| ,
/	l«(y) - u(x)\dy <—	,	dy.
JB(x,r)	Jb)x,t) к У\
This is (21).
2.	Now fix ж € Rn. We apply inequality (21) as follows:
(22)
kk)| < + kk) - u(y)\ dy+ 4-	|u(y)| dy
J B(z,l)	j B(x,l)
-c L( k-^k"1 +
Iх У\
2/	/	\ p—*
<c([ \Dufdy)'[[ ---------------------’ +СНИР)
\JR”	/	|x — J/| Pp-i у
< C'||u||w1.P(Rn)•
The last estimate holds since p > n implies (n — l)^y < n; so that
268
5. SOBOLEV SPACES
As x € Rn is arbitrary, inequality (22) implies
(23)
sup | и I < C'||l4||wl-P(K")-
3. Next, choose any two points x, у € Rn and write r := |x — y\. Let
W := B(x,r) A B(y,r). Then
(24)	|u(r) - u(y)| < + \u(x)-u(z)\dz+-[ \u(y) - u(z)\dz.
J w	J w
But inequality (21) allows us to estimate
|u(z) — w
w
|u(a;) — u(z)| dz
B(x,r)
i/p
(25)
r dz I ₽
B(x,r) |x — Z^ 1^P^i J
Likewise,
= Cr1-p ||Du||LP(Rn).
i \и(У) ~u(z)\dz <Cr* p ||Du||LP(Rn).
J w
Our substituting this estimate and (25) into (24) yields
|u(x) -u(y)| < C'r1“p||JOu||LP(Rn) =C'|x-y|1_p||Du||LP(Rn).
Thus
Г1	-о П«(х) -«(?/)ll ^nn, Il
[«Jc-o.i-n/p^n) — sup < -—	Il-n/p [ - M-kM.LP(R")-
x^y I y\ ' )
This inequality and (23) complete the proof of (20).
Remark. A slight variant of the proof above provides the estimate
/ .	\Vp
|u(y) — u(r)| < Cr1-p I / |Z>u(z)|pdzj
\J B(x,2r)	1
for all и E С1(В(ж, 2r)), у E B(x,r), n < p < oo. By an approximation
the same bound is valid for и E W1,p{B(x,‘2r}'), n < p < oo. We will use
this inequality later in §5.8.2. ( This estimate is in fact valid if on the right
hand side we integrate over B(x,r), instead of B(x, 2r), but the proof is a
bit trickier.)	□
5.6. SOBOLEV INEQUALITIES
269
DEFINITION. We say u* is a version of a given function и provided
и = u* a.e.
THEOREM 5 (Estimates for W1,p, n < p < oo). Let U be a bounded, open
subset ofRn, and suppose dU is C1. Assume n < p < oo, and и € W1,p(t/).
Then и has a version и* € С0’у(и), for 7 = 1 — with the estimate
IIй*llco.7(i/) < C'llullvyl.pfi/).
The constant C depends only on p,n and U.
Remark. In view of Theorem 5, we will henceforth always identify a func-
tion и e W1,p(t/) (p > n) with its continuous version.	□
Proof. Since dU is C1, there exists according to Theorem 1 in §5.4 an
extension Ей = й E W1,P(R”) such that
{й = и in U,
й has compact support, and
INIivi.pfR") < с|Ы1 wi.p([/)•
Since й has compact support, we obtain from Theorem 1 in §5.3 the
existence of functions um € C^R”) such that
(27)	um -4 й in W1,p(Rn).
Now according to Theorem 4, ||um — tq||co,i-n/p(]Rn) < C||um — щ ||yyi.p(Rn)
for all I, m > 1; whence there exists a function u* € C'0,1-n/p(Rn) such that
(28)	um -4 и in C'°’1-n/₽(Rn).
Owing to (27) and (28) we see that и* = и a.e. on (J; so that u* is a ver-
sion of u. Since Theorem 4 also implies ||Wm|lc°.i-n/p(R") — C’llttm||iv1,p(Rn),
assertions (27) and (28) yield:
||«*|Ic,°-1-"/p(R") - C'll^llIV1’P(Rn)•
This inequality and (26) complete the proof.	□
5.6.3. General Sobolev inequalities.
We can now concatenate the estimates established in §§5.6.1 and 5.6.2
to obtain more complicated (and hard-to-remember) inequalities.
270
5. SOBOLEV SPACES
THEOREM 6 (General Sobolev inequalities). Let U be a bounded open
subset ofRn, with a C1 boundary. Assume и € Wk-p(U).
(i)Zf
(29)	k<^
P
then и € Lq(U), where
1 _ 1 _ k
q p n'
We have in addition the estimate
(30)	IMIl9([/) < C'||«||wfe-p(l/),
the constant C depending only on k,p,n and U.
(ii)	V
(31)	k>^,
P
then и E C p (17), where
- + 1 — -, if - is not an integer
7 = < LpJ p p
any positive number < 1, if is an integer.
We have in addition the estimate
(32)	1М1ск_[й]_1,7(-) < C'||“Hwk-p(i/)>
the constant C depending only on k,p,n,^ and U.
Proof. 1. Assume (29). Then since Dau E LFILT) for all |a| = k, the
Sobolev-Nirenberg-Gagliardo inequality implies
I|£>z34Ilp*([/) — C|I7ZIIwk>p(u) if |/?| = & —
and so и E Wk~1,p’ (U). Similarly, we find и E Wk~2,p**(U), where -kr =
p* ~ n = p ~ n' Continuing, we eventually discover after к steps that и E
W0,q(U) = Lq(U), for | = I — The stated estimate (30) follows from
multiplying the relevant estimates at each stage of the above argument.
2.	Assume now condition (31) holds, and is not an integer. Then as
above we see
(33)	и E Wk~l’r(U),
5.7. COMPACTNESS
271
for
(34)
1 _ 1 _ £
r p n’
provided Ip < n. We choose the integer I so that
(35)
n
P
that is, we set I =	. Consequently (34) and (35) imply r =	> n.
Hence (33) and Morrey’s inequality imply that Dau € C0’1_r([7) for all
|a| < k — I — 1. Observe also that 1 — ^ = 1 — + Z =	+ 1 — Thus
и € Ск 1,[p]+1 p (£7), and the stated estimate follows easily.
3.	Finally, suppose (31) holds, with an integer. Set I = — 1 = — 1.
Consequently, we have as above и G Wk~l'r(U) for r =	= n. Hence
the Sobolev-Nirenberg-Gagliardo inequality shows Dau E Lq{U) for all
n < q < oo and all |a| < k — I — 1 = k —	. Therefore Morrey’s inequality
further implies Dau E C0,1-«(U) for all n < q < oo and all |a| < k— — 1.
Consequently и E Ck 1,7(C) for each 0 < 7 < 1. As before, the stated
estimate follows as well.
□
Remark. Various general Sobolev-type inequalities can also be proved us-
ing the Fourier transform: see Problem 18.	□
5.7. COMPACTNESS
We have seen in §5.6 that the Gagliardo-Nirenberg-Sobolev inequality im-
plies the embedding of W1,P(U) into LP*(U) for 1 < p < n, p* = We
will now demonstrate that W1,P(C) is in fact compactly embedded in T9(C)
for 1 < q < p*. This compactness will be fundamental for our applications
of linear and nonlinear functional analysis to PDE in Chapters 6-9.
DEFINITION. Let X and Y be Banach spaces, X c Y. We say that X
is compactly embedded in Y, written
X ECY,
provided
(i) IMY < C||x||x (x E X) for some constant C,
272
5. SOBOLEV SPACES
and
(ii) each bounded sequence in X is precompact in Y.
THEOREM 1 (Rellich-Kondrachov Compactness Theorem). Assume U
is a bounded open subset o/Rn, and dU is C1. Suppose 1 < p < n. Then
W^tU) CC Lq(U)
for each 1 < q < p*.
Proof. 1. Fix 1 < q < p* and note that since U is bounded, Theorem 2 in
§5.6.1 implies
W1,P(W) c Lq{U\ ||u||L,([Z) < C||u||vyl,P([/).
It remains therefore to show that if {um}^=1 is a bounded sequence in
there exists a subsequence {umjwhich converges in Lq(U).
2. In view of the Extension Theorem from §5.4 we may with no loss of
generality assume that U = Rn and the functions {um}^=1 all have compact
support in some bounded open set V C Rre. We also may assume
(1)	sup ||um||wi,P(V) < OO.
m
3.	Let us first study the smoothed functions
u£m:=q£*um (e > 0, m = 1,2,...),
q£ denoting the usual mollifier. We may suppose the functions {«m}m=i
have support in V as well.
4.	We first claim
(2)	u"m —> um in Lq(V") as e —► 0, uniformly in m.
To prove this, we first note that if um is smooth, then
- um(x) = /	r?(y)(um(a; - ey) - um(x)) dy
= [	Т1(У) [ ^Aum(x - ety\) dtdy
J В{0,1)	Jo
= -e [	r?(y) [ Dum(x - ety) -ydtdy.
Jb{o,i) Jo
5.7. COMPACTNESS
273
Thus
[ |г4(ж) - um(x)|da; < £ [	7](y) [ [ \Dum(x - ety)\dxdtdy
Jv	Jb(oX) Jo Jv
<e \Dum(z)\dz.
Jv
By approximation this estimate holds if um 6 W1,P(V). Hence
ll^m um||L'(V) — ell-^um||L1(V) — £C||-^'um||LP(V)!
the latter inequality holding since V is bounded. Owing to (1) we thereby
discover
(3)	u£m —> um in Т1(У), uniformly in m.
But then since 1 < q < p*, we see using the interpolation inequality for
//-norms (§B.2) that
— um||L«(V) — ll^m — um H/,1 (V) ll^m ~	II £,p* (y)»
where j = в +	, 0 < в < 1. Consequently (1) and the Gagliardo-
Nirenberg-Sobolev inequality imply
— 'um||L9(V) < C||um —
whence assertion (2) follows from (3).
5. Next we claim
( for each fixed £ > 0, the sequence
( is uniformly bounded and equicontinuous.
Indeed, if x 6 Rn, then
\u£m(x)\<	T]£(x — y)\um(y)\dy
(J
< ll??e|k°o(R«)ll«m||L1(V) < — < OO
for m = 1,2,... . Similarly
\Du£m(x)\< [	\Dr/£(x-y)\\um(y)\dy
JB(x,e)
(J
< l|-^??e||Loo(Rn)ll'um||L1(V) — £n+l <
274
5. SOBOLEV SPACES
for m = 1,... . Assertion (4) follows from these two estimates.
6.	Now fix 8 > 0. We will show there exists a subsequence {umj C
{um}^=1 such that
(5)	lim sup - umk ||l«(v) < 6.
j,k—>oo
To see this, let us first employ assertion (2) to select e > 0 so small that
(6)	\\u£m-um\\Lq{V)<8/2
for m = 1,2,... .
We now observe that since the functions {tim}“=1, and thus the func-
tions {ttm}m=i> have support in some fixed bounded set V C Rn, we may
utilize (4) and the Arzela-Ascoli compactness criterion, §C.7, to obtain a
subsequence {uem.C	which converges uniformly on V. In
particular therefore
(7)	lim sup ||r4. - u^J|M(v) = 0.
j,k—>oo
But then (6) and (7) imply
lim sup 11um||z,9(v) —
j,k—>oo
and so (5) is proved.
7.	We next employ assertion (5) with 8 = 1, |, |,... and use a standard
diagonal argument to extract a subsequence {umi}/^i C {um}^=1 satisfying
lim SUp ЦПт;	= 0-
l,k—>oo
□
Remark. Observe that since p* > p and p* —> oo as p —► n, we have in
particular
W^U) CC If(U)
for all 1 < p < oo. (Observe that if n < p < oo, this follows from Morrey’s
inequality and the Arzela-Ascoli compactness criterion.) Note also that
W01,p((7) CC U>(U),
even if we do not assume dU to be C1.	□
5.8. ADDITIONAL TOPICS
275
5.8.	ADDITIONAL TOPICS
5.8.1.	Poincare’s inequalities.
We now illustrate how the compactness assertion in §5.7 can be used to
generate new inequalities.
Notation. (u)(7 = J^udy = average of и over U.	□
THEOREM 1 (Poincare’s inequality). Let U be a bounded, connected,
open subset o/Rn, with a C1 boundary dU. Assume 1 < p < oo. Then there
exists a constant C, depending only on n,p and U, such that
(1)	IIй - (n) и HbP(LZ) < C'||Du||Lp((7)
for each function и G W1,p(17).
The significance of (1) is that only the gradient of и appears on the right
hand side.
Proof. We argue by contradiction. Were the stated estimate false, there
would exist for each integer k — 1,... a function Uk G Wi,p(U) satisfying
(2)	||«fc - («*:)[/||lp((7) > kll-DwfcIIlp(!7)•
We renormalize by defining
Uk ~ (Uk)u /,	, X
( )	Vk'~ \\uk - (uk)u\\LP{u) ( "
Then
(vk)u = 0, ||Vfc||bp(L7) = 1;
and (2) implies
(4)	||.Dvfc||хд>(£/) < — (fc = l,2,...).
/С
In particular the functions	are bounded in W1,P(U').
In view of the Remark after the Rellich-Kondrachov Theorem in §5.7,
there exists a subsequence {v^}^ C	and a function v G 1/(11)
such that
(5)	vk] - n in LP(U).
From (3) it follows that
(6)	(v)[7 = 0, ||v||lp(c/) = 1.
276
5. SOBOLEV SPACES
On the other hand, (4) implies for each i = 1,..., n and ф G C£°(U)
that
/ уфх^х= lim / Vk^Xidx = - lim / v^^dx = 0.
JU	kj-^ooJu	kj^ooJu
Consequently v G W1,P(U"), with Dv = 0 a.e. Thus v is constant, since U
is connected (see Problem 10). However this conclusion is at variance with
(6): since v is constant and (v)u = 0, we must have v = 0; in which case
||v||lp(u) = 0- This contradiction establishes estimate (1).	□
A particularly important special case follows.
Notation. (u)x,r — г}и(1у = average of и over the ball B(x, r).	□
THEOREM 2 (Poincare’s inequality for a ball). Assume 1 < p < oo.
Then there exists a constant C, depending only on n and p, such that
(7)	IIй - («)х,г||ьр(В(х,г)) < C'rl|T>u||LP(B(x,r))
for each ball B(x, r) C IRn and each function и G W1,p(B0(x, r)).
Proof. 1. The case U = B°(0,1) follows from Theorem 1. In general, if
и G Ил1’р(В°(ж, r)) write
v(y) := u(x + ry) (?/GB(0,l)).
Then v G VP1,p(B°(0,1)), and we have
||v - (v)o,i||lp(b(o,i)) < C||.Dv||£p(b(o,i)).
Changing variables, we recover estimate (7).	□
Remark. Assume и G Ил1,га(Жп) П L1(Rn), and let В(ж, r) be any ball.
Then Theorem 2 with p = 1 implies
+	\u-{u)x<r\dy <Cr+ \Du\dy
J B(x,r)	J B(x,r)
if	\1/n	/ r	\i/n
<Cr[L |В«|п<й/	<C[ |Bu|nd7/ .
yj B(x,r)	J	vR"	/
Thus и G BMO(R"), the space of functions of bounded mean oscillation in
IRn, with the seminorm
Ывмо(К") = sup If \u — (u)x,r\dy >.
B(x,r)cIRn J B(x,r)
See Stein [SE, Chapter IV] for the theory of BMO.	□
5.8. ADDITIONAL TOPICS
277
5.8.2.	Difference quotients.
When we later apply Sobolev space theory to PDE, we will be forced to
study difference quotient approximations to weak derivatives. Following is
the relevant theory, which the reader may wish to postpone studying until
the need arises, in §6.3.
a. Difference quotients and W1,p.
Assume и : U —> Ж is a locally summable function, and V CC U.
DEFINITIONS.
(i)	The ith -difference quotient of size h is
h
for x G V and h G Ж, 0 < |h| < dist(V, dU).
(ii)	Dhu := (PiU,..., D„u).
THEOREM 3 (Difference quotients and weak derivatives).
(i) Suppose 1 < p < oo and и G Wlp(U). Then for each V CC U
(8)	||Dhu\\LP^v>) < C||Du||lp(c/)
for some constant C and all 0 < |h| < | dist(V,9f7).
(ii) Assume 1 < p < oo, и G L^V), and there exists a constant C such
that
(9)	PMl₽(V) < C
for all 0 < \h\ < j dist(V,3(7). Then
и G W1’P(V), with ||Du||Lp(v) < C.
Remark. Assertion (ii) is false for p = 1 (Problem 11).	□
Proof. 1. Assume 1 < p < oo, and temporarily suppose и is smooth. Then
for each x G V, i = 1,..., n, and 0 < |/i| < | dist(V, dU), we have
u(x + hei) — u(x) = / uXi(x + thei) dt  hei;
Jo
so that	!
\u(x + hei) — u(x)\ < h / \Du(x + thei)\dt.
Jo
278
5. SOBOLEV SPACES
Consequently
( \Dhu\pdx<C\2 [ [ \Du(x + thei)\pdtdx
V	Jv Jo
n fl f
= СУ2 / / \Du{x+ thei)\pdxdt.
~{Jo Jv
Thus
[ \Dhu\pdx<C [ \Du\p dx.
Jv	Ju
This estimate holds should и be smooth, and thus is valid by approximation
for arbitrary и G Ил1,р(17).
2. Now suppose estimate (9) holds for all 0 < |/i| < |dist(V, dU) and
some constant C. Choose i = l,...,n, ф G	and note for small
enough h that
ф(х) dx;
that is,
(10)
This is the “integration-by-parts” formula for difference quotients. Estimate
(9) implies
sup||rrSi||LP(V) < oo;
h
and therefore, since 1 < p < oo, there exists a function Vi G I/tV) and a
subsequence hk —► 0 such that
(11)	D~hku —*• Vi weakly in LP(V').
(See §D.4 for weak convergence.) But then
/ ифХг dx = I ифХг dx — lim / uD^k ф dx
Jv	Ju	hk~>0Ju
= — lim I D~hk,u^dx
hk^o Jv
= — Viфdx = — I Viфdx.
Jv	Ju
Thus Vi = uXi in the weak sense (г = 1,... , n), and so Du G ^(V). As
и G L^V), we deduce therefore that и G JT1,P(V).	□
5.8. ADDITIONAL TOPICS
279
Remark. Variants of Theorem 3 can be valid even if it is not true that
V CC U. For example if U is the open half-ball B°(0,1) П {xn > 0},
V — B°(0,1/2) A {xn > 0}, we have the bound fv \D^u\p dx < ]uXilpdx
for i — 1,..., n — 1. The proof is similar to that just given.
We will need this comment in Chapter 6, §6.3.2.
b. Lipschitz functions and W1,o°.
THEOREM 4 (Characterization of W1,o°). Let U be open and bounded,
with dU of class C1. Then и : U —> R is Lipschitz continuous if and only if
и e W1’00^).
Proof. 1. First assume U = Rn and и has compact support.
Suppose и G W1,oc'(Rn). Then u£ := T]£*u, where r)£ is the usual mollifier,
is smooth and satisfies
f u£ —► и uniformly as £ —> 0,
I l|B>Ue||Loo(Rn) < pu||Loo{Rn).
Choose any two points x, у € Rn, x y. We have
u£ (xj - u£ (y) = [ -^-u£(tx + (1 - t)y) dt
Jo dt
= I Du£(tx + (1 — tfy) dt • (x — y);
Jo
and so
|u£(x) - ue(y)| < ||Bue||Loo(Kn)|ж - y\ < IlDullboo^)|ж - y|.
We let £ —> 0 to discover
|u(x) - u(y)| < ||r)u||Loo(Rn)|ж - y|.
Hence и is Lipschitz continuous.
2.	On the other hand assume now и is Lipschitz continuous; we must
prove that и has essentially bounded weak first derivative. Since и is Lip-
schitz, we see
II	< Lip(u),
and thus there exists a function Vi G L°°(lRn) and a subsequence hk —► 0
such that
(12)	D~hku —> Vi weakly in Lfoc(R”).
280
5. SOBOLEV SPACES
Consequently
/ u<bXi dx = lim / uD^k ф dx
jRn	hk^0j^n
= — lim / D^hku(/>dx = — / Vi<pdx
hk^O	JR"
by (12). The above equality holds for all ф G C'^o(Rn), and so V{ = uXi in
the weak sense (i = 1,..., n). Consequently и E W'1,oo(IR.n).
3.	In the general case that U is bounded, with dU of class (71, we as
usual extend и to Ей — й and apply the above argument.	□
Remark. The argument above adapts easily to prove that for any open set
U, и E (17) if and only if и is locally Lipschitz continuous in U. There
is no corresponding characterization of the spaces W1,p for 1 < p < oo. If
n < p < oo, then each function и E IT1’7’ belongs to C0,1~n/p, but on the
other hand a function Holder continuous with exponent less than one need
not belong to any Sobolev space Ж1,р.	□
5.8.3.	Differentiability a.e.
Next we examine more closely the connections between weak partial
derivatives and partial derivatives in the usual calculus sense.
DEFINITION. A function и : U —> R. is differentiable at x E U if there
exists a E Rn such that
(13)	u(y) = u(x) + a  (у - x) + о(|г/ - ж|) asy-+x.
In other words,
lim Щу) - Ц(ж) - а- (т/-ж)| = Q
У~>х	\y — x|
Notation. It is easy to check that a, if it exists, is unique. We henceforth
write
Du(x)
for a and call Du the gradient of u.	□
To be sure that this notation is consistent, we need to study the rela-
tionships between the various notions of derivatives:
THEOREM 5 (Differentiability almost everywhere). Assume и E W^(U)
for some n < p < oo. Then и is differentiable a.e. in U, and its gradient
equals its weak gradient a.e.
Recall we always identify и with its continuous version.
5.8. ADDITIONAL TOPICS
281
Proof. 1. Assume first n < p < oo. From the Remark after the proof of
Theorem 4 in §5.6.2, we recall Morrey’s estimate
/ .	\ Vp
(14)	|v(y) — v(x)| < (7r1_p [ / \Dv(z)\pdz ] (yGB(x, r)),
\JB(x,2r)	J
valid for any (71 function v and thus, by approximation, for any v G IV1,P.
2.	Choose и G W^£(U). Now for a.e. x G U, a version of Lebesgue’s
Differentiation Theorem (§E.4) implies
(15)	7" \Du(x) — Du(z)\pdz—> 0
J B{x,r)
asr —> 0, Du denoting as usual the weak derivative of u. Fix any such point
x and set
v(y) := u(y) - u(x) -Du(x)  (у - ж)
in estimate (14), where
(16)	r = \x-y\.
We find
|u(y) - u(x) - Du(x) • (у - ж)|
/	\ i/p
<	(7ri-n/p ( I \Du(x) - Du(z)\pdz ]
\JB(x,2r)	J
/ .	\Vp
<	Cr | +	|_Du(x) — Du(z)\pdz I
B(x,2r)	J
=	o(r) by (15)
=	o(|x-y|) by (16).
Thus и is differentiable at x, and its gradient equals its weak gradient at x.
3. In case p = oo, we note	C W^£(U) for all 1 < p < oo, and
apply the reasoning above.	□
Finally, in view of Theorem 5, we obtain
THEOREM 6 (Rademacher’s Theorem). Let и be locally Lipschitz con-
tinuous in U. Then и is differentiable almost everywhere in U.
282
5. SOBOLEV SPACES
5.8.4. Fourier transform methods.
Next we employ the Fourier transform (§4.3) to give an alternate charac-
terization of the spaces _Hfc(IRn). For this section all functions are complex-
valued.
THEOREM 7 (Characterization of Hk by Fourier transform).
Let к be a nonnegative integer.
(i)	A function и G Т2(ЖП) belongs to Hk(Wl) if and only if
(17)	(1 + \y\k)u G L2(Rn).
(ii)	In addition, there exists a positive constant C such that
(18)	Q1Ы1 tf*(R") < IK1 + |y|fc)«||L2(R„) < С||и||Як(Кп)
for each и G J7fc(Rn).
Proof. 1. Assume first и G _Hfc(Rn). Then for each multiindex |a| < к, we
have Dau G L2(Rn). Now if и G Ck has compact support, we have
(19)	Dau = (iy)au
according to Theorem 2 in §4.3.1. Approximating by smooth functions we
deduce formula (19) provided и G Hk(JRn). Thus (iy)°u G L2(Rn) for each
|a| < k. In particular choosing a = (k, 0,..., 0), (0, k,..., 0),..., (0,..., k),
we deduce
f |?/|2A:|й|2</2/ < C [ \Dku\2dx < oo.
JR"	JR"
Thus
JU1 + Ы‘")21*1Ч < СМЯ-ЧПР),
and so (1 + |y|fc)u G L2(Rn).
2.	Suppose conversely (1 + |7/|fc)u G L2(Rn) and |a| < k. Then
(20)	||(z2/)%||L2(Rn) < [ |у|21“1|й|Ч, < C||(l + |y|fc)«||L2(Rn).
JR"
Set
Ua := ((zy)%)v.
Then for each ф G C'^!O(IRn)
f (раф)й<1х = f (D^)udy = [ (iy}a^>udy
JR"	JR"	JR"
= (—l)l“l [ фйо^х.
JR"
5.9. OTHER SPACES OF FUNCTIONS
283
Thus ua = Dau in the weak sense and, by (20), D°u G L2(Rn). Hence
и G Hk(U), as required.	□
It is sometimes useful to define also fractional Sobolev spaces.
DEFINITION. Assume 0 < s < oo and и G L2(Rn). Then и G H^W1)
if (1 + |i/|s)u G L2(Rn). For noninteger s, we set
Hl№(R") := 11(1 + |y|S)«||L2(Rn).
5.9. OTHER SPACES OF FUNCTIONS
5.9.1. The space H-1.
As we will see later in our systematic study in Chapters 6 and 7 of linear
elliptic, parabolic and hyperbolic PDE, it is important to have an explicit
characterization of the dual space of Hq. (See Appendix D for definitions.)
DEFINITION. We denote by H~\U) the dual space to H^U).
In other words f belongs to Я-1(17) provided f is a bounded linear
functional on Hq(U).
Notation. We will write ( , ) to denote the pairing between И-1(С7) and
W)-	□
DEFINITION. Iff G H-^U), we define the norm
11/11/7-1(17) = sup {(/,«) | U G Я01(С/),||и||Я1{[/) < 1|.
THEOREM 1 (Characterization of U-1).
(i) Assume f G 7?-1(17). Then there exist functions /°,/1, ...,fn in
L2(U) such that
(1)	(M)= [ f°v + ±fVxidx (vG#0W
i=l
(ii) Furthermore,
f satisfies (1) for f°,... ,fn G L2(U)
284
5. SOBOLEV SPACES
Notation. We write f = f° — fXi whenever (1) holds.
□
Proof. 1. Given u, v 6 Hq (U), we define the inner product (u, v) := Du-
Dv + uvdx. Let f G B-1(17). We apply the Riesz Representation Theorem
(§D.3) to deduce the existence of a unique function и E Hq(U") satisfying
B[u, v] = (f,v) for all v E Hq(U); that is,
(2)	/ Du  Dv + uvdx = (f, v)
Ju
for each v E Hq(U). This establishes (1) for
( f° — и
l Г = uXi (i =
2. Assume now f E
(4)	(f, v) = [ g°v + Y}glvXl dx
Ju i=i
for 5°,51,... ,gn 6 L2(U"). Setting v = и in (2) and using (4), we deduce
Thus (3) implies
(5)
3. From (1) it follows that
if ||v||#i((7) < 1. Consequently
/ n	\ 1/2
нли-.р) < I / Li/T*
\Jui-0	J
Setting v = ццу -— in (2), we deduce that in fact
/ n	\ V2
(6)	11/11я-.ст = (/ Ёи2*)
\Ju i=0	/
Assertion (ii) follows now from (4)-(6).
□
5.9. OTHER SPACES OF FUNCTIONS
285
5.9.2. Spaces involving time.
We study next some other sorts of Sobolev spaces, these comprising
functions mapping time into Banach spaces. These will prove essential in
our constructions of weak solutions to linear parabolic and hyperbolic PDE
in Chapter 7 and to nonlinear parabolic PDE in Chapter 9.
Let X denote a real Banach space, with norm || ||. The reader should
first of all read §E.5 about measure and integration theory for mappings
taking values in X.
DEFINITION. The space
17(0,T-, X)
consists of all measurable functions u : [0, T] —► X with
f Гт \ Vp
(i)	I|u||lp(0,T;X) = \Jo ||u(t)||pdtj <oo
for 1 < p < oo, and
(ii)	:= ess sup||u(t)|| < oo.
0<t<T
DEFINITION. The space
C([0,T];X)
comprises all continuous functions u : [0, T] —► X with
l|u||c([0,T];X) := om^||u(t)|| < oo.
DEFINITION. Let u G Lr(0,T\X). We say v G Lx(0,T;X) is the weak
derivative of u, written
u' = V,
provided
[ </>'(t)u(t) dt = — [ </>(t)v(t) dt
Jo	Jo
for all scalar test functions ф G C£°(0,T).
286
5. SOBOLEV SPACES
IIUIO-P(0,T;X)
DEFINITIONS, (i) The Sobolev space
W^p^T-X)
consists of all functions u G 1^(0, T; X) such that u' exists in the weak sense
and belongs to LP(f},T',Xf Furthermore,
(joT|lu(OII₽ + llu'(*)ll₽d*) /P (l<f><oo)
ess sup(||u(f)|| + ||u'(t)||) (p = oo).
0<t<T
(ii)	We write H\Q,T-X) = W^O^X).
THEOREM 2 (Calculus in an abstract space). Let u G Wrl,p(0, T; X) for
some 1 < p < oo. Then
(i)	u G C([0, T];X) (after possibly being redefined on a set of measure
zero), and
(ii)	u(t) = u(s) + ff u'(r) dr for all 0 < s <t <T.
(iii)	Furthermore, we have the estimate
(7)	max||u(t)|| < C||u||iyi,P(0)T;X),
the constant C depending only on T.
Proof. 1. Extend u to be 0 on (—oo, 0) and (T, oo), and then set ue = 7?e*u,
7?e denoting the usual mollifier on R1. We check as in the proof of Theorem 1
in §5.3.1 that ufc = /)£*u' on (e,T — e).
Then as e —> 0,
inLP(0,T;X),
t (O' -> uz	inP(0,T;I).
Fixing 0 < s < t < T, we compute
/•t
ue(t) = ue(s) + J ue'(r) dr.
Thus
(9)	u(t) = u(s) + f u'(r) dr
J s
for a.e. 0 < s < t < T, according to (8). As the mapping t ff u'(r) dr is
continuous, assertions (i), (ii) follow.
2. Estimate (7) follows easily from (9).	□
The next two propositions concern what happens when u and u' lie in
different spaces.
5.9. OTHER SPACES OF FUNCTIONS
287
THEOREM 3 (More calculus). Suppose u G Z2(0, T; Hq(UY), with u' G
L2(0,T;H-\U)).
(i)	Then
ueC([0,T]-,L2(U))
(after possibly being redefined on a set of measure zero).
(ii)	The mapping
t I—> ||u(i)||ь2(Г/)
is absolutely continuous, with
^Пи(*)НЬ(с/) = 2M*),U(*))
for a.e. 0 < t < T.
(iii)	Furthermore, we have the estimate
(1°)	о“^^,Ни(О11ь2(С7) < С'(Ни||£2(0,Т;Яо1(!7)) +	>
the constant C depending only on T.
Proof. 1. Extend u to the larger interval [—a, T + a] for a > 0, and define
the regularizations ue = r]e * u, as in the earlier proof. Then for e, 8 > 0,
^l|ueW -	= 2(иЛ*) - u5'(t),ue(t) - u5(t))L2([/).
Thus
||ue(t) - u5(t)||£2([Z) = ||ue(s) - u5(s)||22([/)
(n)	л ,
+ 2 J (ue (r) — u5 (r), ue(r) — u^r)) dr
for all 0 < s, t < T. Fix any point s G (0, T) for which
ue(s) —> u(s)	in L2(U}.
Consequently (11) implies
limsup sup ||ue(t)-u5(t)||^2([Z) <
e,5—>0 0<t<T	'

+ Цие(т) - u5(r)||^i([/) dr
= 0.
288
5. SOBOLEV SPACES
Thus the smoothed functions {ue}o<e<i converge in C([0,T]; L2(J7)) to a
limit v G C([0,7'];Z2(?7)). Since we also know ue(t) —> u(t) for a.e. t, we
deduce u = v a.e.
2. We similarly have
= l|ue(s)|lb(t/) +2 [ (ue'(-r),ue(-r))dT,
J s
and so, identifying u with v above,
(12)	l|u(t)H12([;) = ||u(s)||£2([Z) + 2 I (u'(r), u(t)) dr
J 8
for all 0 < s,t < T.
3. To obtain (10), we integrate (12) with respect to s, recall the inequal-
ity |(uz, u)| < ||uz||fl-i((7) ||u||//i((/), and make some simple estimates. □
For use later in the regularity theory for second-order parabolic and hy-
perbolic equations in Chapter 7, we will also need this extension of Theorem
3.
THEOREM 4 (Mappings into better spaces). Assume that U is open,
bounded, and dU is smooth. Take m to be a nonnegative integer.
Suppose u G Т2(0,Т;Ят+2(Я)), with uz G T2(0,T;
(i) Then
uG С([0,Т];Нт+1(иУ)
(after possibly being redefined on a set of measure zero).
(ii) Furthermore, we have the estimate
(13)	тЮ^||и(4)||я"Ч-1([/) < C(||u||L2(0,T;Hm+2(U)) + Ни/||ь2(0,Т;Я’"(С/))),
the constant C depending only on T, U, and m.
Proof. 1. Suppose first that m = 0, in which case
u G Т2(0,Т;Я2(С/)), uz G Z2(0,T; Z2(C/)).
We select a bounded open set V DD U, and then construct a corre-
sponding extension й = Eu, as in §5.4. In view of estimate (10) from that
section, we see
Й G Т2(0,Т;Я2(У)),
5.10. PROBLEMS
289
and
(14)	||й||ь2(0,Т;Я2(У)) < C'IIuIIl2(0,T;H2(!7)),
for an appropriate constant C. In addition, u' G Z2(0,T;Z2(V)), with the
estimate
(15)	||й/||£2(0,Т;Ь2(У)) < C'||u'||l2(0)T;L2((7)).
This follows if we consider difference quotients in the t-variable, remember
the methods in §5.8.2, and observe also that E is a bounded linear operator
from L2(U) into _L2(V).
2.	Assume for the moment that ii is smooth. We then compute
\-t([ |-0u|2 dx) | = 2| f Du -Du'dx\ = 2\ f Дйй'б1ж|
dt J у	J у	Jv
— С(||й|1я2(у) + ||Й ||£2(y}) •
There is no boundary term when we integrate by parts, since the extension
ii = Eu has compact support within V. Integrating and recalling (14), (15),
it follows that
(16)	om^c^||u(t)||j/i(CZ) < С'(||и||Ь2(0;Т;Я2([/)) + ||и'Ь(0>Т;Ь2([/))).
We obtain the same estimate if u is not smooth, upon approximating by
u£ := T)£ * u, as before. As in the previous proofs, it also follows that
ugC([0,T];№(1/)).
3.	In the general case that m > 1, we let a be a multiindex of order
|a| < m, and set v := Dau. Then
v G Z2(0, T; Я2(17)), v' G L2(0, T; L2(U)).
We apply estimate (16), with v replacing u, and sum over all indices |a| < m,
to derive (13).
□
5.10. PROBLEMS
In these exercises U always denotes an open subset of Rn.
1. Suppose k G {0,1,... }, 0 < 7 < 1. Prove C'fc,7(t7) is a Banach space.
290
2.
3.
4.
5.
6.
7.
8.
9.
5. SOBOLEV SPACES
Let U, V be open sets, with V CC U. Show there exists a smooth
function C such that ( = 1 on V, ( = 0 near dU. (Hint: Take
V CC W CC U and mollify xw.)
Assume 0 < 0 < 7 < 1. Prove the interpolation inequality
||г4||с°’'>'(!7) — llullc^./3((7) IMU(U)'
Assume U is bounded and U CC	Show there exist C°°
functions Ci (г = 1,..., TV) such that
0< Ci < 1, sptCi C Vi (i = l,...,N)
iXi Ci = 1 on U.
The functions {G}^=i form a partition of unity.
Prove that if n = 1 and и G JT1,P(0,1) for some 1 < p < 00, then и is
equal a.e. to an absolutely continuous function, and u' (which exists
a.e.) belongs to 7/(0,1).
Prove directly that if и G Ж1,₽(0,1) for some 1 < p < 00, then
|«(ж) — u(y)I < |ж -	(Jq	/ for a.e. x, у G [0,1].
Denote by U the open square {rGR2 | |a?i| < 1, |жг| < 1}. Define
' 1	— a?i	if xi	> 0,	|жг|	<	xi
, x 1	+ <Ei	if xi	< 0,	1ж21	<	-xi
u(x) =
1	- X2	if X2	> 0,	|a?i I	<	X2
, 1	+ X2	if X2	< 0,	|xi|	<	— X2-
For which 1 < p < 00 does и belong to W1,P(U)?
Integrate by parts to prove the interpolation inequality:
/	/ r \ W2 / r \ 1/2
j |£hi|2(fa < C yj u2dxj |D2u|2dxJ
for all и G	By approximation, prove this inequality if и G
H2(C/)nH01(C/).
Integrate by parts to prove:
r	/ Г \ ^2 / Г	\V2
j \Du\pdx < C (j ]ulPdx ] (J \D2ulPdx}
5.10. PROBLEMS
291
for 2 < p < oo and all и € W2'P(U) A Wq’p(U). (Hint: fv \Du\Pdx =
Y,i=i Ju uXiuXi\Du\P-2dx.)
10.	Suppose U is connected and и E IV1,P(17) satisfies
Du = 0 a.e. in U.
Prove и is constant a.e. in U.
11.	Show by example that if we have ||.Dhu||Li(y) < C for all 0 < |/i| <
j dist(V, dU), it does not necessarily follow that и E
12.	Give an example of an open set U C Rn and a function и E Ил1,0°(17),
such that и is not Lipschitz continuous on U. (Hint: Take U to be the
open unit disk in R2, with a slit removed.)
13.	Verify that if n > 1, the unbounded function и = log log (1 +
belongs to Жг-П(17), for U = B°(0,1).
14.	Let U be bounded, with a C1 boundary. Show that a “typical” func-
tion и E LP(U) (1 < p < oo) does not have a trace on dU. More
precisely, prove there does not exist a bounded linear operator
T : If(U) - Lp(dU)
such that Tu = u\gu whenever и E C(U) A //(B).
15.	Fix a > 0 and let U = B°(0,1). Show there exists a constant C,
depending only on n and a, such that
f u2dx < C [ |Bu|2dx ,
Ju Ju
provided
|{жеВ|и(ж) = 0}| >a , p EH^U).
16.	Assume F : R —> R is C1, with F' bounded. Suppose U is bounded
and и E W1,P(U) for some 1 < p < oo. Show
v := F(u) E W^U) and vXi=F'(u)uXi (a = l,...,n).
17. Assume 1 < p < oo, and U is bounded.
(i) Prove that if и E JV1,P(B), then |u| E W1,P(U).
292
5. SOBOLEV SPACES
(ii) Prove и G Ил1,р(С7) implies u+,u € Wrl,p(C7), and
Du+ =
Du =
Du
0
0
—Du
a.e. on {n > 0}
a.e. on {« < 0},
a.e. on {u > 0}
a.e. on {u < 0}.
(Hint: u+ = lim^o Fe(u), for
Fe(2) :=
e if z > 0
if z < 0.)
(iii) Prove that if и G Ил1,р(17), then
Du = 0 a.e. on the set {u = 0}.
18. Use the Fourier transform to prove that if и G №(Rn) for s > n/2,
then и G Z°°(]Rn), with the bound
||«||loo(]R") < C'||“ll№(R"),
the constant C depending only on s and n.
5.11. REFERENCES
Sections 5.2-8 See Gilbarg-Trudinger [G-T, Chapter 7], Lieb-Loss [L-L],
Ziemer [Z] and [E-G] for more on Sobolev spaces.
Section 5.	5 W. Schlag showed me the proof of Theorem 2.
Section 5.	6 J. Ralston suggested an improvement in the proof of Theo-
rem 4.
Section 5.	9 See Temam [ТЕ, pp. 248-273].
Chapter 6
SECOND-ORDER
ELLIPTIC
EQUATIONS
6.1	Definitions
6.2	Existence of weak solutions
6.3	Regularity
6.4	Maximum principles
6.5	Eigenvalues and eigenfunctions
6.6	Problems
6.7	References
This chapter investigates the solvability of uniformly elliptic, second-
order partial differential equations, subject to prescribed boundary condi-
tions. We will exploit two essentially distinct techniques, energy methods
within Sobolev spaces (§§6.1-6.3) and maximum principle methods (§6.4).
(1)
6.1.	DEFINITIONS
6.1.1.	Elliptic equations.
We will in this chapter mostly study the boundary-value problem
Lu = f in U
и = 0 on dU,
where U is an open, bounded subset of Rn and и : U —> Ж is the unknown,
и = и(ж). Here f : U —► R is given, and L denotes a second-order partial
293
294
6. SECOND-ORDER ELLIPTIC EQUATIONS
differential operator having either the form
n	n
(2)	Lu = -	(а4 (x)uxf)x, + У7 b\x)uXi + c(x)u
i,j=l	i=l
or else
n	n
(3)	Zu = - У2	+'^Ьг(х)иХг + с(ж)и,
i,j=l	i=l
for given coefficient functions агз,Ьг,с (i,j = 1,..., n).
We say that the PDE Lu — f is in divergence form if L is given by (2),
and is in nondivergence form provided L is given by (3). The requirement
that и = 0 on dU in (1) is sometimes called Dirichlet’s boundary condition.
Remark. If the highest order coefficients av (г, j = 1,..., n) are C1 func-
tions, then an operator given in divergence form can be rewritten into non-
divergence structure, and vice versa. Indeed the divergence form equation
(2) becomes
n	n
(2')	Lu = - alj(x)uXiXj + ^bl(x)uXi + c(r)u
i,j=l	i=l
for b1 ~bl — 52”=1 dxj (i = 1, •  •, n), and (2') is obviously in nondivergence
form. We will see, however, there are definite advantages to considering
the two different representations of L separately. The divergence form is
most natural for energy methods, based upon integration by parts (§§6.1—
6.3), and the nondivergence form is most appropriate for maximum principle
techniques (§6.4).	□
We henceforth assume as well the symmetry condition
a4 = aJt (i, j = 1,..., n) .
DEFINITION. We say the partial differential operator L is (uniformly)
elliptic if there exists a constant в > 0 such that
n
(4)	£ a^x^j > 0|£|2
i,j=l
for a.e. x E U and all £ E IRn.
Ellipticity thus means that for each point x E U, the symmetric n x n
matrix А(ж) = ((аи(ж))) is positive definite, with smallest eigenvalue greater
than or equal to в.
6.1. DEFINITIONS
295
An obvious example is = 6ij, bl = 0, c = 0, in which case the operator
L is —Д. Indeed we will see that solutions of the general second-order elliptic
PDE Lu = 0 are similar in many ways to harmonic functions. However,
for these partial differential equations we do not have available the various
explicit formulas developed for harmonic functions in Chapter 2: we must
instead work directly with the PDE. Readers should continually be alert in
the following calculations for uses of the structural condition of ellipticity
(4).
Physical interpretation. As just noted, second-order elliptic PDE gener-
alize Laplace’s and Poisson’s equations. As in the derivation of Laplace’s
equation set forth in §2.2, и in applications typically represents the density
of some quantity, say a chemical concentration, at equilibrium within a re-
gion U. The second-order term A : D2u =	=1 a4 uXiXj represents the
diffusion of и within U, the coefficients ((a4)) describing the anisotropic,
heterogeneous nature of the medium. In particular, F := — A.Du is the
diffusive flux density, and the ellipticity condition implies
F • Du < 0;
that is, the flow is from regions of higher to lower concentration. The first-
order term b • Du = bluXi represents transport within U, and the
zeroth-order term cu describes the local creation or depletion of the chemical
(owing, say, to reactions). A careful analysis of these interpretations requires
the probabilistic study of diffusion processes.
Nonlinear second-order elliptic PDE also arise naturally in the calculus
of variations (as the Euler-Lagrange equations of convex energy integrands)
and in differential geometry (as expressions involving curvatures). We will
encounter some such nonlinear equations later, in Chapters 8 and 9.	□
6.1.2.	Weak solutions.
Let us consider first the boundary-value problem (1) when L has the
divergence form (2). Our overall plan is first to define and then construct
an appropriate weak solution и of (1), and only later to investigate the
smoothness and other properties of u.
We will assume in the following exposition that
(5)	aijXcGL°°(f7) (?, j = 1,... ,n)
and
(6)
f G L\U).
296
6. SECOND-ORDER ELLIPTIC EQUATIONS
Motivation for definition of weak solution. How should we define
a weak or generalized solution? Assuming for the moment и is really a
smooth solution, let us multiply the PDE Lu = f by a smooth test function
v E C£°(U), and integrate over U, to find
r n	n	*
(7)	/ У2 aNuXivx + \\bluxv + cuvdx = / fvdx,
Ju~^	Ju
where we have integrated by parts in the first term on the left hand side.
There are no boundary terms since v = 0 on dU. By approximation we
find the same identity holds with the smooth function v replaced by any
v E Hq(U), and the resulting identity makes sense if only и E Hq(U). (We
choose the space Hq(U') to incorporate the boundary condition from (1) that
“u = 0 on dW.)
DEFINITIONS, (i) The bilinear form B[ , ] associated with the diver-
gence form elliptic operator L defined by (2) is
p n	n
(8)	B[u, и] := / a^uXivXi + У^ bzuXiv + cuv dx
Ju ij=i	i=i
for u,v E Hq(U).
(ii)	We say that и E Hq(U) is a weak solution of the boundary-value
problem (1) if
(9)	B[u,v] = (J,v)
for all v E Hq(U), where ( , ) denotes the inner product in L2(U).
Remark. The identity (9) is sometimes called the variational formulation
of (1). This terminology will be explained later, in Example 2 of §8.1.2.
□
More generally, let us consider the boundary-value problem
и = 0	on dU,
where L is defined by (2) and fl E L2(U) (i = 0,... ,n). In view of the
theory set forth in §5.9.1 we see that the righthand term f = f0 — fXi
belongs to H’-1(17), the dual space of Hq (17).
6.2. EXISTENCE OF WEAK SOLUTIONS
297
DEFINITION. We say и E Hq(U) is a weak solution of problem (10)
provided
B[w,v] = (f,v)
for all v E Hq(U), where (f,v) = f°v + ftv^i dx and {,) is the
pairing of H^^lU) and Hq(U).
Remark. We will hereafter, as above, focus attention exclusively on the
case of zero boundary conditions, but in fact a problem with prescribed,
nonzero boundary values can easily be transformed into this setting. We
spell this out by supposing now that dU is C1 and и E H^U) is a weak
solution of
( Lu = f in U
1 и = g on dU.
This means that и = g on dU in the trace sense, and furthermore that the
bilinear form identity (9) holds for all v E Hq(U). That this be possible,
it is necessary for g to be the trace of some H1 function, say w. But then
й := и — w belongs to Hq(U), and is a weak solution of the boundary-value
problem
f Lu — f in U
( й = 0 on dU,
where f := f — Lw E H^^U).
See Problems 2, 3 to learn how to cast some other sorts of PDE and
boundary conditions into weak formulations.	□
6.2. EXISTENCE OF WEAK SOLUTIONS
6.2.1. Lax—Milgram Theorem.
We now introduce a fairly simple abstract principle from linear func-
tional analysis, which will later in §6.2.2 provide in certain circumstances
the existence and uniqueness of a weak solution to our boundary-value prob-
lem.
We assume for this section H is a real Hilbert space, with norm || || and
inner product ( , ). We let ( , ) denote the pairing of H with its dual space.
Readers should review as necessary the basic Hilbert space theory described
in §D.2-3.
THEOREM 1 (Lax-Milgram Theorem). Assume that

298
6. SECOND-ORDER ELLIPTIC EQUATIONS
is a bilinear mapping, for which there exist constants a, (3 > 0 such that
(i)	|B[u,u]| < a||u|| ||v|| (и,ибЯ)
and
(ii)	/3||u||2 < B[u,u] (uGH).
Finally, let f : H —> R be a bounded linear functional on H.
Then there exists a unique element и E H such that
(1)	B[u,v] = {f,v)
for all v E H.
Proof. 1. For each fixed element и E H, the mapping v >—> B[u, v] is a
bounded linear functional on H; whence the Riesz Representation Theorem
(§D.3) asserts the existence of a unique element w E H satisfying
(2)	B[u, г;] = (w, v) (v E H).
Let us write Au = w whenever (2) holds; so that
(3)	B[u,v] = (Au,v) (u,v E H).
2.	We first claim A : H —> H is a bounded linear operator. Indeed if
Ai, A2 E R. and ui,U2 G H, we see for each v E H that
(A(Ami + A2U2),v) = B[Aiui + A2u2, v] by (3)
= AiBfyi, v] + A2B[u2,u]
= Ai(Aui,u) + A2(Au2,u) by (3) again
= (AMui + A2Au2, v).
This equality obtains for each v E H, and so A is linear. Furthermore
||Au||2 = (Au, Au) = B[u, Au] < o||u|| ||Au||.
Consequently ||Au|| < a||u|| for all и E H, and so A is bounded.
3.	Next we assert
( A is one-to-one, and
(4)
( R(A), the range of A, is closed in H.
To prove this, let us compute
/?IMI2 < B[u,u] = (Au,u) < IIAu|| ||<
6.2. EXISTENCE OF WEAK SOLUTIONS
299
Hence /3||u|| < ||-Au||. This inequality easily implies (4).
4.	We demonstrate now
(5)	R(A) = H.
For if not, then, since R(A) is closed, there would exist a nonzero element
w G H with w G RtAyL. But this fact in turn implies the contradiction
/3||w||2 < B[w,w] = (Aw, w) = 0.
5.	Next, we observe once more from the Riesz Representation Theorem
that
(f, v) = (w, v) for all v G H
for some element w G H. We then utilize (4) and (5) to find и G H satisfying
Au = w. Then
B[u, v] = (Au,v) = (w,v) = (f, v) (v G H),
and this is (1).
6.	Finally, we show there is at most one element и G H verifying (1). For
if both B[u,v] = (f,v) and В[й, v] = (f, v}, then B(u — it,v] = 0 (v G H).
We set v = и — u to find /3||u — й||2 < B[u — й, и — й] = 0.	□
Remark. If the bilinear form B[ , ] is symmetric, that is, if
B[u, v] = B[v, u] (u, v G H),
we can fashion a much simpler proof by noting ((u,v)) B[u,v] is a new
inner product on H. to which the Riesz Representation Theorem directly
applies. Consequently, the Lax-Milgram Theorem is primarily significant in
that it does not require symmetry of , ].	□
6.2.2. Energy estimates.
We return now to the specific bilinear form B[ , ], defined in §6.1.2 by
the formula
г n	n
B[u,v] = / aNuxvx + /7 blux v + cuv dx
i=i
for u, v G Hq(U), and try to verify the hypothesis of the Lax-Milgram
Theorem.
300
6. SECOND-ORDER ELLIPTIC EQUATIONS
THEOREM 2 (Energy estimates). There exist constants a,(3 > 0 and
7 > 0 such that
№,v]| < а|М|Яо1(с/)Н|Н1(с/)
and
(ii)	/?HuIIh01(1/) - B\-u>+ 7lMlb(t;)
for all u,v G Hq(U).
Proof. 1. We readily check
|B[u,v]| <	||«v ||lo° / |-Du| \Dv\dx
t,j=i	u
+	/ |E>u| |v| tta + ||c||Loo / |u| \v\dx
Ju	Ju
а11и11н1(1/)Пи11н^(1/),
for some appropriate constant a.
2. Furthermore, in view of the ellipticity condition (4) from §6.1 we have
Now from Cauchy’s inequality with £ (§B.2), we observe
We insert this estimate into (6) and then choose £ > 0 so small that
^SiibiiL" <|-
Z = 1
Thus
6.2. EXISTENCE OF WEAK SOLUTIONS
301
for some appropriate constant C. In addition we recall from Poincare’s
inequality in §5.6.1 that
IIuIIl2(C/) < ^II^ML\U) •
It easily follows that
/3||u|Ih01(i/) <	+ 7||«II12{[7)
for appropriate constants (3 > 0, 7 > 0.
Observe now that if 7 > 0 in these energy estimates, then B[ , ] does not
precisely satisfy the hypotheses of the Lax-Milgram Theorem. The following
existence assertion for weak solutions must confront this possibility:
THEOREM 3 (First Existence Theorem for weak solutions). There is a
number 7 > 0 such that for each
(7)
F > 7
and each function
f e L\U),
there exists a unique weak solution и G Hq(U) of the boundary-value problem
(8)
Lu + pu = f
и — 0
in U
on dU.
Proof. 1. Take 7 from Theorem 2, let /z > 7, and define then the bilinear
form
B^[u, v] := B[u, v] + p(u, v) (u,v G Hq (£7)),
which corresponds as in §6.1 to the operator L^u := Lu + pu. As before ( , )
means the inner product in L2([7). Then B^[ , ] satisfies the hypotheses of
the Lax-Milgram Theorem.
2. Now fix f G L2(U) and set (/, v) := (J, v)L2^y This is a bounded
linear functional on L2(?7), and thus on Hq(LT).
We apply the Lax-Milgram Theorem to find a unique function и G
Hq(U) satisfying
BfAu,v] = (f,v)
for all v G Hq(U)', u is consequently the unique weak solution of (8).	□
302
6. SECOND-ORDER ELLIPTIC EQUATIONS
Remark. We can similarly show that for all
fleL\U) (i = 0, ...,n),
there exists a unique weak solution и of the PDE
zqx	j Lu + pu = f(> - fx. in U
'	| и = 0	on dU.
Indeed, it is enough to note (/, v) = f°v + flvXi dx is a bounded
linear functional on as previously discussed in §5.9.1.
In particular, we deduce that the mapping
Ь^.= Е + ц1-.Н^и^Н-\и)
is an isomorphism.	□
Examples. In the case Lu = —Nu, so that B[u,v] = Du  Dv dx, we
easily check using Poincare’s inequality that Theorem 2 holds with 7 = 0. A
similar assertion holds for the general operator Lu = —	(.а^их,)х. +
cu, provided c > 0 in U.	□
6.2.3. Fredholm alternative.
We next employ the Fredholm theory for compact operators (discussed in
§D.5) to glean more detailed information regarding the solvability of second-
order elliptic PDE.
DEFINITIONS, (i) The operator L*, the formal adjoint of L, is
n	n	n
L*v := ~ 22	+ (c ~
ij=l	г=1	г=1
provided bl E C'1(t7) (? = 1,... ,n).
(ii)	The adjoint bilinear form
В* : H x H R
is defined by
B*[v,u] = B[u, v]
for all u,v E
(iii)	We say that v E Hq(U) is a weak solution of the adjoint problem
( L*v = f in U
( v = 0 on dU,
provided
B*[v, u] = (f,u)
for alluE H^U).
6.2. EXISTENCE OF WEAK SOLUTIONS
303
THEOREM 4 (Second Existence Theorem for weak solutions).
(i)	Precisely one of the following statements holds:
either
{for each f G L2(U) there exists a unique
weak solution и of the boundary-value problem
( Lu = f inU
(10)	( и = 0 on dU
or else

' there exists a weak solution и 0 of
the homogeneous problem
Lu = 0	in U
и = 0	on dU.
(ii)	Furthermore, should assertion (/3) hold, the dimension of the sub-
space N C Hq(U) of weak solutions of (11) is finite and equals the
dimension of the subspace N* C Hq(LT) of weak solutions of
(12)
L*v = 0
v = 0
in U
on dU.
(iii)	Finally, the boundary-value problem (10) has a weak solution if and
only if
(f, v) = 0 for all v G N*.
The dichotomy (a), (/3) is the Fredholm alternative.
Proof. 1. Choose p — 7 as in Theorem 3 and define the bilinear form
v] := v] + 7(u, v),
corresponding to the operator L-u := Lu + 'yu. Then for each g G L2(U')
there exists a unique function и G Hq(U) solving
(13)	B-f[u, v] = (<7, v) for all v G Hq(U}.
Let us write
(14)	и = L~xg
whenever (13) holds.
2. Observe next и G HffiU) is a weak solution of (10) if and only if
(15)	By[u, v] = (7U + f, v) for all v G Hq(U);
304
6. SECOND-ORDER ELLIPTIC EQUATIONS
that is, if and only if
(16)	и = L“1(7u+ /).
We rewrite this equality to read
(17)	u-Ku = h,
for
(18)	Ku := 7£-1u
and
(19)	h := L-'f.
3. We now claim К : L2(U) —> L2(U) is a bounded, linear, compact
operator. Indeed, from our choice of 7 and the energy estimates from §6.2.2
we note that if (13) holds, then
= ($>“) < Hsr|lb2(C7)ll^l|b2(tz) < Ilsr||b2(t/)
so that (18) implies
(1/) C||5IIl2(cq (5 Ь2([У))
for some appropriate constant C. But since Hq(U) CC L2(U) according
to the Rellich-Kondrachov compactness theorem, we deduce that К is a
compact operator.
4. We may consequently apply the Fredholm alternative from §D.5:
either
(20)
(a)
' for each h G L2(U} the equation
< и — Ku — h
. has a unique solution и G L2(U)
or else
(21)	(/?)
the equation
и — Ku = 0
has nonzero solutions in L2(U).
Should assertion (a) hold, then according to (15)-(19) there exists a
unique weak solution of problem (10). On the other hand, should assertion
6.2. EXISTENCE OF WEAK SOLUTIONS
305
(/?) be valid, then necessarily 7 / 0 and we recall further from §D.5 that
the dimension of the space N of the solutions of (21) is finite and equals the
dimension of the space N* of solutions of
(22)	v — K*v = 0.
We readily check however that (21) holds if and only if и is a weak solution
of (11); and (22) holds if and only if v is a weak solution of (12).
5. Finally, we recall (20) has a solution if and only if
(23)	(h,v) = 0
for all v solving (22). But from (18), (19) and (22) we compute
(h,v) = —(Kf,v) = —(f,K*v) = -(/,v) .
7	7	7
Consequently the boundary-value problem (10) has a solution if and only if
(/, v) — 0 for all weak solutions v of (12).	□
THEOREM 5 (Third Existence Theorem for weak solutions).
(i) There exists an at most countable set S C IR. such that the boundary-
value problem
..	( Lu = Xu + f in U
' '	( и = 0 on dU
has a unique weak solution for each f G L2(U) if and only if A S.
(ii) If S is infinite, then S = {Afc}^, the values of a nondecreasing
sequence with
Afc +00.
DEFINITION. We call S the (real) spectrum of the operator L.
Note in particular the boundary-value problem
J Lu = Xu in U
[ и = 0 on dU
has a nontrivial solution w 0 if and only if A G S, in which case A is called
an eigenvalue of L, w a corresponding eigenfunction. The partial differential
equation Lu = Xu for L = —A is sometimes called Helmholtz’s equation.
□
306
6. SECOND-ORDER ELLIPTIC EQUATIONS
Proof. 1. Let 7 be the constant from Theorem 2 and assume
(25)	A > -7.
Assume also with no loss of generality that 7 > 0.
2. According to the Fredholm alternative, the boundary-value problem
(24) has a unique weak solution for each f G Т2([У) if and only if и = 0 is
the only weak solution of the homogeneous problem
( Lu = Xu in U
[ и = 0 on dU.
This is in turn true if and only if и = 0 is the only weak solution of
( Lu + 7U = (7 + A)u in U
' '	[ и = 0	on dU.
Now (26) holds exactly when
(27)	и = L~\^ + A)u = Ku,
7
where, as in the proof of Theorem 4, we have set Ku = 7-L“1u. Recall also
from that proof that К : L2(C) —> L2(C) is a bounded, linear, compact
operator.
Now if и = 0 is the only solution of (27), we see
у
(28)	----r is not an eigenvalue of K.
7 + A
Consequently we see the PDE (24) has a unique weak solution for each
f E L2([7) if and only if (28) holds.
2. According to Theorem 6 in §D.5 the collection of all eigenvalues of К
comprises either a finite set or else the values of a sequence converging to
zero. In the second case we see, according to (25) and (27), that the PDE
(24) has a unique weak solution for all f E L2(U), except for a sequence
A*, —► +00.	□
Finally, we explicitly note:
THEOREM 6 (Boundedness of the inverse). If A S, there exists a
constant C such that
(29)	IMIl2(c/) < C'II/IIl2^),
whenever f E L2(U) and и E Hq(U) is the unique weak solution of
Lu = Xu + f in U
и — 0	on dU.
The constant C depends only on X, U and the coefficients of L.
This constant will blow up if A approaches an eigenvalue.
6.2. EXISTENCE OF WEAK SOLUTIONS
307
Proof. If not, there would exist sequences {fk}kLi C L2(U) and	C
Hq (t7) such that
( Luk = Xuk + fk in U
[ Uk = 0	on dU
in the weak sense, but
ll^fc IIl2((7) > k\\fk ||l2((7)

As we may with no loss suppose ||иА;||ь2(Я) = 1, we see fk —* 0 in L2(U).
According to the usual energy estimates the sequence	is bounded
in Hq(U). Thus there exists a subsequence }J2=i C	such that
(30)
( Ukj	и
I Ukj	U
weakly in Hq(U),
in L2(U).
(See §D,4 for weak convergence.)
Then и is a weak solution of
( Lu = Xu
[ и = 0
in U
on dU.
Since A S, и = 0. However (30) implies as well that ||и||£2(Я) = 1, a
□
contradiction.
Complex solutions. The foregoing theory extends easily to include com-
plex-valued solutions. Given complex-valued u,v G H’1(17), write
(u, v)£2([/} := / uvdx,
(и,и)Я1([/) := / Du • Dv + uv dx,
and set
p n	n
B[u, v] := / azjuXivXj +	+ cuvdx,
Ju i,j=i	i=i
where “ denotes complex conjugate. We check
|B[u,v]| < a||u||Hi(a)||v||Hi([;),
/3IIuIIh01(P) Re u] + 'vl|w||^2(C7) (u,v G H^(Uf)
for appropriate constants a,/3 > 0, 7 > 0. Complex variants of the Lax-
Milgram Theorem and Fredholm alternative lead to analogues of Theo-
rems 3-6 above.	□
308
6. SECOND-ORDER ELLIPTIC EQUATIONS
6.3. REGULARITY
We now address the question as to whether a weak solution и of the PDE
(1)	Lu = f in U
is in fact smooth: this is the regularity problem for weak solutions.
Motivation: formal derivation of estimates. To see that there is some
hope that a weak solution may be better than a typical function in Hq(U),
let us consider the model problem
(2)	-Au = f in Rn.
We assume for heuristic purposes that и is smooth and vanishes sufficiently
rapidly as |rr| —► oo to justify the following calculations. We then compute
Thus we see the T2-norm of the second derivatives of и can be estimated
by (and in fact equals) the T2-norm of f. Similarly, we can differentiate the
PDE (2), to find
-ДЙ = f,
for й := uXk and f := fXk (k = 1,... , n). Applying the same method, we
discover that the T2-norm of the third derivatives of и can be estimated by
the first derivatives of f. Continuing, we see the T2-norm of the (m + 2)"d
derivatives of и can be controlled by the L2-norm of the mth derivatives of
f, for m = 0,1,... .	□
These computations suggest that for Poisson’s equation (2), we can ex-
pect a weak solution и G Hq to belong to Hm+2 whenever the inhomoge-
neous term f belongs to Hm (m — 1,...). Informally we say that и has “two
more derivatives in L2 than f has”. This will be particularly interesting for
m = oo, in which case и will belong to Hm for all m = 1,..., and thus will
be in C°°.
Observe, however, the calculations above do not really constitute a proof.
We assumed и was smooth, or at least say C3, in order to carry out the
6.3. REGULARITY
309
calculation (3); whereas if we start with merely a weak solution in Hq we
cannot immediately justify these computations. We will instead have to rely
upon an analysis of certain difference quotients.
The following calculations are often technically difficult, but eventually
yield extremely powerful and useful assertions concerning the smoothness of
weak solutions. As always, the heart of each computation is the invocation
of ellipticity: the point is to derive analytic estimates from the structural,
algebraic assumption of ellipticity.
6.3.1. Interior regularity.
We as always assume that U C Rn is a bounded, open set. Suppose also
и G Hq(U) is a weak solution of the PDE (1), where L has the divergence
form
n	n
(4)	Lu = -	(alJ(X)uxt)Xj + '^b\x')uXl + c(x)u.
i,j=l	i=l
We continue to require the uniform ellipticity condition from §6.1.1, and
will as necessary make various additional assumptions about the smoothness
of the coefficients а4, b\c.
THEOREM 1 (Interior ^-regularity). Assume
(5)	aij EC\U), Ь\сеЬ°°(и) (i,j = l,...,n)
and
(в)	f£L\U).
Suppose furthermore that и G is a weak solution of the elliptic PDE
Lu = f in U.
Then
(7)	и G Я12ос(СП ;
and for each open V CC U we have the estimate
(8)	||u||н2(у) < C (||/||i2([z) + IMIl2(c/)) ,
the constant C depending only on V, U, and the coefficients of L.
310
6. SECOND-ORDER ELLIPTIC EQUATIONS
Remarks, (i) Note carefully that we do not require и E Hq(U')', that is,
we are not necessarily assuming the boundary condition и = 0 on dU in the
trace sense.
(ii) Observe additionally that since и E 71^(17), we have
Lu = f a.e. in U.
Thus и actually solves the PDE, at least for a.e. point within U. To see
this, note that for each v E Cff>(U), we have
B[u,v] = (/,v).
Since и E	we can integrate by parts:
B[u, v] = (Lu, v).
Thus (Lu — f,v) = 0 for all v E С^°(и), and so Lu = f a.e.
□
Proof. 1. Fix any open set V CC U, and choose an open set W such that
V CC W CC U. Then select a smooth function ( satisfying
( C = 1 on V, C = 0 on Rn - W,
t 0<C<l.
We call C a cutoff function: its purpose in the subsequent calculations will
be to restrict all expressions to the subset W, which is a positive distance
away from dU. This is necessary as we have no information concerning the
behavior of и near dU. (As an interesting technical point, notice carefully in
the following calculations why we put “£2”, and not just in (11) below.)
2.	Now since и is a weak solution of (1), we have B[u, v] = (f, v) for all
и E Hq(U). Consequently
(9)
I fv dx,
и
where
n
(10)	f := f - blu^i ~ cu-
i=l
3.	Now let |h| > 0 be small, choose к E {1,. .. , n}, and then substitute
(И)
v := -D-h(C2Dhku)
6.3. REGULARITY
311
into (9), where as in §5.8.2 D%u denotes the difference quotient
Di:u(x) — ------------------- (h G R, h 0).
h
We write the resulting expression as
(12)	A = B,
for
(13)	A := V [ aljuXivx dx
i,j=lJu
and
(14)	В := [ fvdx.
Ju
4.	Estimate of A. We have
(17)	Dhk(Vw) = vhDhkw + wDhkv,
for vh(x') := v(x + he).).
312
6. SECOND-ORDER ELLIPTIC EQUATIONS
Returning now to (15), we find
a^hD^uXtD^uXjedx
ij=1 JU
(18)	+ E f [alj'hDhkuXiDhku2^X] + (phka^ их^ких^2
i,j=iJu
+ (D^u^D^^dx
=: Af + A?.
The uniform ellipticity condition implies
(19)	Ai > 6 [ £2\DkDu\2dx.
Ju
Furthermore we see from (5) that
|A2| <C [ C\DhkDu\ \Dhku\ + (\DhkDu\ |£»u| + (Ж \Du\dx,
Ju
for some appropriate constant C. But then Cauchy’s inequality with e (§B.2)
yields the bound
|Л21 < e [ (2\DkDu\2dx + - [ \D%u\2 + \Du\2dx.
Ju	e Jw
We choose e = | and further recall from Theorem 3,(i) in §5.8.2 the estimate
[ \D^u\2 dx<C [ |7?и|2 dx,
Jw	Ju
thereby obtaining the inequality
|A2| <	[ C,2\DkDu\2dx + C [ \Du\2dx.
2 Ju	Ju
This estimate, (19) and (18) imply finally
(20)	A>^ [ C2\D%Du\2dx-C [ \Du\2dx.
2 Ju	Ju
5.	Estimate of B. Recalling now (10), (11), and (14), we estimate
(21)	\B\<C [ (|/| + |Z>u| + |u|)Hdx.
Ju
6.3. REGULARITY
313
Now Theorem 3,(i) in §5.8.2 implies
[ \v\2dx <C [ \D(£2Dku)\2dx
Ju Ju
<C [ \Dku\2 + C,2\DkDu\2dx
Jw
<C [ \Du\2 + (2\DkDu\2dx.
Ju
Thus (21) and Cauchy’s inequality with e imply
|B| < e [ (,2\DkDu^dx + — [ f2 + u2dx + — [ \Du\2dx.
Ju	6 Ju	e Ju
Select e = |, to obtain
(22)	|B| < °- [ (2\DkDu\2dx + C [ f2 + u2 + \Du\2dx.
4 Ju	Ju
6.	We finally combine (12), (20) and (22), to discover
[ \DkDu\2dx < [ ^2\DkDu\2dx < C [ f2 + u2 + \Du\2dx
Jv	Ju	Ju
for к = 1,..., n and all sufficiently small |h|	0.
In view of Theorem 3,(ii) in §5.8.2, we deduce Du E H\OC(U), and thus
и E Hioc(U), with the estimate
(23)	ll‘ull//2(V) < C (||/||l2([7) + ||и||Я1(с/)) •
7.	We now refine estimate (23) by noting that if V CC W CC U, then
the same argument shows
(24)	1Ы1я2(Г) < С (II/IIl2(w) + П^Няцио) >
for an appropriate constant C depending on V, W, etc. Choose a new cutoff
function £ satisfying
Г C = 1 on W, spt С C U,
l о < c < 1.
Now set v = £2u in identity (9) and perform elementary calculations, to
discover
[ (2\Du\2dx < C [ f2 + u2dx.
Ju	Ju
314
6. SECOND-ORDER ELLIPTIC EQUATIONS
Thus
ll^llni(w) < С (Н/Нь2(а) + IMIl2(c/)) •
This inequality and (24) yield (8).
□
Our intention next is to iterate the argument above, thereby deducing
our weak solution lies in various higher Sobolev spaces (provided the coef-
ficients are smooth enough and the righthand side lies in sufficiently good
spaces).
THEOREM 2 (Higher interior regularity). Let m be a nonnegative inte-
ger, and assume
(25)	aij,b\ce Cm+1(JT) (i,j = l,...,n)
and
(26)	f G Hm(Uf
Suppose и G H’1([7) is a weak solution of the elliptic PDE
Lu — f in U.
Then
(27)	и G H-+2([/);
and for each V CC U we have the estimate
(28)	||и||я™+2(У) < С'(||/||я"’(С7) + IMIl2(C7)),
the constant C depending only on m,U,V and the coefficients of L.
Proof. 1. We will establish (27), (28) by induction on m, the case m = 0
being Theorem 1 above.
2. Assume now assertions (27) and (28) are valid for some nonnegative
integer m and all open sets U, coefficients aN, bl,c, etc., as above. Suppose
then
(29)	а^,Ь\сЕ Cm+2(U),
(30)
f G Ят+1(17),
6.3. REGULARITY
315
and и G Я1(1/) is a weak solution of Lu = f in U. By the induction
hypotheses, we have
(31)	и g Я-+2(С/),
with the estimate
(32)	||'“Ня’"+2(1У) С'(11/11я"1((7) + llulk2((7)),
for each W CC U and an appropriate constant C, depending only on W,
the coefficients of L, etc. Fix V CC W CC U.
3.	Now let a be any multiindex with
(33)	|a| = m + 1,
and choose any test function v G	Insert
v := (-l)lQ|£>“v
into the identity B[u, v] = (/, v)L2^, and perform some integrations by
parts, eventually to discover
(34)	B[u,v] = (f,v)
for
(35)	й := Dau G H\W)
and
f :=	- E (fl) [- Y
{3<a У ' L M=1
(36)	№
+ ^2 Ва-^В0иХг + Da~0cD0u .
i=i
Since the identity (34) holds for each v G	we see that й is a weak
solution of
Lu = f in W.
In view of (29)-(32) and (36), we have f G L2(W"), with
(37)	I|7||l2(IV) < С'(П/11я’"+1(£/) + IIuIIl2((7))-
316
6. SECOND-ORDER ELLIPTIC EQUATIONS
4.	In light of Theorem 1 then, we see й G H2(V), with the estimate
IN|h2(V) < C(||7||L2(HZ) + ||й||£2(Н/))
< С,(П/11ят+1 (a) + IMIl2(c/))-
This inequality holds for each multiindex a with |a| = m +1, and й = D°u
as above. Consequently и G Hm+3(V), and
||u||/pn+3(V) < Cdl/H^+qc/) + IIuIIl2(c/))-
□
We can now repeatedly apply Theorem 2 for m = 0,1,2,... to deduce
the infinite differentiability of u.
THEOREM 3 (Infinite differentiability in the interior). Assume
aij ,Ь\сЕ(Т°(и) (i,j = l,...,n)
and
f e
Suppose и E H2(U) is a weak solution of the elliptic PDE
Lu = f in U.
Then
и G C°°(t7).
We are again making no assumptions here about the behavior of и on
dU. Therefore, in particular, we are asserting that any possible singularities
of и on the boundary do not “propagate” into the interior.
Proof. According to Theorem 2, we have и G HffjU') for each integer
m = 1,2,.... Hence Theorem 6 in §5.6.3 implies и G Ck(U) for each
k = 1,2,... .	□
6.3.2. Boundary regularity.
Now we extend the estimates from §6.3.1 to study the smoothness of
weak solutions up to the boundary.
6.3. REGULARITY
317
( Lu = f in U
( и = 0 on dU.
dU is C2.
и G H2(U),
THEOREM 4 (Boundary //^-regularity). Assume
(38)	aijG C\U), b\cE L™(fJ) (ij = 1,..., n)
and
(39)	f G L2(U).
Suppose that и G //q(C/) is a weak solution of the elliptic boundary-value
problem
(40)
Assume finally
(41)
Then
and we have the estimate
(42)	1Ы1я2(с/) < C(||/||l2(c/) + ||u||l2(c/)),
the constant C depending only on U and the coefficients of L.
Remarks, (i) If и G Hq(U) is the unique weak solution of (40), estimate
(42) simplifies to read
||tx||jf2([7) < C'||/||L2([/).
This follows from Theorem 6 in §6.2.
(ii)	Observe also that in contrast to Theorem 1 in §6.3.1, we are now
assuming и = 0 along dU (in the trace sense).	□
Proof. 1. We first investigate the special case that U is a half-ball:
(43)	/7 = B°(0,1) П R”.
Set V •.= B°(0, |) П R”. Then select a smooth cutoff function Q satisfying
( C = 1 on B(0, |), C = 0 on R” — B(0,1),
t 0 < c < i.
So ( e 1 on V and £ vanishes near the curved part of dU.
318
6. SECOND-ORDER ELLIPTIC EQUATIONS
2.	Since и is a weak solution of (3), we have Bfit, v] = (/, u) for all
v G Hq(U); consequently
(44)	У2 / aljuXivXj dx = / fvdx,
i,j=iJu	Ju
for
(45)	f := f -	“ cu-
i—1
3.	Now let h > 0 be small, choose k G {1,.. •, n — 1}, and write
v := -P-\<2P^).
Let us note carefully
v(x) = -^Dk h(C2(x)[u(x + hek) - w(x)])
= --^2 (C2(x “ hei)[u(x) - u(x - hek)]
- С2(ж)[и(ж + hek) ~ u(z)])
if x G U. Now since и = 0 along {rcn = 0} in the trace sense and £ = 0 near
the curved portion of dU, we see v e Bq(U).
We may therefore substitute v into the identity (44), and write the re-
sulting expression as
(46)	A = B,
for
n f
(47)	A := 52 / aliuXivXj dx
i,j=l JU
and
(48)	В := I fv dx.
Ju
4.	We can now estimate the terms A and В in almost exactly the same
way that we estimated their counterparts in the proof of Theorem 1. After
some calculations we find
(49)	A>®- f ()2\DkDu\2dx — C [ \Du\2dx
2 Ju	Ju
6.3. REGULARITY
319
and
(50)	|5|<j [ C2\D^Du\2dx + C [ f2 + u2 + \Du\2dx,
4 Ju	Ju
for appropriate constants C. We then combine (46), (49), and (50) to dis-
cover
[ \D%Du\2dx <C [ f2 + u2 + \Du\2dx
Jv	Ju
for к = 1,..., n — 1. Thus recalling the Remark after Theorem 3 in §5.8.2,
we deduce
uXkEH\V) (fc = l,...,n-l),
with the estimate
n
(51)	52	C'dl/IL^c/) + I Ml#1 (c/))-
fc,Z=l
k+l<2n
5.	We must now augment (51) with an estimate of the T2-norm of uXnXn
over V. For this we recall from the Remarks after Theorem 1 that Lu = f
a.e. in U. Remembering the definition of L, we can rewrite this equality into
nondivergence form, as
(52)	- aljuXiXj + 52 bluXi +cu = f,
i,j=l	2=1
for Ьг := Ьг — alj (i = 1,..., n). So we discover
n	n
(53)	annuXnXn = - ^2	+ 52 ^Uxi Ycu- f.
i,j=l	2=1
i+j<2n
Now according to the uniform ellipticity condition, XL"J=i	>
0|£|2 for all x G U, £ E Rn. We set £ = en = (0,..., 0,1), to conclude
(54)	ann(x) > 6 > 0
for all x E U. But then (38), (53) and (54) imply
(П	4
52	+ Pui + iui + 1/1)
M=1	'
i+j<2n
320
6. SECOND- ORDER ELLIPTIC EQ UATIONS
in U. Utilizing this estimate in inequality (51), we conclude и G H2(V), and
(56)	|1и11я2(у) < C'(||/||L2([/) + ||и||я1(1/))
for some appropriate constant C.
6. We now drop the assumption that U is a half-ball and so has the
special form (43). In the general case we choose any point x° G dU and
note that since dU is C2, we may assume—upon relabeling the coordinate
axes if needs be—that
U П B(x°,r) = {re	| xn > y(xi,... ,xn-i)}
for some r > 0 and some C2 function 7 : Rn-1 —> R. As usual, we change
variables utilizing §C.l and write
(57)
у = Ф(ж), X = Ф(у).
7. Choose s > 0 so small that the half-ball U7 := B°(0, s) П {yn > 0}
lies in Ф(и П U(x°,r)). Set V' := B°(0, s/2) П {yn > 0}. Finally define
(58)	u'(y) := и(Ф(у)) (у E U').
It is straightforward to check
(59)	u'GtfW)
and
(60)	и — 0 on dU' П {yn = 0}
in the trace sense.
8.	We now claim u' is a weak solution of the PDE
(61)	L'u = f in U',
for
(62)	Ш := /(Ф(у))
and
(63)	L'u' := - £ (AU, + L + A
fc,Z=l	fc=l
6.3. REGULARITY
321
where
n
(64)	akl(y) := £ ^(ММММ (fc,/ = l,...,n),
r,s=l
(65)	b'k(y) := £ Ь7Ф(у))ФХ(Ф(у)) (fc = 1,..., n),
r=l
and
(66)	cz(y) := с(Ф(у))
for у G U1, k,l = 1,..., n.
If v' G Hq(U') and B'{ , ] denotes the bilinear form associated with the
operator L', we have
(67)	B'[u',v'] = I aklu'ykv'yi +^b'ku'ykv' + c'u'v' dy.
Ju> k,l=i	k=i
Now define
v(z) := ?/(Ф(х)).
Then from (67) we calculate
Now according to (64), we find for each i, j = 1,..., n that
n	n n
к,1=1	r,s=lk,l=l
since ОФ = (ОФ) 4 Similarly for i = 1,..., n, we have
n	n n
fc = l	fc=l r=l
322
6. SECOND-ORDER ELLIPTIC EQUATIONS
Substituting these calculations into (68) and changing variables yields, since
| det .РФ | = 1,
p n	n
B'[u',v'] = / E a^Ux-Vx - + E blux v + cuv dx
i=1
= B[u, v] = (/,u)L2([/) =
This establishes (61).
9.	We now check that the operator L' is uniformly elliptic in U'. Indeed
if у G U' and ( GRn, we note that
E «“ш = E E
.	4	k,l=l	r,s=l fc,Z=l
(69)
= £	> 0|»j|2,
r,s=l
where r? = £РФ; that is, 7]r = £X=1 $xr£k (r = 1,... ,n). But then, since
РФРФ = I, we have £ = т]ЕФ; and so |£| < C|7?| for some constant C.
This inequality and (69) imply
(70)	E a'kl(y^ > 0'l£|2
fc,Z=l
for some ff > 0 and all у G U', £ G Rn.
Observe also from (64) that the coefficients a kl are C1, since Ф and Ф
are C2.
10.	In view of (61) and (70), we may apply the results from steps 1-5
in the proof above to ascertain that u' G H2(V), with the bound
IKIIh2(V') < C'(ll//|k2(u') + IMIl2^'))-
Consequently
(71)	ll'ull#2(v) < C’(||/||L2(c/) + Hull/y^))
for V := Ф(У').
Since dU is compact, we can as usual cover dU with finitely many sets
Ц,..., Vn as above. We sum the resulting estimates, along with the interior
estimate, to find и G H2(U), with the inequality (42).	□
Now we derive higher regularity for our weak solutions, all the way up
to dU.
6.3. REGULARITY
323
THEOREM 5 (Higher boundary regularity). Let m be a nonnegative in-
teger, and assume
(72)	а^,Ь\се Cm+1(U) (ij = l,...,n)
and
(73)
Suppose that и G Hq(U) is a weak solution of the boundary-value problem
<«)
v '	( и = 0 on dU.
Assume finally
(75)	dUisCm+2.
Then
(76)	и G Hm+2(U),
and we have the estimate
(77)	llullнт+2(и) < C (H/ll#™^) + HullL2(t/)) ,
the constant C depending only on m,U and the coefficients of L.
Remark. If и is the unique solution of (74), then estimate (77) simplifies
to read
||u||//m+2(t/) < С||/||Я’П((7)-
□
Proof. 1. We first investigate the special case
(78)	[/:=Во(0,«)ПГ;
for some s > 0. Fix 0 < t < s and set V := B°(0,t) П R”.
2.	We intend to prove by induction on m that whenever и = 0 along
{xn = 0} in the trace sense, (72) and (73) imply
(79)	и G Hm+2(V),
with the estimate
(80)	IMIHm+2(V) < C (||/||#m(C7) + IMIz^p)) ,
324
6. SECOND-ORDER ELLIPTIC EQUATIONS
for a constant C depending only on U, V and the coefficients of L. The case
m = 0 follows as in the proof of Theorem 4 above.
Suppose then
а”,Ъ*,с(Е Cm+2(U),
(82)
and и is a weak solution of Lu = f in U, which vanishes in the trace sense
along {rcn = 0}. Fix any 0 < t < r < s, and write W := _B°(0,r) П R”. By
the induction assumption we have
(83)	и G Hm+2(W),
with the estimate
(84)	II'UIIh™+2(W) < С (||/||ят(С7) + IMI/y^)) .
Furthermore according to the interior regularity Theorem 2, и 6 H^3(U).
3.	Next, let a be any multiindex with
(85)	|a| = m + 1
and
(86)	an = 0.
Then
(87)	й := Dau
belongs to 771(С7), and vanishes along the plane {rcn = 0} in the trace sense.
Furthermore, as in the proof of Theorem 2, й is a weak solution of Lu = f
in U, for
0/a
+ Da~/3biD/3uXi + D^cD^u
i=i
6.3. REGULARITY
325
In view of (72), (73), (82) and (84), we see f G L2(W), with
(88)	ll/lll2(w) < С (П/Пн^+чг/) + IIuIIl2(c/)) •
Consequently the proof of Theorem 4 shows й G H2(V), with the estimate
ll«llя2(У) < C (ll/llL2(w) + II^IIl2(iv))
< С (||/||н™+1(с/) + IIй IIl2(c/)) •
In light of (85)~(88), we thus deduce
(89)	II-c>/3'uIIl2(v) < c (II/IIh^+t^) + IIuIIl2(c/))
for any multiindex /3 with |/3| = m + 3 and
(90)	/Зп = 0,1, or 2.
4.	We must extend estimate (89) to remove the restriction (90). For
this, let us suppose by induction
(91)	ll^^lll2(v) < C (II/IIh^+Vc/) + IIuIIl2(c/))
for any multiindex /3 with |/3| = m + 3 and
(92)	/3n = 0,l,...,j,
for some j G {2,..., m + 2}. Assume then |/3| = m + 3,
(93)	(3n=j + l.
Let us write /3 = 7+6, for 6 = (0,..., 2) and |y| = m+1. Since и G H^3(U)
and Lu = f in U, we have Z>7Lu = D~'f a.e. in U. Now
WLu = annD^u+ { sum of terms involving at most j
derivatives of и with respect to xn, and
at most m + 3 derivatives in all }.
Since ann > в > 0, we thus find by utilizing (91), (92) that
(94)	IID^uW^^y) < С (||/||нт+1(р) + ||u|Il2(J7))
provided |/3| = m + 3 and /3n = j + 1. By induction on j then, we have
IMIЯт+3((7) < C (||/||#m+i(p) + Hull/y^)) .
This estimate in turn completes the induction on m, begun in step 2.
5.	We have now shown that (72) and (73) imply (79) and (80), provided
U has the form (78). The general case follows once we straighten out the
boundary, using the ideas explained in the proof of Theorem 4.	□
We finally iterate the foregoing estimates to obtain
326
6. SECOND-ORDER ELLIPTIC EQUATIONS
THEOREM 6 (Infinite differentiability up to the boundary). Assume
аг\Ь\сеС°°(й) (i,j = l,...,n)
and
f € C°°(t7).
Suppose и G Hq(U) is a weak solution of the boundary-value problem
( Lu = f in U
( и = 0 on dU.
Assume also that dU is C00. Then
и G C°°(t7).
Proof. According to Theorem 5 we have и G Hm(U) for each integer m =
1,2,.... Thus Theorem 6 in §5.6.3 implies и G Ck(U) for each к = 1,2,... .
□
The computations in this section have basically been repeated appli-
cations of “energy” methods to higher and higher partial derivatives. The
basic tool of integration by parts has eventually taken us from weak solutions
(belonging merely to Hq(U)') to smooth, classical solutions.
6.4. MAXIMUM PRINCIPLES
This section develops the maximum principle for second-order elliptic partial
differential equations.
Maximum principle methods are based upon the observation that if a
C2 function и attains its maximum over an open set U at a point xq e U,
then
(1)	Du(xo) = 0, D2(xq) < 0,
the latter inequality meaning that the symmetric matrix D2u = ((u^))
is nonpositive definite at x0. Deductions based upon (1) are consequently
pointwise in character, and are thus utterly different from the integral-based
energy methods set forth in §§6.1-6.3.
In particular we will need to require that our solutions и are at least C2,
so that it makes sense to consider the pointwise values of Du, D2u. (In view
of the regularity theory from §6.3 we know however that a weak solution is
6.4. MAXIMUM PRINCIPLES
327
this smooth, at least provided the coefficients, etc. are sufficiently regular.)
As we will shortly learn, it is also most appropriate now to consider elliptic
operators L having the nondivergence form
71	П
(2)	Lu = - У7 ^iuxiXj + bluXi + cu,
i,j=l	i=l
where the coefficients аУ ,b\ c are continuous and—as always— the uniform
ellipticity condition (4) in §6.1 holds. We continue also to assume, without
loss of generality, the symmetry condition aiJ = aJl (i,j = 1,... ,n).
6.4.1. Weak maximum principle.
First, we identify circumstances under which a function must attain its
maximum (or minimum) on the boundary. We always assume U C IR” is
open, bounded.
THEOREM 1 (Weak maximum principle). Assume и € C2(t7) n C(IT)
and
c = 0 in U.
(i) V
(3)	Lu < 0 in U,
then
maxu = maxu.
и du
(П) If
(4)	Lu > 0 in U,
then
min и = min u.
U dU
Remark. A function satisfying (3) is called a subsolution. We are thus
asserting a subsolution attains its maximum on dU. Similarly, if (4) holds,
и is a supersolution and attains its minimum on dU.	□
Proof. 1. Let us first suppose we have the strict inequality
(5)	Lu <0 in U,
and yet there exists a point xq E U with
(6)	ufyo) = maxu.
и
328
6. SECOND-ORDER ELLIPTIC EQUATIONS
Now at this maximum point xo, we have
(7)	Du(xq) = 0,
and
(8)	Z>2iz(xo) < 0.
2.	Since the matrix A = ((av(x0))) is symmetric and positive definite,
there exists an orthogonal matrix О = ((ojj)) so that
(9)	OAOT = diag(di,...,dn), OOT = I,
with dk > 0 (k = 1,... ,n). Write у = xq + O(x — xq). Then x — xo =
OT\y — xo), and so
n	n
uxt ~ У7 UVk°ik 1 uXiXj = 5 ] иУкУ1°гк°.Ц (b J' = 1> • •  j n)-
fc=l
Hence at the point xq,
n	n n
57 a^uXiXj — 57 57 a^uykyi°ik°jl
i,j=l	k,l=li,j=l
(10)
= 57 dkuykyk by (9)
fc=i
<0,
since dk > 0 and иУкУк(хо) <0 (fc = 1,... ,n), according to (8).
3. Thus at xo
71	П
Lu = - 57 a^UxiXj + У bluXi > 0,
i,j=i	i=i
in light of (7) and (10). So (5) and (6) are incompatible, and we have a
contradiction.
4. In the general case that (3) holds, write
ize(x) := iz(x) + eeXxi (x &U),
where Л > 0 will be selected below and c > 0. Recall (as in the proof of
Theorem 4 in §6.3.2) that the uniform ellipticity condition implies агг(х) > в
(i = 1,..., n, x e U). Therefore
Lue = Lu + eL(eXxi)
< eeXxi [-A2an + Ab1]
<бел-1[_А20+||Ь||ЛооА]
<0 in U,
6.4. MAXIMUM PRINCIPLES
329
provided we choose Л > 0 sufficiently large. Then according to steps 1 and
2 above max^u6 = max^f/t/. Let c —> 0 to find max^ и — rn.ax.du и. This
proves (i).
5. Since — и is a subsolution whenever и is a supersolution, assertion (ii)
follows.
We next modify the maximum principle to allow for a nonnegative
zeroth-order coefficient c. Remember from §A.3 that u+ = max(u, 0), u~ =
— min(u,0).
THEOREM 2 (Weak maximum principle for c > 0). Assume и e C2(J7)D
C(U) and
in U.
(i) If
Lu < 0
in U,
then
(И)
maxu
и
max
du
(ii) Likewise, if
then
Lu > 0
in U,
(12)
min и > — max и
U	dU
Remark. So in particular, if Lu = 0 in U, then
(13)
max |u| = max lul.
£7 dU ' '
Proof. 1. Let и be a subsolution and set V := {rc € U j u(x) > 0}. Then
Ku := Lu — cu
—cu <0 in V .
The operator К has no zeroth-order term and consequently Theorem 1 im-
plies maxp и — max^yu = max^u+. This gives (11) in the case that
V 0. Otherwise и < 0 everywhere in U, and (11) likewise follows.
2. Assertion (ii) follows from (i) applied to — u, once we observe that
(—u)+ = и
330
6. SECOND-ORDER ELLIPTIC EQUATIONS
6.4.2. Strong maximum principle.
We next substantially strengthen the foregoing assertions, by demon-
strating that a subsolution и cannot attain its maximum at an interior point
of a connected region at all, unless и is constant. This statement is the strong
maximum principle, which depends on the following subtle analysis of the
outer normal derivative at a boundary maximum point.
LEMMA (Hopf’s Lemma). Assume и G C2(t/) П and
c = 0 in U.
Suppose further
Lu <0 in U,
and there exists a point x° 6 dU such that
(14)	u(x°) > u(x) for all xEU.
Assume finally that U satisfies the interior ball condition atx°; that is, there
exists an open ball В CU with x° G dB.
(i)	Then
where v is the outer unit normal to В at x°.
(H) Zf
c > 0 in U,
the same conclusion holds provided
u(x°) > 0.
Remark. The importance of (i) is the strict inequality: that ^(x°) > 0 is
obvious. Note that the interior ball condition automatically holds if dU is
C2.	□
Proof. 1. Assume c > 0 and u(rc°) > 0. We may as well further assume
В = B°(0, r) for some radius r > 0. Define
v(x) := е-А|ж|2 - e-Ar2 (x G B(0, r))
6.4. MAXIMUM PRINCIPLES
331
for Л > 0 as selected below. Then using the uniform ellipticity condition,
we compute:
n	n
Lv = - ^ a4vXiXj + ^2 b%v^i + cv
ij=l	i=l
= e-^l2 aij (-4X2xiXj + 2A<5°)
m=i
- е-лИ2 £b’2Axj + cte-^l2 - e-Ar2)
i=i
< е-лИ2(_40Д2|x|2 + 2Atr A + 2A|b||x[ + c),
for A = ((av)), b = (b1,..., bn). Consider next the open annular region
R := B°(0, r) - B(0, r/2). We have
(15)	Lv < e~x^2(~e\2r2 + 2Atr A + 2A|b|r + c) < 0
in R, provided A > 0 is fixed large enough.
2.	In view of (14) there exists a constant e > 0 so small that
(16)	u(x°) > iz(x) + ev{x) (ж G ЭВ(0, r/2)).
In addition note
(17)	и(ж°) > u{x) + (x G dB(0, r)),
332
6. SECOND-ORDER ELLIPTIC EQUATIONS
since v = 0 on dB(0,r).
3. From (15) we see
L{u + ev — u(x0)) < — cu(x°) <0 in R,
and from (16), (17) we observe
и + ev — п(ж°) <0 on dR.
In view of the weak maximum principle, Theorem 1, u + ev —u(x0>) < 0 in R.
But u(x°) + eu(x°) — u(rr°) = 0, and so
Consequently
x-(^°) > — б—^(ж°) = — ~Dv(xQ)  xQ = 2Лсге Лг2 > 0,
av	av	r
as required.
Hopf’s Lemma is the primary technical tool in the next proof:
THEOREM 3 (Strong maximum principle). Assume и G C2(U) П C(U)
and
c = 0 in U.
Suppose also U is connected, open and bounded.
(i) If
Lu <0 in U
and и attains its maximum over U at an interior point, then
и is constant within U.
(ii) Similarly, if
Lu>0 inU
and и attains its minimum over U at an interior point, then
и is constant within U.
6.4. MAXIMUM PRINCIPLES
333
Proof. Write M max^ и and C := {rc G U | u(x) = M}. Then if и M,
set
V :={xeU\ u(x) < M}.
Choose a point у G V satisfying dist(j/,C) < dist(y,dU), and let В denote
the largest ball with center у whose interior lies in V. Then there exists
some point x° G C, with ж0 G dB. Clearly V satisfies the interior ball
condition at ж0; whence Hopf’s Lemma, (i), implies ^fy°) > 0. But this is a
contradiction: since и attains its maximum at xQ G U, we have Du(x0~) = 0.
□
If the zeroth-order term c is nonnegative, we have this version of the
strong maximum principle:
THEOREM 4 (Strong maximum principle with c > 0). Assume и G
C2(U) CC(U) and
c > 0 in U.
Suppose also U is connected.
(i) If
Lu < 0 in U
and и attains a nonnegative maximum over U at an interior point,
then
и is constant within U.
(ii) Similarly, if
Lu >0 inU
and U attains a nonpositive minimum over U at an interior point,
then
и is constant within U.
The proof is like that above, except that we use statement (ii) in Hopf’s
Lemma.
6.4.3. Harnack’s inequality.
Harnack’s inequality states the values of a nonnegative solution are com-
parable, at least in any subregion away from the boundary. We assume as
usual that
n	n
Lu= - ^2 aljuXiXj + bluXi + cu.
i,j=l	i=l
334
6. SECOND-ORDER ELLIPTIC EQUATIONS
THEOREM 5 (Harnack’s inequality). Assume и > 0 is a C2 3 solution of
Lu = 0 in U,
and suppose V CC U is connected. Then there exists a constant C such that
(18)	supu < Cinf u.
у	V
The constant C depends only on V and the coefficients of L.
If the coefficients are smooth, the proof is a (much easier) special case
of the calculations to be presented later for the parabolic Harnack inequal-
ity, in §7.1.4. For the general situation of merely bounded and measurable
coefficients, see Gilbarg-Trudinger [G-T],
6.5. EIGENVALUES AND EIGENFUNCTIONS
We consider in this section the boundary-value problem
.	( Lw = Xw in U
( '	( w = 0 on dU,
where U is open, bounded, and recall that Л is an eigenvalue of L provided
there exists a nontrivial solution w of (1). From the theory developed in
§6.2 we recall that the set S of eigenvalues of L is at most countable.
The theorems in §6.5.1 below are analogues for elliptic PDE of the stan-
dard linear algebra assertion that a real symmetric matrix possesses real
eigenvalues and an orthonormal basis of eigenvectors. Similarly, the results
in §6.5.2 are PDE versions of the Perron-Frobenius theorem that a matrix
with positive entries has a real, positive eigenvalue and a corresponding
eigenvector with positive entries (cf. Gantmacher [GA]).
6.5.1. Eigenvalues of symmetric elliptic operators.
For simplicity, we consider now an elliptic operator having the divergence
form
(2)	Lu = - £ («4,), ,
M=1
where аг-} G C°°(U) (г, j = 1,..., n). We suppose the usual uniform elliptic-
ity condition to hold, and as usual suppose
(3)	au = (i,j = 1,..., n).
The operator L is thus formally symmetric, and in particular the associated
bilinear form £?[ , ] satisfies B[u, u] = B[v, u] (u,v G Hq(U)). Assume also
U is connected.
6.5. EIGENVALUES AND EIGENFUNCTIONS
335
THEOREM 1 (Eigenvalues of symmetric elliptic operators).
(i)	Each eigenvalue of L is real.
(ii)	Furthermore, if we repeat each eigenvalue according to its (finite)
multiplicity, we have
S = {Afc}£°=i,
where
0 < Ai < Аг < A3 < •  • ,
and
At —> 00 as к —> oo.
(iii)	Finally, there exists an orthonormal basis	of L2(U), where
Wk € Hq(U) is an eigenfunction corresponding to Xk:
( Lwk = Afcwfc in U
' '	[ Wk — 0 on dU,
for к = 1,2,....
Remark. Owing to the regularity theory in §6.3, Wk € C°°(U) (and Wk €
C°°(i7) if dU is smooth), for к = 1,2,... .	□
Proof. 1. As in §6.2,
S := L-1
is a bounded, linear, compact operator mapping L2(U) into itself.
2. We claim further that S is symmetric. To see this, select f,gE L2{U).
Then Sf = и means и e Hq(U) is the weak solution of
( Lu = f in U
[ и = 0 on dU,
and likewise Sg = v means v 6 H}fiU) solves
( Lv = g mU
[ v = 0 on dU
in the weak sense. Thus
(Sf,g) = (u,g) = B[v,u]
and
(/, Sg) = (f,v) = B[u,v].
336
6. SECOND-ORDER ELLIPTIC EQUATIONS
Since B[u, v] = B[v,u], we see (Sf,g) = (/,8д) for all f,g € L2(U). There-
fore S is symmetric.
3. Notice also
(Sf,f) = (u,f) = B[u,u]>0 (/€£»)•
Consequently the theory of compact, symmetric operators from §D.6 implies
that all the eigenvalues of S are real, positive, and there are corresponding
eigenfunctions which make up an orthonormal basis of L2(U). But observe
as well that for g 0, we have Sw — три if and only if Lw = Aw for A = |.
The theorem follows.	□
We next scrutinize more carefully the first eigenvalue of L.
DEFINITION. We call Ai > 0 the principal eigenvalue of L.
THEOREM 2 (Variational principle for the principal eigenvalue).
(i) We have
(5)	Ai = min{B[u,u] | и e Hq(U), ||u||i,2 = 1}.
(ii) Furthermore, the above minimum is attained for a function w\, pos-
itive within U, which solves
( Lw\ = Aiwi in U
[ wi = 0 on dU.
(iii) Finally, ifuE H^(U') is any weak solution of
( Lu = Ai« in U
[ и = 0 on dU,
then и is a multiple of wi.
Remarks, (i) Assertion (iii) says the principal eigenvalue Ai is simple. In
particular
0 < Ai < Аг < A3 < • • • .
(ii) Expression (5) is Rayleigh’s formula, and is equivalent to the state-
ment
.	. B[u, u]
Ai = min -jj—1,2----•
иен^и) Ilulll2(t7)
□
6.5. EIGENVALUES AND EIGENFUNCTIONS
337
Proof. 1. In view of (4) we see
(6)	B[wfc,wfc] = AfcHwfcll^^j = Afc)
and
(7)	B[wk,wi] = Xk(wk,wi) = 0
for к, I = 1,2,..., к 7^ I.
2.	As	is an orthonormal basis of L2(U), if и E Hq(U") and
IIuIIl2((7) — 1, we can write
OO
(8)	и =
fc=i
for dk = (u, Wfc)£2(;j), the series converging in L2(U\. see §D.2. In addition
OO
(9)	= 11и1112((7)= 1-
t=i
Г	1 00
3.	Furthermore from (6) and (7) we see that < -7^ f is an orthonor-
l	J fc=i
mal subset of endowed with the new inner product B[ , ].
Г 00
We claim further that < f is in fact an orthonormal basis of
I A* J fc=i
Hq(U), with this new inner product. To see this, it suffices to verify that
B[wfc,u] = 0 (A: = 1,2,...)
implies и = 0. But this assertion is clearly true, since the identities
B[wk,u] = Xk(wk,u) =0 (Ar = l,...)
force и = 0, as	is a basis of L2(U}. Consequently
OO
U = Z^^
t=i Afc
for Hk = В и, , the series converging in Hq(U). But then according to
xk J
1 /2
(8), /Lik = dkXf! ; and so the series (8) in fact converges also in H}}(U).
338
6. SECOND-ORDER ELLIPTIC EQUATIONS
4.	Thus (6) and (8) imply
00
B[u, u] = 52^Afc>Ai by (9).
fc=i
As equality holds for и = uq, we obtain formula (5).
5.	We next claim that if и € Hq(U} and ||u||i,2(i/) = 1, then и is a weak
solution of
(10)
if only only if
Lu = Aiu
и = 0
in U
on dU
(И)
B[u, u] = Ai-
Obviously (10) implies (11). On the other hand, suppose (11) is valid.
Then, writing dk = (u,Wfc) as above, we have
(12)	52 4A1 = Ai = B[u, «] = 52 dkXk- k=l	k=l
Hence	00
(13)	£(Afc-Ai)d2=0. t=i
Consequently	dfc = (u,wfc) = 0 if Afc > Ai.
Since Ai has finite multiplicity, it follows that
(14)	m U = fc=l
for some m, where Lwk = Aiw^. Therefore
(15)	m Lu = ^2(u,Wk)Lwk = Ain. fc=i
This proves (10).
6.	Next we will show that if и E Hq(W) is a weak solution of (10), и 0,
then either
(16)
и > 0 in U
6.5. EIGENVALUES AND EIGENFUNCTIONS
339
or else
(17)
и < 0 in U.
To see this, let us assume without loss that ||«||i,2 = 1, and note
(18)
« + /?=!
for
a := [ (u+)2dx, /3 '= [ (u~)2dx.
Ju	Ju
Furthermore since u± € with
. f Du a.e. on {u > 0}
Du+ = (	г .
( 0 a.e. on {u < 0},
f 0 a.e. on {u > 0}
Du = <
( —Du a.e. on {u < 0}
(cf. Problem 17 in Chapter 5), we have B[u+,u-] = 0. Accordingly
Ai = B[u, u] = B[u+, u+] + B[u“, u-]
> Ai||u+	+ Ai||u	by (5)
= (a + /?) Ai = Ai.
But then we see that the inequality above must in fact be an equality, and
so
B[u+,u+] = Ai||u+||22{[/), B[u“,u“] = Ai||u“||^2([/).
Therefore the claim proved in step 5 asserts
(19)
and
(20)
Lu+= Aiu+
u+= 0
Lu = Aiu
u~ = 0
in U
on dU
in U
on dU
in the weak sense.
7.	Next, since the coefficients av are smooth, we deduce from (19) that
u+ € C°°(U) and
Lu+ = Aiu+ > 0 in U.
The function u+ is therefore a supersolution. Thus the strong maximum
principle implies either u+ > 0 in U or else u+ = 0 in U. Similar arguments
apply to u~, and so (16) and (17) hold.
340
6. SECOND-ORDER ELLIPTIC EQUATIONS
8.	Finally assume that и and й are two nontrivial weak solutions of (10).
In view of steps 6 and 7 above
й dx ф 0,
and so there exists a real constant x such that
(21)
и — x^ dx = 0.
But since u — yh is also a weak solution of (10), steps 6, 7, and the equality
(21) imply и = хй in U. Hence the eigenvalue Ai is simple.	□
6.5.2. Eigenvalues of nonsymmetric elliptic operators.
We will now consider a uniformly elliptic operator L in the nondivergence
form:
n	n
Lu = - aNuXiXj + bluXi + cu.
i,j=l	i=l
Let us for simplicity assume au, Ьг,с € C'oo([7), U is open, bounded, con-
nected, and dU is smooth. We suppose also a’-7 = адг (i,j = l,...,n),
and
(22)	c > 0 in U.
Notice however that in general the operator L will not equal its formal
adjoint. We therefore cannot invoke as above the abstract theory from §D.6.
And in fact L will in general have complex eigenvalues and eigenfunctions.
Remarkably, however, the principal eigenvalue of L is real, and the cor-
responding eigenfunction is of one sign within U.
THEOREM 3 (Principal eigenvalue for nonsymmetric elliptic operators).
(i)	There exists a real eigenvalue Ai for the operator L, taken with zero
boundary conditions. Furthermore if A € C is any other eigenvalue,
we have
Re(A) < Ai.
(ii)	There exists a corresponding eigenfunction wi, which is positive within
U.
(iii)	The eigenvalue Ai is simple; that is, if и is any solution o/(l), then
и is a multiple ofwy.
6.5. EIGENVALUES AND EIGENFUNCTIONS
341
Proof*. 1. Choose m = [§] + 2 and consider the Banach space X =
Hm(U) D H^U). According to Theorem 3 in §5.3.2 X c C2(t7). We define
the linear, compact operator A : X —> X by setting Af = u, where и is the
unique solution of
(23)
( Lu = f
( и = 0
in U
on dU.
Next define the cone
C = {u G X | и > 0 in U}.
According to the maximum principle, A : C —> C.
2.	Hereafter fix any function w e C, w 0. Employing the strong
maximum principle and Hopf’s Lemma, we deduce
dv
(24)	v > 0 in U, — < 0 on dU
du
for v = A(w).
Remember that w = 0 on dU. So in view of (24) there exists a constant
p > 0 so that
(25)
pv > w in U.
3.	Fix e > 0, rj > 0, and consider then the equation
(26)	и = pA[u + ew]
for the unknown и E C. We claim that
(27)
if (26) has a solution u, then rj < p.
To verify this assertion, suppose in fact и E C solves (26). We compute
u > rjAlew] = rjev > — €W,
according to (25). Hence
. т?2е .	t]2€
и > riAu > -----Aw - ------ v
Д	p
€W.
*Omit on first reading.
342
6. SECOND-ORDER ELLIPTIC EQUATIONS
Continuing, we deduce
/ x к
u> I — j ew (k = 1,...),
\P J
a contradiction unless r] < p. This observation confirms the assertion (27).
4.	Define
Se := {« € C | there exists 0 < r] < 2p such that и = T]A[u + ew]}.
We next assert
(28)	St is unbounded in X.
For otherwise we could apply Schaefer’s fixed point theorem (to be proved
later, as Theorem 4 in §9.2.2), to deduce that the equation
и = 2/иА[и + ew]
has a solution, in contradiction to (27).
5.	Owing to (28), there exist
(29)	0 < r]€ < 2p
and ve G C, with ||ve||x > 1, satisfying
(30)	vt = r]eA[ve + ew].
Renormalize by setting
Using (29)-(31) and the compactness of the operator A, we obtain a subse-
quence ед, —> 0 so that
r]tk —> tj and uek —> и in X.
Then (31) implies
(32)	||«|| x =l,uEC.
Since ue = rjfA [14 + j , we deduce in the limit that и = tjAu. In view
of (32), rj > 0. We may consequently rewrite the above to read
Lwi = Aiwi in U
wi = 0 on dU,
6.5. EIGENVALUES AND EIGENFUNCTIONS
343
for Ai = rj, и = wi- Thus Ai is a real eigenvalue for the operator L, taken
with zero boundary conditions, and wi > 0 is a corresponding eigenfunction.
In view of the strong maximum principle and Hopf’s Lemma, we have
(33)	wi > 0 in U, < 0 on dU.
dv
Additionally, we know wi is smooth, owing to the regularity theory in §6.3.
6.	All expressions occurring in steps 1-5 above are real. Suppose now
A € C and и is a complex-valued solution of
(34)	(L“ = )“ “L
v ’	( и = 0	on dU.
Now choose any smooth function w : U —> R, with w > 0 in U, and set
v w- We compute
Av = — L(yw) by (34)
(35)	W 2 A
= Lv - cv---У <T3wx vXi H-----Lw.
w	J w
i,j=i
Writing
n	n
+ 12 b'iVxi
i,j=l	i=l
for Ъ'г := Ьг — 52j=i a^wXj (1 < i < n), we deduce from (35) that
(36)	Kv + |	— A | v = 0 in U.
\ w J
Take complex conjugates:
(37)	Kv + ( — - A ) v = 0 in U.
\ w /
Next we compute
(38)	AT(|v|2) = K(vv) = vKv + vKv — 2	< vKv + vKv,
i,j=l
since
n	n
12 «ij(& = £ «wMHm&Mj)) > о
j,j=l	i,j=l
344
6. SECOND-ORDER ELLIPTIC EQUATIONS
for £ e C”. Combining (36)-(38), we discover
tf(|v|2) < 2 (ReA- —|n|2.
\ w /
Now choose
(39)	w :=
for 0 < c < 1. Then
(1 — c) e(l — e)	j_e .	,.
Lw =--------—Lwi + —Гт— > a3wi Xiwi Xi+€cw-, > (1 - c)Aiw.
1	1 l,J=l
Consequently
JC(|v|2) < 2(Re A — (1 — c)Ai)|v|2 inf/.
Thus if Re(A) < (1 — e)Ai, then K(|v|2) < 0 in U. As v = 0 on dU, according
to (33) and (39), we deduce from the maximum principle that v =	= 0
in U. Thus и = 0 in U and so A cannot be an eigenvalue. This conclusion
obtains for each e > 0, and so Re A > Ai if A is any complex eigenvalue.
7. Finally, let и be any (possibly complex-valued) solution of
. .	(Lu — X\u in U
[ и = 0 on dU.
Since Re(u) and Im(u) also solve (40), we may as well suppose from the
outset и is real-valued. Replacing и by — и if needs be, we may also suppose
и > 0 somewhere in U. Now set
(41)	x := sup{/z > 0 | wi — p,u > 0 in U}.
Then 0 < x < Write v = wi — yn; so that v > 0 in U and
( Lv = Ain >0 in U
[ v = 0 on dU.
Now if v is not identically zero, the strong maximum principle and Hopf’s
Lemma imply
dv
v > 0 in U, — < 0 on dU.
du
Thus
v — eu > 0 in U for some e > 0,
and so
wi — (y + e)u >0 in U,
a contradiction to (41). Hence v = 0 in U, and so и is a multiple of wi. □
6.6. PROBLEMS
345
6.6.	PROBLEMS
In the following exercises we assume the coefficients of the various PDE are
smooth and satisfy the uniform ellipticity condition. Also U C Rn is always
an open, bounded set, with smooth boundary.
1.	Let
n
Lu = - У7 {al3ux^x_ + cu.
ij=l
Prove that there exists a constant p. > 0 such that the correspond-
ing bilinear form B[ , ] satisfies the hypotheses of the Lax-Milgram
Theorem, provided
c(x) > — Ц (x € U).
2.	A function и e is a weak solution of this boundary-value prob-
lem for the biharmonic equation
/ ч	( A2u = f in U
\u = ^= Q on dU
provided
/ ДпДv dx = / fv dx
Ju	Ju
for all v €	Given f € L2(U), prove that there exists a unique
weak solution of (*).
3.	Assume U is connected. A function и G Я1([7') is a weak solution of
Neumann’s problem
, .	( —&u = f in U
W	|	= 0 on dU
if
/ Du  Dv dx = I fv dx
Ju	Ju
for all v E B1([7'). Suppose f E L2(U). Prove (*) has a weak solution
if and only if
[ f dx = 0.
Ju
4.	Let и E	have compact support and be a weak solution of the
semilinear PDE
-Ди + с(и) = / in R",
346
6. SECOND-ORDER ELLIPTIC EQUATIONS
where f G £2(R”) and c : R —> R is smooth, with c(0) = 0 and c' > 0.
Prove и G №(Rn).
(Hint: Mimic the proof of Theorem 1 in §6.3.1, but without the cutoff
function £.)
5.	Let и be a smooth solution of Lu = — 52Fj=i a^uXiXj = 0 in U. Set
v := |£>u|2 + Au2. Show that
Lv < 0 in U. if A is large enough.
Deduce
ll^lk°°(U) < C(||£M£°°(5J7) + llulk°°(5J7))-
6.	Assume u is a smooth solution of Lu = — a^uxtxj = f in U,
u = 0 on dU, where f is bounded. Fix ж0 € dU. A barrier at x° is a
C2 function w such that
Lw > 1 in U, w(x°) = 0, w > 0 on dU.
Show that if w is a barrier at x°, there exists a constant C such that
7.	Assume U is connected. Use (a) energy methods and (b) the maximum
principle to show that the only smooth solutions of the Neumann
boundary-value problem
( —Ли = 0 in U
I g=o on ас/
are u = constant.
8.	Assume u G H{(U) is a bounded weak solution of
- (avuXi)x. = 0 in U.
i,j=l
Let ф : R —> R be convex and smooth, and set w — ф(и). Show w is a
weak subsolution; that is,
B[w, v] < 0
for all v G Hq(U), v > 0.
6.7. REFERENCES
347
9.	(Courant minimax principle). Let L = — Y^ij=Aa^uXi)xjy where
((av)) is symmetric. Assume the operator L, with zero boundary
conditions, has eigenvalues 0 < Ai < A2 < • • • . Show
Xk — max min B[u,u] (fc = l, 2,...).
ues1-
ll“llb2=l
Here Sfc-i denotes the collection of (k — l)-dimensional subspaces of
Я,».
10.	Let Ai be the principal eigenvalue of the uniformly elliptic, nonsym-
metric operator
n	n
Lu = - atjuXiXj + У bluXt + cu,
i,j=l	1=1
taken with zero boundary conditions. Prove the “max-min” represen-
tation formula:
A1=s„pinfl^,
и x U(X)
the “sup” taken over functions и € С00(Я) with и > 0 in U, и = 0
on dU, and the “inf” taken over points x E U. (Hint: Consider the
eigenfunction corresponding to Ai for the adjoint operator £*.)
6.7.	REFERENCES
Section 6.1 See Gilbarg-Trudinger [G-T, Chapters 5, 8].
Section 6.	3 Consult Gilbarg-Trudinger [G-T], Krylov [KR] and Lady-
zhenskaya-Ur altseva [L-U] for more on regularity theory for
elliptic PDE.
Section 6.	4 Gilbarg-Trudinger [G-T, Chapter 3]. Protter and Wein-
berger [P-W] is a good reference for further maximum prin-
ciple methods.
Section 6.	5 See Smoller [S, pp. 122-125]. The last part of the proof
of Theorem 3 is modified from Protter-Weinberger [P-W,
§2.8]. O. Hald simplified my proof of Theorem 2.
Chapter 7
LINEAR
EVOLUTION
EQUATIONS
7.1	Second-order parabolic equations
7.2	Second-order hyperbolic equations
7.3	Hyperbolic systems of first-order equations
7.4	Semigroup theory
7.5	Problems
7.6	References
This long chapter studies various linear partial differential equations
that involve time. We often call such PDE evolution equations, the idea
being that the solution evolves in time from a given initial configuration.
We will study by energy methods general second-order parabolic and hyper-
bolic equations, and also certain first-order hyperbolic systems. The Fourier
transform, utilized in §7.3.3, and the semigroup technique, discussed in §7.4,
provide alternative approaches.
7.1.	SECOND-ORDER PARABOLIC EQUATIONS
Second-order parabolic PDE are natural generalizations of the heat equa-
tion (§2.3). We will study in this section the existence and uniqueness of
appropriately defined weak solutions, their smoothness and other properties.
349
350
7. LINEAR EVOLUTION EQUATIONS
7.1.1.	Definitions.
a.	Parabolic equations.
For this chapter we assume U to be an open, bounded subset of Rn, and
as before set Ut = U x (0, T] for some fixed time T > 0.
We will first study the initial/boundary-value problem
' ut + Lu = f
< и = 0
„ u = 9
(1)
in Ut
on dU x [0, T]
on U x {t = 0},
where f : Ut —> R and g : U —> R are given, and и : Ut —> R is the
unknown, и = u(x,t). The letter L denotes for each time t a second-order
partial differential operator, having either the divergence form
n	n
(2)	Lu = - У7	+ У? b\x, fjuXi + c(x, Qu
ij=l	i=l
or else the nondivergence form
n	n
(3)	Lu = - У7 QuxiXj + У7 t^Uxi + c(x, Qu,
i,j=l	i=l
for given coefficients aV ,bl,c (i.j = 1,..., n).
DEFINITION. We say that the partial differential operator ^, + L is (uni-
formly) parabolic if there exists a constant 0 > 0 such that
(4)
n
L > 0|CI2
i,j=l
for all (x, Q E Ut, £ E Rn.
Remark. Note in particular that for each fixed time 0 < t < T the operator
L is a uniformly elliptic operator in the spatial variable x.	□
An obvious example is = 6^,Ьг = c = / = 0; in which case L = —Д
and the PDE + Lu becomes the heat equation. We will see in fact that
solutions of the general second-order parabolic PDE are similar in many
ways to solutions of the heat equation.
General second-order parabolic equations describe in physical applica-
tions the time-evolution of the density of some quantity u, say a chemical
7.1. SECOND-ORDER PARABOLIC EQUATIONS
351
concentration, within the region U. As noted for the equilibrium setting (i.e.,
second-order elliptic PDE, in §6.1.1), the second-order term 52ij=i
describes diffusion, the first-order term b'LuXi describes transport, and
the zeroth-order term cu describes creation or depletion.
The Fokker-Planck and Kolmogorov equations from the probabilistic
study of diffusion processes are also second-order parabolic equations.
b.	Weak solutions.
Mimicking the developments in §6.1.2 for elliptic equations, we consider
first the case that L has the divergence form (2) and try to find an appro-
priate notion of weak solution for the initial/boundary-value problem (1).
We assume for now that
(5)	агЛ6г,се£°°([7т) (i,j = l,...,n),
(6)	/еЬ2й),
(7)	g e L\U).
We will also always suppose av = aJi (г, j = 1,..., n).
Let us now define, by analogy with the notation introduced in Chapter
6, the time-dependent bilinear form
(8)	B[u,v;t]:= /	t)uXivX} +	6г(-, f)uXiv + c(-, t)uv dx
i,j=i	i=i
for u, v e Hq (C7) and a.e. 0 < t < T.
Motivation for definition of weak solution. To make plausible the
following definition of weak solution, let us first temporarily suppose that
и = u(x,t) is in fact a smooth solution of our parabolic problem (1). We
now switch our viewpoint, by associating with и a mapping
и t [0,T]Hq(U)
defined by
[u(t)](x) := u(x,t) (x EU, 0 < t < T).
In other words, we are going to consider и not as a function of x and t
together, but rather as a mapping u of t into the space Hq(U) of functions
of x. This point of view will greatly clarify the following presentation.
Returning to problem (1), let us similarly define
f : [0,T] —> L2(?7)
352
7. LINEAR EVOLUTION EQUATIONS
by
[f(t)](z) := f(x,t) (xeU, 0<t<T).
Then if we fix a function v G Hq(U), we can multiply the PDE ^t+Lu- f
by v and integrate by parts, to find
(9)	(u',v) + B[u,v;t] = (f,v)	(' =
\	“t /
for each 0 < t < T, the pairing ( , ) denoting inner product in L2(U).
Next, observe that
n
(10)	Ut^ g°+ ^g3X3 in UT
1=1
for g° := f - 527=1	“ cu and 9j := EXi aIjuXi (j = 1,... ,n). Con-
sequently (10) and the definitions from §5.9.1 imply the right hand side of
(10) lies in the Sobolev space Я-1([7'), with
/ n	\i/2
IMh-1^) - ( 52 H^IIl2(i7) ) - c (IIuIIh0i((7) + II/IIl2((7)j •
X J=o	7
This estimate suggests it may be reasonable to look for a weak solution with
u' G H~X(U} for a.e. time 0 < t < T; in which case the first term in (9) can
be reexpressed as (u', v), ( , ) being the pairing of Я-1([7') and H^tU). □
All these considerations motivate the following
DEFINITION. We say a function
u G L2(0,T-,H^U)), with u' G Т2(0,Т;Я-1(С7)),
is a weak solution of the parabolic initial/boundary-value problem (1) pro-
vided
(i)	(u', v) + B[u,v;t] = (f,v)
for each v G Hq(U') and a.e. time 0 < t <T, and
(ii)	u(0) = g.
Remark. In view of Theorem 3 in §5.9.2, we see u G C([0, T]; L2(Uf), and
thus the equality (ii) makes sense.	□
7.1. SECOND-ORDER PARABOLIC EQUATIONS
353
7.1.2.	Existence of weak solutions.
a. Galerkin approximations.
We intend to build a weak solution of the parabolic problem
(И)
' ut + Lu = f
< и = 0
и = 9
in Ut
on dU x [0, T]
on U x {t = 0}
by first constructing solutions of certain finite-dimensional approximations
to (11) and then passing to limits. This is called Galerkin’s method.
More precisely, assume the functions = Wfe(x) (k = 1,...) are
smooth,
(12)	is an orthogonal basis of Hq(U),
and
(13)	is an orthonormal basis of L2(W).
(For instance, we could take	to be the complete set of appropriately
normalized eigenfunctions for L = —Д in	see §6.5.1.)
Fix now a positive integer m. We will look for a function um : [0,T]^
Hq (U) of the form
m
(14)	um(t) :=52d^(t)wfc,
fe=i
where we hope to select the coefficients d^(t) (0 < t < T, к = 1,..., m) so
that
(15)	d^(0) = (</,wfc) (fc=l,...,m)
and
(16)	(u^Wfc) + B[um,wfc;t] = (f,wfc) (0 < t < T, к = 1,... ,m).
(Here, as before, ( , ) denotes the inner product in L2([7’).)
Thus we seek a function um of the form (14) that satisfies the “projec-
tion” (16) of problem (11) onto the finite dimensional subspace spanned by
WZU-
354
7. LINEAR EVOLUTION EQUATIONS
THEOREM 1 (Construction of approximate solutions). For each integer
m = 1, 2,... there exists a unique function um of the form (14) satisfying
(15), (16).
Proof. Assuming um has the structure (14), we first note from (13) that
(17)	(u'm(i),wfc) = d^'(t).
Furthermore	m
(18)	B[um,wk;t] = ^ekl(t)dlm(t), i=i
for eki(t) :=	(k,l = l,...,m). Let us further write fk(t) :=
(f(t),Wfc) (k = 1,... ,m). Then (16) becomes the linear system of ODE
m
(19) O) + £ ek\Qdlm(Q = fk(t) (k = 1,..., m),
z=i
subject to the initial conditions (15). According to standard existence theory
for ordinary differential equations, there exists a unique absolutely contin-
uous function dm(t) = (dm(t),..., dj^(t)) satisfying (15) and (19) for a.e.
0 < t < T. And then um defined by (14) solves (16) for a.e. 0 < t < T.
□
b. Energy estimates.
We propose now to send m to infinity and to show a subsequence of our
solutions um of the approximate problems (15), (16) converges to a weak
solution of (11). For this we will need some uniform estimates.
THEOREM 2 (Energy estimates). There exists a constant C, depending
only on U,T and the coefficients of L, such that
llUm(t)llb2(C7) + |lum||L2(0;T.Hi((7)) + ||u(„||L21 (I/))
< C'dlfIIl2(0,T;L2((7)) + llfz||l,2((7))
for m = 1,2,... .
Proof. 1. Multiply equation (16) by d^(t), sum for к = 1,..., m, and then
recall (14) to find
(21)	(Wn’Wn) 4" D[um,um;t] = (f,um)
7.1. SECOND-ORDER PARABOLIC EQUATIONS
355
for a.e. 0 < t < T. We proved in §6.2.2 that there exist constants /? > 0,
7 > 0 such that
(22)	i] + 7 ||um |Il2((7)
for all 0 < t < T, m = 1,.... Furthermore |(f,um)| <	+
|||um||22([/), and «,um) = ^(|||um||22([/)) for a.e. 0 < t < T. Con-
sequently (21) yields the inequality
(23)	— (||um+ 2/?||um||^i< Ci||um||jr,2([/) + C*2||fIIl2((7)
for a.e. 0 < t < T, and appropriate constants Ci and C2.
2.	Now write
(24)	r)(t) := ||um(t)||22((7)
and
(25)	C(i) := 1Ж1ь2([/).
Then (23) implies
for a.e. 0 < t < T. Thus the differential form of Gronwall’s inequality (§B.2)
yields the estimate
(26)	T}(t) < eClt (77(0)+C2 [\(s)ds\ (0<t<T).
X Jo J
Since 7?(0) = ||um(0)||22((7) < ||s||£2([7) by (15), we obtain from (24)-(26)
the estimate
0<t<r llUm(^lli2(^) —	H^Hl2(0,T;L2((7))) ’
3.	Returning once more to inequality (23), we integrate from 0 to T and
employ the inequality above to find
2	fT 2
HUmllL2(O,T;Ho1(i7)) = / HUrzlll^(C7)
< C (||5|Il2((7) + ||f II £,2(0 ,T;L2((7))) '
4. Fix any v E with ||v||££i(CZj < 1, and write v = v1 +v2, where
v1 E span{wfc}J(L1 and (v2,Wfe) = 0 (k = 1, ...,m). Since the functions
356
7. LINEAR EVOLUTION EQUATIONS
{wfc}^0 are orthogonal in H^(U), Цп1 ||Я1(с/) < ||v||Hi(c/) < 1. Utilizing
(16), we deduce for a.e. 0 < t < T that
(u^,!?1) + В[ит:1'Ч] = (f,^1).
Then (14) implies
(<4,v) = (<4^) = (umV) = (fV) -B[umy;t].
Consequently
since	< 1- Thus
Ни^Ня-1^) < C'(llflk2(C/) + 11ит||я1(С/))’
and therefore
||иш11я-1(Я) — C Jv II^IIl2({/) + llumll//(j({/)
< C'dlsllb^) + Н^Нь2(0,Т;Ь2(я)))-
□
c. Existence and uniqueness.
Next we pass to limits as m -> oo, to build a weak solution of our
initial/boundary-value problem (11).
THEOREM 3 (Existence of weak solution). There exists a weak solution
of Qi).
Proof. 1. According to the energy estimates (20), we see that the se-
quence {uTO}“=1 is bounded in Z2(0, T; Hq(U)), and {u(n}“=1 is bounded
in L2(0,T-,H~\U)).
Consequently there exists a subsequence {um/}gj C {uTO}“=1 and a
function u € T2(0,T; Bq(U)), with u'€ T2(0,T; H-1(U)), such that
U
(27)
um;
weakly in T2(0, T; Hq(U))
weakly in T2(0,T; B-1(U)).
(See §D.4 and Problem 4.)
7.1. SECOND-ORDER PARABOLIC EQUATIONS
357
2.	Next fix an integer N and choose a function v € C1([0,T];
having the form
N
(28)	v(t) = ^dk(t)wk,
fc=i
where {dfc}^L1 are given smooth functions. We choose m > N, multiply
(16) by dfc(t), sum к = 1,..., N, and then integrate with respect to t to find
(29)	[	+ B[um,v;t\dt = [ (f,v)dt.
Jo	Jo
We set m = mi and recall (27), to find upon passing to weak limits that
(30)	[ (uz,v) + B[u,v,t\dt = [ (f, v)dt.
Jo	Jo
This equality then holds for all functions v € B2(0, T; Hq(U)), as functions
of the form (28) are dense in this space. Hence in particular
(31)	(u',u) + B[u, v;t] = (f, u)
for each v G Hq (17) and a.e. 0 < t < T. From Theorem 3 in §5.9.2 we see
that furthermore u G C([0, T]; L2(t7)).
3.	In order to prove u(0) = g, we first note from (30) that
(32)	[ —(v',u) + B[u, v;i] dt = [ (f, v)dt + (u(0), v(0))
Jo	Jo
for each v G C1([0,T];with v(T) = 0. Similarly, from (29) we
deduce
(33)	[ -(v',uTO) + B[uTO, v;i] dt = [ (f, v) dt + (uTO(0), v(0)).
Jo	Jo
We set m = mi and once again employ (27) to find
(34)	[ —(v',u) + B[u,v;i] dt — [ (f,v) dt + (g,v(0)),
Jo	Jo
since uTO!(0) —> g in L2(U). As v(0) is arbitrary, comparing (32) and (34),
we conclude u(0) = g.	□
358
7. LINEAR EVOLUTION EQUATIONS
THEOREM 4 (Uniqueness of weak solutions). A weak solution of (11) is
unique.
Proof. It suffices to check that the only weak solution of (11) with f = g = 0
is
(35)
u = 0.
To prove this, observe that by setting v = u in identity (31) (for f = 0) we
learn, using Theorem 3 in §5.9.2, that
(36)
4 f IIIu||2L2(u)) + B[u, u; t] = (u', u) + B[u, u; t] = 0.
QiT \ Zt	j
Since
B[u,u;t] >/3||u||^i(f/) -7||u||22({/) > -7||u||22([/),
Gronwall’s inequality and (36) imply (35).
□
7.1.3. Regularity.
In this section we discuss the regularity of our weak solutions u to the
initial/boundary-value problem for second-order parabolic equations. Our
eventual goal is to prove that u is smooth, provided the coefficients of the
PDE, the boundary of the domain, etc. are smooth. The following presen-
tation mirrors that from §6.3.
Motivation: formal derivation of estimates, (i) To gain some insight
as to what regularity assertions could possibly be valid, let us temporarily
suppose и = u(x, t) is a smooth solution of this initial-value problem for the
heat equation:
(37)
ut - Au = f
u = g
in K” x (0,T]
on Rn x {t = 0},
and assume also и goes to zero as |x| —» oo sufficiently rapidly to justify the
following computations. We then calculate for 0 < t < T:
Rn
[ u2t
(38)
— 2Nuut + (A«)2 dx
= u2 + 2Du  Dut + (Au)2 dx.
7.1. SECOND-ORDER PARABOLIC EQUATIONS
359
Now 2Du  Dut = (|Du|2), and consequently
n2Du  Dut dxds = [ |Du|2
p	JR"	S
Furthermore, as demonstrated in §6.3,
[ (Au)2 dx = [ \D2u\2dx.
JR"	JR"
We utilize the two equalities above in (38) and integrate in time, to obtain
sup / |Du|2 dx + /	/ u2 + \D2u\2 dxdt
(39)	•'° •'Rn
< C ( [ [ f2 dxdt + f |£)g|2 dx j .
\Jo JR"	JR" /
We therefore see that we can estimate the L2-norms of ut and D2u within
Rn x (0,T), in terms of the Z2-norm of f on Rn x (0,T) and the T2-norm
of Dg on Rn.
(ii) Next differentiate the PDE with respect to t and set й := Ut- Then
, ,	( щ - Ай = f in Rn x (0, T]
[ й = g on Rn x {t = 0},
for f := ft, g := ut(-,0) = /(-,0) + A#. Multiplying by й, integrating by
parts and invoking Gronwall’s inequality, we deduce:
(41)
sup i |ut|2 dx + [ i \Dut\2 dxdt
0<t<T jRn	Jo
<c([ [ f2 dxdt+[ \D2g\2 + f(-,0)2dx
\J0 JR"	JR"
But
(42) тЮС ||/(-,t)||L2(]Rn) < C'(||/||i2(Knx(0T)) + ||/t||L2(R"x(0,T)))>
according to Theorem 2,(iii) of §5.9.2. Furthermore, writing —Au = f —Ut,
we find as in §6.3 that
(43)
u2 dx.
360
7. LINEAR EVOL UTION EQ UATIONS
Combining (41)-(43) leads us to the estimate
sup i |ut|2 + |D2u|2 dx + [ i \Dut\2 dxdt
/ .	0<t<T Л"	JO JRn
(44)	- " T
<C[ [ [ ft+f2 dxdt + [ \D2g\2 dx ) ,
\Jo JRn	JR"	/
for some constant C.	□
The foregoing formal computations suggest that we have estimates cor-
responding to (39) and (44) for our weak solution to a general second-order
parabolic PDE. These calculations do not constitute a proof however, since
our weak solution of (11), constructed in §7.1.2, is not smooth enough to
justify the foregoing computations.
We will instead calculate using the Galerkin approximations. To stream-
line the presentation, we hereafter assume that {wfc}^5=1 is the complete col-
lection of eigenfunctions for —A on LQ(U), and that U is bounded, open,
with dU smooth. We furthermore suppose that
f the coefficients av, Ьг, c (i.j = 1,..., n) are smooth on
(45)	V	’
( U and do not depend on t.
THEOREM 5 (Improved regularity).
(i) Assume
geH^U), feL2(0,T;L2(U)).
Suppose also и G T2(0, T; Hq (t/)), with u' € Т2(0, T; Я-1(17)), is the weak
solution of
{Ut + Lu = f in Ut
и = 0 on dU x [0, T\
и - g on U x {t = 0}.
Then in fact
u G T2(0,T; №([/)) П L°°(0,T; 7Jq(17)), u' G T2(0,T;L2(D)),
and we have the estimate
ess sup||u(t)||Hi+ ||u||L2(0iT.H2(u)) + llu'll/^O’T;/^))
(46)	°-4-T
< C (l|f IIl2(o,T;L2(C/)) + Ill'll
the constant C depending only on U,T and the coefficients of L.
7.1. SECOND-ORDER PARABOLIC EQUATIONS
361
(ii) If, in addition,
g £ H2(U), f' G L2(0,T;L2(t/)),
then
u e L°°(0, T; H2(Uf), u' G L°°(0, T; L2(Uf) П Т2(0, Г;
u" G T2(0,T;
with the estimate
ess^sup(||u(t)||n2([/) + ||u'(£)||l2(c/)) + Пи'П^^я^я))
(47)	+IIu//||l2(o,t;h-i(t/)) < C'dlf|1я!(0,Т;Ь2(^)) + 11^Ня2(*У))-
Remark. Assertions (i), (ii) of Theorem 5 are precise versions of the formal
estimates (39), (44) (for the heat equation on 17 = Rn).	□
Proof. 1. Fixing m > 1, we multiply equation (16) in §7.1.2 by d^{f) and
sum к = 1,..., m, to discover
(um, <4) + SK = (f> um)
for a.e. 0 < t < T. Now
=: A + B.
Since a’J = aJZ (?, j = 1,... ,n) and these coefficients do not depend on t,
we see A = ^(|A[uTO,uTO]), for the symmetric bilinear form
A[u,v] := [ aljuXivx dx (u,v e
Furthermore,
|B| < —	+ е||итНь2(Я), l(f>um)l llf II/?((,') + e 11 u^zz IIЬ2 (t7)
for each e > 0.
362
7. LINEAR EVOLUTION EQUATIONS
2.	Combining the above inequalities, we deduce
llumIL2(C7)
—	~(lluvi||//l([/) + 11^ llz,2(C7)) + 2б||ит•
Choosing e = | and integrating, we find
fT
/ HumllL2(C/)^+ SUP A[uTO(t),Um(t)]
Jo	0<t<T
<	C ^A[uTO(0), um(0)] + / l|um||//i([/) + ||f||L2([/)d^
<	cdi^iin^c/) + i^Hl2(0]T;L2(C/))) ,
according to Theorem 2 in §7.1.2, where we estimated ||uTO(0)||h^(P)
Цй’Ця^с/)- As > 0 Ju \Du\2dx for each и G H^U'), we find that
(48)	^sup^ ||um(t)||^i([/) < C'(||fl'||^i(cz) + ||fHl2(0,T;L2(C/)))-
Passing to limits as m = mi —» oo, we deduce u G L00(0,T;Ho(I7)),
u' G Т2(0, T; L2(U)), with the stated bounds; cf. Problem 5.
3.	In particular, for a.e. t we have the identity
(uz, v) + B[u, u] = (f, u)
for each v G Hq(U). This equality we rewrite to read
B[u, u] = (h, u)
for h := f — u'. Since h(t) G L2([Z) for a.e. 0 < t < T, we deduce from the
elliptic regularity Theorem 4 in §6.3.2 that u(t) G B2(C) for a.e. 0 < t < T,
with the estimate
НиНя'2^) <
(49)	< Cdlfll^^ + llu'll^z^j + ||u||22(c/}).
Integrating and utilizing the estimates from step 2, we complete the proof
of (i)-
4.	The goal next is to establish higher regularity for our weak solution.
So now suppose g G H2(U) Г) Hq(U), f G Tf^O.T; T2(t/)). Fix m > 1 and
7.1. SECOND-ORDER PARABOLIC EQUATIONS
363
differentiate equation (16) in §7.1.2 with respect to t. Owing to (45), we
find
(50)	(u'm,wk) +B[um,wk\ = (f\wk) (k = l,...,m),
where iiTO := u^. Multiply (50) by d^f) and sum к = 1,..., m:
(й!т, um) + um] = (fz, йто).
Employing Gronwall’s inequality, we deduce
sup ||<4(i)||i2([/) + [ ||u^||^i([/) dt
0<t<T	v ’ Jo	°
( )	— C,(llUm(0)llL2([/) + ||f |1ь2(017’.£2([/)))
C'GIf IIh1(0,T;L2(C/)) + llu^(0) |ln2((7))’
We employed (16) in the last inequality.
5.	We must estimate the last term in (51). Remember that we have
taken	to be the complete collection of (smooth) eigenfunctions for
—A on Hq(U). In particular, AuTO = 0 on dU. Thus
llum(0)Hn2(C/) — ^11 Aum(0)l|2 2(CZ) = C(um(0), A2Um(0)).
Since Д2ито(0) G span-Jwfc}^ and (uTO(0), wk) = (g,wk) for к = 1,..., m,
we have
1|ит(0)||2я2(с/) < C(g, Д2ито(0)) = CW, AuTO(0))
— 211и”г(^)Ия2(с7) +
Hence ||ит(0)||Я2(с/) < С1Ы1я2(я)- Therefore (51) implies
,co. sup ||u(n(t)||22(c/) + / ||l4||21(
(52) o<t<T	v ’ Jo () ’
- C’dlfПя1(0,Т;^2(я)) + llsll/f2^)).
6. Now
B[um,Wfc] = (f-i4,Wfc) (k = 1,... ,m).
Let A*, denote the kth eigenvalue of —A on H^U). Multiplying the above
identity by Afcd^(t) and summing к = 1,... ,m, we deduce for 0 < t < T
that
(53)
AuTO] — (f uTO, AuTO).
364
7. LINEAR EVOLUTION EQUATIONS
Since AuTO = 0 on dU, we see B[uTO)—AuTO] = (Lum,— AuTO). Next we
invoke the inequality
(54)	/?||«Ня2(р) < (Lu, -Au) + 7||u||22([/) (u G H2(U) П H^U))
for constants /3 > 0, 7 > 0. See Problem 8 and also the Remark following
the proof.
We thereupon conclude from (53) that
Нит|1я2(С7) < C(||f||b2(C7) + llumll.L2(C/) + II иттг || Z,2 (CZ)) -
This inequality, (52), (48) and Theorem 3 in §5.9.2 imply
0^PT(HUmWllL2([/) + 1|ит(<)|1я2([/)) +	||u^||2ffi({/) dt
— С,(11^11я1(0,Т;^2([/)) + Н^Няг^))-
Passing to limits as m = mi —» oo, we deduce the same bound for u.
7. It remains to show u" 6 B2(0, T; B-1^)). To do so, take v € Hq(U),
with ||u||Hi(c/) < 1> and write v = u1 + v2, as in the proof of Theorem 2 in
§7.1.2. Then for a.e. time 0 < t < T:
(um,v) = «,«) = (um4;1) = (f',^1)
according to (50), since = й(„. Consequently
|K,u)|<C(||f'||L2(c/) + |Kllnol([/)),
since Н^Ня^р) < 1- Thus
Нит11н-1(Я) < C'(llfZ|lL2(C/) + ll^m II Яд ((/)) ’
and so u"n is bounded in L2(0,T;B-1(B)). Passing to limits, we deduce
that u/z € L2(0, T; H~1(U)), with the stated estimate.	□
Remark. If L were symmetric, we could alternatively have taken {u^}^
to be a basis of eigenfunctions of L on (U), and so avoided the need for
inequality (54).	□
Let us now build upon the previous regularity assertion:
7.1. SECOND-ORDER PARABOLIC EQUATIONS
365
THEOREM 6 (Higher regularity). Assume
g£H2m+1(U),	£ L2(0,T;ff2m~2k(U)) (к = 0,... ,m).
dtK
Suppose also the following mth-order compatibility conditions hold:
(g0:=ge	91 := f(0) - Lg0 £
I-.., 9m-= 5^(0) - Lgm-i £
Then
€ Z2(0, T- H2m+2-2k(Uf) (к = 0,..., m + 1);
and we have the estimate
(55)
m-f-1
fc=0
dku
dtk
L2(p,T-H2m+2-2k{U))
dkf
dt^
+ ||5||я2’п+1(Р)
L2\0T\H2rrl '2k(U))
< c
the constant C depending only on m, U, T and the coefficients of L.
Remark. Taking into account Theorem 4 in §5.9.2, we see that
f(0) £ H2m~\U), f'(0) £ H2m~\U), ..., f(TO_1)(0) £ H\U),
and consequently
go £ H2m+1(U), gi £ H2m~\U), ..., gm£ H^U).
The compatibility conditions are consequently the requirements that, in
addition, each of these functions equals 0 on dU, in the trace sense. □
Proof. 1. The proof is an induction on m, the case m = 0 being Theo-
rem 5,(i) above.
2. Assume now the theorem is valid for some nonnegative integer m,
and suppose then
dkf
(56) g£H2m+\U),	£L2{0,T-H2m+2~2k(U^ (k = 0,..., m + 1),
and the (m + l)t/l-order compatibility conditions hold. Now set ii := u'.
Differentiating the PDE with respect to t, we check that u is the unique
weak solution of
{ut + Lu = f in Ut
u = 0 on dU x [0,T]
v, = g on U x {t = 0},
366
7. LINEAR EVOLUTION EQUATIONS
for f := ft, g := /(•, 0) — Lg. In particular, for m = 0 we rely upon Theorem
5,(ii) to be sure that й e T2(0, T; H^IU)), й' € Т2(0,Т;Я-1(17)).
Since f and g satisfy the (m + l)t/l-order compatibility conditions, it
follows that f and g satisfy the mi/l-order compatibility condition. Thus
applying the induction assumption, we deduce
HkVl
—Г e T2(0, T; H2m+2"2fc(t/)) (k = 0,..., m + 1)
and
m+1
dku
fc=0
dtk
L2(0,T;H2rn+2~2k(U))
dkf
dtk
L2((>T:H2rrl 2к(Щ)
for f := f'. Since й = и', we can rewrite the foregoing:
e T2(0, T; H2m+i-2k(U}) (k = 1,..., m + 2),
(58)
dku
dtk
L2(0,T;H2rn+4~2k(Uy)
dkf
dtk
+ l|f(0)llH2m+1(tr) + Н^Нн^+Ч^)
L2(0,T;H2m+2~2k(U))
+ 1Ы1я2т+3(С/) I •
L2(0,T;H2m+2-2fc(C/))	/
Here we used the estimate
(59)	||f(0)||n2m+l(C/) < C'(||f||L2(0)T;H2m+2([/)) + ||f/||L2(0T;n2m(C/))),
which follows from Theorem 4 in §5.9.2.
3. Now write for a.e. 0 < t < T: Lu — f - uz =: h. According to
Theorem 5 in §6.3.2, we have
Пи||я2т+4(С7) < C'(||h||H2m+2({/) + ||и||^2([/))
< C(llf||я2тп+2(С7) + Ни/||я2т+2(С7) + IIuIIl2(C/))-
7.1. SECOND-ORDER PARABOLIC EQUATIONS
367
Integrating with respect to t from 0 to T and adding the resulting expression
to (58), we deduce
dku
dtk
Ь2(0,Т-Н2т+*~2к(УУ)
(60)
dfcf	. ..	,, l|2	\
ТД7	+ lldllH2m+3(U) + IIulll2(0,t-,l2(СЛ) I •
at L2(0,T;H2m+2-2k(U)~)	/
Since
IIuI|l2(0,T;L2(C/)) < С(||Г IIl2(0,T;L2(C/)) + 119IIL2 (U) ),
we thereby obtain the assertion of the theorem for m + 1.
THEOREM 7 (Infinite differentiability). Assume
g e C°°(t7), f € C°°(t7T),
and the mtfl-order compatibility conditions hold for m = 0,1,... . Then the
parabolic initial/boundary-value problem (11) has a unique solution
и e C°°(UT).
Proof. Apply Theorem 6 for m = 0,1,... .
As we did for elliptic operators in Chapter 6, we have succeeded in
repeatedly applying fairly straightforward “energy” estimates to produce a
smooth solution of our parabolic initial/boundary-value problem (1). This
assertion requires the compatibility conditions (53) hold for all m, and it is
easy to see that these conditions are necessary for the existence of a solution
smooth on all of Ut-
Remark. Interior estimates, analogous to those developed for elliptic PDE
in §6.3.1, can also be derived, and these in particular do not require the
compatibility conditions.	□
7.1.4.	Maximum principles.
This section develops the maximum principle and Harnack’s inequality
for second-order parabolic operators.
368
7. LINEAR EVOLUTION EQUATIONS
a.	Weak maximum principle.
We will from now on assume that the operator L has the nondivergence
form
n	n
(61)	Lu = - aljuXlXj +	+ cu,
i=l
where the coefficients a’J, bl,c are continuous. We will always suppose the
uniform parabolicity condition from §7.1.1, and also that alJ = a31 (i,j =
1,..., n). Recall also that the parabolic boundary of Ut is Гт = Ur — Ur-
THEOREM 8 (Weak maximum principle). Assume и € Cf (Ut) OC(Ut)
and
(62)	c = 0 in Ut-
(i)	V
(63)	ut + Lu < 0 in Ut,
then
max и = max u.
UT гт
(ii)	Likewise, if
(64)	Ut + Lu > 0 in Ut,
then
minu = minri.
UT гт
Remark. A function и satisfying the inequality (63) is called a subsolution,
and so we are asserting that a subsolution attains its maximum on the
parabolic boundary Гт- Similarly, и is a supersolution provided (64) holds,
in which case и attains its minimum on Г^.	□
Proof. 1. Let us first suppose we have the strict inequality
(65)	Ut + Lu < 0 in Ut,
but there exists a point (яоЛо) € Ut with u(xo,to) = тах.цт и.
2.	If 0 < to < T, then (xo,to) belongs to the interior of Ut and conse-
quently
(66)	ut = 0 at (xo,to),
7.1. SECOND-ORDER PARABOLIC EQUATIONS
369
since и attains its maximum at this point. On the other hand Lu > 0 at
(xo,to), as explained in the proof of Theorem 1 in §6.4. Thus щ + Lu >
0 at (xo,to), a contradiction to (65).
3.	Now suppose to = T. Then since и attains its maximum over Ut at
(xo, to), we see
ut > 0 at (xo,to).
Since we still have the inequality Lu > 0 at (a?o,to), we once more deduce
the contradiction
ut+Lu>0 at (xo,to).
4.	In the general case that (63) holds, write ue(x, t) := u(x, t) — et where
e > 0. Then
u\ + Lue = Ut + Lu — e < 0 in Ut,
and so max(7T u~ = maxrT ue. Let e —» 0 to find тах.цт и = maxrT и. This
proves assertion (i).
5.	As —u is a subsolution whenever и is a supersolution, assertion (ii)
follows at once.	□
Next we allow for zeroth-order terms.
THEOREM 9 (Weak maximum principle for c > 0). Assume и G Cf (Ut)
Г1 C(Ur) and	c > 0 in Ut-
(i) V	ut + Lu < 0 in Ut,
then	max a < maxu+. uT	uT
(h) V	щ + Lu > 0 in Ut,
then	min a > — maxu". Ut	Гт
Remark. In particular, if ut + Lu = 0 within Ut, then
max ltd = max ltd.
UT	It
□
370
7. LINEAR EVOLUTION EQUATIONS
Proof. 1. Assume и satisfies
(67)	Ut + Lu < 0 in Ut
and attains a positive maximum at a point (xq, to) € Ut- Since u(xq, to) > 0
and c > 0, we as above derive the contradiction
ut + Lu > 0 at (xq, to).
2.	If instead of (67), we have only
ut + Lu <0 in Ut,
then as before ue(x, t) := u(x, t) — et satisfies
uft + Lue <0 in Ut-
Furthermore if и attains a positive maximum at some point in Ut, then ue
also attains a positive maximum at some point belonging to Ut, provided
e > 0 is small enough. But then, as in the previous proof, we obtain a
contradiction.
3.	Assertion (ii) follows similarly.	□
Remark. Unlike the situation for elliptic PDE, various versions of the max-
imum principle obtain for parabolic PDE, even if the zeroth-order coefficient
is negative: see Problem 7.	□
b.	Harnack’s inequality.
Harnack’s inequality states that if a is a nonnegative solution of our
parabolic PDE, then the maximum of и in some interior region at a positive
time can be estimated by the minimum of и in the same region at a later
time.
THEOREM 10 (Parabolic Harnack inequality). Assume и G C'i(U’r)
solves
(68)
ut + Lu — 0 in Ut,
and
и >0 in Ut-
7.1. SECOND-ORDER PARABOLIC EQUATIONS
371
Suppose V CC U is connected. Then for each 0 < ti < t? <T, there exists
a constant C such that
(69)	supti(-,ii) < <7infu(-,<2)-
v	v
The constant C depends only on V, t\,t2, and the coefficients of L.
This is true if the coefficients are continuous, or even merely bounded
and measurable; see Lieberman [LB], We will however provide a proof only
for the special case that bl = c = 0 and the aZJ are smooth (г, j — 1,..., n).
The following computations are elementary, but fairly tricky.
Proof*. 1. We may assume и > 0 in Ut, for otherwise we could apply the
result to и + e and then let e —» 0+.
Set
(70)	v := logu in Ut-
Using (68), we compute
n
(71)	vt = 57	+ aljvxiVXj in Ut-
i,j=l
Define
n	n
(72)	w := 52 aiJvxiXj, w := 52 ^^x^xf,
i,j=l	i,j=l
so that (71) reads
(73)	vt = w + w.
2.	We calculate using (72), (73) that for k, I = 1,..., n:
n
Vx^xit = ,ujxkxi "k 5 у	jvXiXkXlvXj + 2a vXiXkvXjxt + R,
i,j=l
where the remainder term R satisfies an estimate of the form
(74)
|Я| < e|D2u|2 + C(e)|Du|2 + C
*Omit on first reading.
372
7. LINEAR EVOLUTION EQUATIONS
for each e > 0. Thus
n
wt= aklvwt + atlvxkxt
k,l=^l
n	n
— 5 Cl 1^xkxi + 2^ ' й 3VxjWXi
к, 1=1
+ 2 5	a ia, vXiXkvx^Xl + R,
i,j,k,l=l
where R now denotes another remainder term satisfying estimate (74).
Therefore choosing e > 0 small enough in (74) and remembering the uniform
parabolicity condition, we discover
n	n
(75)	Wt - 52 aklw^t + 52 bkw^ 02\D2v\2 - C\Dv\2 - C,
k,l=l	k=l
where
(76)	bk :=-2^aklvXl (k = l,...,n).
l=i
3.	Estimate (75) is a differential inequality for w, and our task next is to
obtain a similar inequality for w. Indeed, using (72) and (71) we compute
n	n	Z	n	\
fht ' a wX)cxi — 2 aNvXi (i>tXj	о Vxkxixj J
k,l=l	ij = l	' k,l=l	'
n
2 5 a <1 vXiXkVxjXi T R)
i,j,k,l=l
R yet another remainder term satisfying (74) for each e > 0. Recalling (71)
and (76), we simplify to discover:
n	n
wt — 52a	w^xi	+ 52
(77)	k,l=l	k=l
> -C\D2v\2 - C\Dv\2 -C in UT-
4.	Next set
(78)	w := w + kw,
7.1. SECOND-ORDER PARABOLIC EQUATIONS
373
к > 0 to be selected below. Combining (75) and (77), we deduce
n	n	„2
(79)	wt- ^aklwXkXi+^bkwXk>-\D2v\2-C\Dv\2-C
k,i=i	fc=i
provided 0 < к < | is now fixed to be sufficiently small.
5.	Suppose next V CC U is an open ball and 0 < ti < Q < T. Choose
a cutoff function £ € C°°(Ut) such that
0 < £ < 1, ( = 0on Гт,
(e Ion V X [ii,t2]-
Note that Q vanishes along {t = 0}.
Let p be a positive constant (to be adjusted below), and assume then
(4w + pt attains a negative minimum
at some point (xo,to) E U x (0,T].
Consequently
(82)	£wXk +4(Xkw = 0 at (xo,to) (fc = 1,... ,n).
(81)
In addition
0 > (C4w + pQt - 52 akl(C4w + pt)XkXl at (x0, to)-
k,l=l
Hence at (xo,to):
(83)	0 > p + <4	- £ aktwXkXl) - 2 52 АС4)Л + R,
' k,l=l	' k,l=l
where
(84)	|R| < C<2|w|.
Recall now (79) and (82):
/д2	n	\
0>m + C4 -\D2v\2-C\Dv\2-С-^ь^к +R,
\ 2	fc=i	/
R another remainder term satisfying estimate (84). Utilizing (82) and (76),
we deduce
/Д2	\
(85)	0>M + <4 (-\D2v\2 -C\Dv\2 -С +R,
\ "	z
374
7. LINEAR EVOLUTION EQUATIONS
where now
(86)	|R| <CC2|w| + CC3|r»w||w|.
Remember that (85), (86) are valid at the point (xo, to) where £4w + /j-t
attains a negative minimum. In particular, at this point w = w + kw < 0.
Recalling the definition (72) of w, w, we deduce
(87)	|Z>u|2 < C|Z>2u|,
and so
|w| < C|£>2u| at (xo,to)-
Consequently (86) implies the estimate
(88)	|R| < CC2\D2v\ + CC3|Z»2v|3/2 < eC4|Z»2u|2 + C(e),
where we employed Young’s inequality with e from §B.2. Making use of
(85), (87), (88) we at last discover a contradiction to (81), provided /j, is
large enough.
6.	Therefore (4w + /j-t > 0 in Ut, and so in particular
w + /zt > 0 in V x [ti, Q].
Using (73), we deduce then that
(89)	vt > a|Du|2 - [3 in V x
for appropriate constants a,(3 > 0.
7.	The differential inequality (89) for v = log a now leads us to the
Harnack inequality, as follows. Fix xi,X2 € V, Q > t\. Then
r1 d
v(X2,t2) ~	= / —U(SX2 + (1 - s)xi,St2 + (1 - s)U)ds
Jo ds
= / Dv • (X2 - Xi) + Vt(t2 - tQds
Jo
> [ -|£>v| |X2 - X1| + (<2 - ti)[a|Du|2 - /3] ds by (89)
Jo
>
where 7 depends only upon a, (3, |xi — X2I, |ii — <21- Thus (70) implies
logu(x2,t2) > logu(xi,ii) -7,
and so
«(^2,^2) > e-7u(xi,ti).
This inequality obtains for each xi,X2 € V, and so (69) is valid in the case
V is a ball. In the general case we cover V CC U with balls and repeatedly
apply the estimate above.	□
7.1. SECOND-ORDER PARABOLIC EQUATIONS
375
Parabolic strong maximum principle
c. Strong maximum principle.
Now we employ Harnack’s inequality to establish
THEOREM 11 (Strong maximum principle). Assume и € Cf(UT) Г)
C(UT) and
c = 0 in Ut-
Suppose also U is connected.
(i) If
щ + Lu < 0 in Ut
and и attains its maximum over Ut at a point (xo,to) € Ut, then
и is constant on Ut0.
(ii) Likewise, if
ut + Lu>0 in Ut
and и attains its minimum over Ut at a point (xq, to) € Ut, then
и is constant on Ut0.
Remark. Thus our uniformly parabolic partial differential equations sup-
port “infinite propagation speed of disturbances”.	□
We will for the following proofs assume that и and the coefficients of L
are in fact smooth.
376
7. LINEAR EVOLUTION EQUATIONS
Proof. 1. Assume ut + Lu < 0 in Ut, and и attains its maximum at some
point (xo,to) € Ur-
Select a smooth, open set W CC U, with xq € W. Let v solve
vt + Lv = 0 in Wt
v = и on Дт,
where Дт denotes the parabolic boundary of Wt-
Then by the weak maximum principle и < v. Since
и < v < M,
for M := тах.дт u, we deduce that v = M at (xo,to).
2. Now write v := M — v. Then, since c = 0, we have
(90)	vt + Lv = 0, v > 0 in Wt-
Choose any V CC W with xq € V, V connected. Let 0 < t < to- Then
owing to the Harnack inequality,
(91)	maxn(-,t) < C'infv(-,to).
But infyn(-,to) < n(xo,to) = 0. As v > 0, (91) therefore implies v = 0 on
V x {t}, for each 0 < t < to- This deduction holds for each set V as above,
and so v = 0 in Wt0. But therefore v = M in Wt0. As v = и on Дг, we
conclude и = M on dW x [0, to]-
This conclusion holds for all sets W as above, and therefore и = M on
Ut0-	□
THEOREM 12 (Strong maximinn principle for c > 0). Assume и 6
^(^Пфт) and
c > 0 in Ut-
Suppose also U is connected.
(i) If
щ + Lu < 0 in Ut
and и attains a nonnegative maximum over Ut at a point (zo,to) €
Ut, then
и is constant on Ut0-
(ii) Similarly, if
щ + Lu > 0 in Ut
and и attains a nonpositive minimum over Ur at a point (xo,to) 6
Ut, then
и is constant on Ut0.
7.2. SECOND-ORDER HYPERBOLIC EQUATIONS
377
Proof. 1. As above, set M := max^ и. Assume M > 0, щ + Lu < 0 in
Ur, and и attains this maximum of M at some point (xq,M € Ut-
If M = 0, the foregoing proof directly applies, as then
vt + Lv = 0, v > 0 in Wt-
2.	Assume instead that M > 0. Select as in the previous proof a smooth,
open set W CC U, with xq € W. Now let v solve
vt + Kv = 0
v = u+
in Wt
on Ar,
where
Kv := Lv — cv.
Note 0 < v < M. Since щ + Ku = —cu < 0 on {u > 0}, we deduce
from the weak maximum principle that и < v. As before it follows that
v = (xo,to)-
2.	Now write v := M—v. Then, since the operator К has no zeroth-order
term, we have
vt + Kv = 0, v > 0 in Wt-
Select any V CC W with xq € V, V connected. Let 0 < t < to- Then the
Harnack inequality implies as above that v = u+ = M on dW x [0, to]- Since
M > 0, we conclude that и = M on dW x [0, to].
This deduction is valid for all sets W as above, and therefore и = M on
Ut0-
7.2. SECOND-ORDER HYPERBOLIC EQUATIONS
Second-order hyperbolic equations are natural generalizations of the wave
equation (§2.4). We will build in this section appropriately defined weak
solutions, and study their uniqueness, smoothness and other properties. It
is interesting, given the utterly different physical character of second-order
parabolic and hyperbolic PDE, that we can provide rather similar functional
analytic constructions of weak solutions.
7.2.1.	Definitions.
a.	Hyperbolic equations.
As in §7.1 we write Ut = U x (0,T], where T > 0 and U C R” is an
open, bounded set.
378
7. LINEAR EVOLUTION EQUATIONS
We will next study the initial/boundary-value problem
(1)
' иц + Lu = f in Ut
< и = 0 on dU x [0, T]
yn = д,щ = h on U x {t = 0},
where f : Ut —* R, g, h : U —* R are given, and и : Ut —* R is the unknown,
и — u(x,t). The symbol L denotes for each time t a second-order partial
differential operator, having either the divergence form
(2)	Lu = -	+'^^b\x,t')uXi + c(x,t)u
i,j=l	i=l
or else the nondivergence form
n	n
(3)	Lu = - atJ (x, t)uXiXj + У Ьг(х, t)uXi + c(x, t)u
i,j=l	i=l
for given coefficients a^,bl,c (iff = 1,...,ri).
DEFINITION. We say the partial differential operator + L is (uni-
formly) hyperbolic if there exists a constant в > 0 such that
n
(4)
i,J=l
for all (x,t) € Ut, £ € Rn.
If au = hij, Ьг = c = f = 0, then L = —A and the partial differential
equation becomes the wave equation, already studied in Chapter 2. General
second-order hyperbolic PDE model wave transmission in heterogeneous,
nonisotropic media.
b.	Weak solutions.
As before, in §6.1.2 and §7.1.1, we first assume L has the divergence
form (2) and look for an appropriate notion of weak solution for problem
(1). We will suppose initially that
(5)	а1\Ь\сЕС\ит) (i,j	= l,...,n),
(6)	f € L2(UT),
(7)	geH^(U), h<EL2(U),
and always assume au = aJl (iff = 1,... ,ri).
As in §7.1.1, let us also introduce the time-dependent bilinear form
(8)	B[u,v;t]:= / У^ alj(-,t)uXivXi + b*(-,t)uXiv + c(-, t)uv dx
Ju i,j=l	i=l
for и, v € Hq (tZ) and 0 < t < T.
7.2. SECOND-ORDER HYPERBOLIC EQUATIONS
379
Motivation for definition of weak solution. We begin by supposing и =
u(x, t) to be a smooth solution of (1) and defining the associated mapping
u : [0, T] —► Я()(Я),
by
[u(t)](x) := u(x,t) (x € U, 0 < t < T).
We similarly introduce the function
f : [0,T]	L2(t7)
defined by
[f(t)](x) := /(x,t) (x € U, 0 < t < T).
Now fix any function v E multiply the PDE utt + Lu = f by v,
and integrate by parts, to obtain the identity
(9)	(u", v) + 2?[u, v; £] = (f, v)
for 0 < t < T, where ( , ) denotes the inner product in L2(U). Almost
exactly as in the parallel discussion for parabolic PDE in §7.1.1, we see from
the PDE u^ + Lu = f that
Utt = +
1=1
for g° := f - £"=1 bluXi - cu and g3 := £"=i (j = 1,... ,n). This
suggests that we should look for a weak solution и with u" G H'^U) for
a.e. 0 < t < T, and then reinterpret the first term of (9) as (u",v), { , )
denoting as usual the pairing between Я-1(С7) and Hq(U).	□
DEFINITION. We say a function
и G Z2(0, T; Яд(С7)), with u' G Z2(0, T; Z2(£/)), u" G Z2(0, Г; Я-1 (£/)),
is a weak solution of the hyperbolic initial/boundary-value problem (1) pro-
vided
(i)	(u", v} + B[u, v; t] = (f, v)
for each v G Hq(U} and a.e. time 0 < t <T, and
(ii)	u(0) = g, u'(0) = h.
380
7. LINEAR EVOLUTION EQUATIONS
Remark. In view of Theorem 2 in §5.9.2, we know u € C([0, T]; L2(I7))
and u' € C([0,T]; 7/-1(f7)). Consequently the equalities (ii) above make
sense.	□
7.2.2.	Existence of weak solutions.
a.	Galerkin approximations.
By analogy with the approach taken in §7.1.2 we will construct our weak
solution of the hyperbolic initial/boundary-value problem
(10)
utt + Lu — f
и = 0
и = д,щ = h
in Ur
on dU x [0, T]
on U x {t = 0}
by first solving a finite dimensional approximation. We thus once more
employ Galerkin’s method by selecting smooth functions Wk = Wfc(s) (k =
1,...) such that
(11)	is an orthogonal basis of Hq(U)
and
(12)	{wfc}fc^=i is an orthonormal basis of T2(tZ).
Fix a positive integer m, and write
m
(13)	Um(t) ' =
fc=l
where we intend to select the coefficients d^ft) (0 < t < T, к = 1,..., m)
to satisfy
(14)	dkm(U) = (g,wk) (к = l,...,m),
(15)	d^'(0) = (/i,wfc) (k = l,...,m),
and
(16)	(u^, wfc) + B[um,wk-, t\ = (f, wk) (0 < t < T, к = 1,..., m).
THEOREM 1 (Construction of approximate solutions). For each integer
m = 1,2,..., there exists a unique function um of the form (13) satisfying
(14)-(16).
7.2. SECOND-ORDER HYPERBOLIC EQUATIONS
381
Proof. Assuming um to be given by (13), we observe using (12)
(17)	«, (<),»,.)=
Furthermore, exactly as in the proof of Theorem 1 in §7.1.2, we have
= ^ekl(t)dlm(t)
i=i
for efc/(t) := B[wi,Wk',t\ (k,l = l,...,m). We also write /fe(t) := (f(t),Wfc)
(k = 1,... , m). Consequently (16) becomes the linear system of ODE
m
(18)	4"(t) + £efc/(t)4(t) = /fc(t) (0<t<T, fc = l,...,m),
/=i
subject to the initial conditions (14), (15). According to standard theory for
ordinary differential equations, there exists a unique C2 function dm(t) =
(4(t),..., d^ftf), satisfying (14), (15), and solving (18) for 0 < t < T.
□
b.	Energy estimates.
Our plan is hereafter to send m —► oo, and so we will need some esti-
mates, uniform in m.
THEOREM 2 (Energy estimates). There exists a constant C, depending
only on U,T and the coefficients of L, such that
max
0<t<T
(19)

for m = 1,2,....
Proof. 1. Multiply equality (16) by 4(t), sum k =	and recall
(13) to discover
(20)	(i4,14) + B[um, i4; t] = (f, i4)
for a.e. 0 < t < T. Observe next (t4,i4) =	(|||i4n. Ilb2(cz))• Furthermore,
we can write
7?[um,
n
>2 alJum,Xiu'm dx
(21)
f П
+ / У2 blum,Xiu'm + cumi4 dx
Jui^i
=’• Bi + B2.
382
7. LINEAR EVOLUTION EQUATIONS
Since	(i,j = 1,... ,n), we see
Z1	\	i /* . л
(22)	| —A[um, иш; t] I — / \ J Um,xi^m,xj dx,
dt\2	J	2Ju~\
for the symmetric bilinear form
г n
A[u, v,t] := / У2 al-'uXivXj dx (u,v € Hq(U)}.
The equality (22) implies
(23)	Bi > — A[um, um; — С||ит||^1^р
and we note also
(24)	|B2|< Cdlu^H^
(U) + HUmllz,2((7)) •
Combining estimates (20)-(24), we discover
(llumIIL2(U) +^lum:Um^]j < C ^||umIIl2((7) + IIUmII#*(U) + Н^11л2(£/))
(25)	< C ^||umllL2(t/) + ^4[um> um51] + ||f Ц/,
where we used the inequality
(26)	0 [ \Du\2dx<A[u,u-t] (ueH^U)),
Ju
which follows from the uniform hyperbolicity condition.
2.	Now write
(27)	7?(t) :=	+ A[um(t), um(t); t]
and
(28)	C(t) := ||f(OlU=(t/)-
Then inequality (25) reads
ri\t) <01^ + 02^
7.2. SECOND-ORDER HYPERBOLIC EQUATIONS
383
for 0 < t < T and appropriate constants C\,C2- Thus Gronwall’s inequality
(§B.2) yields the estimate
(29)	7?(t) <eClt ^(0) + С2	(0<t<T).
\ Jo /
However
*1(0) = ||u^(0)||^2(t/) + A[um(0),um(0);0]
- C	+ ||9|1н1(сг)) >
according to (14) and (15) and the estimate ||u7n(0)||/fi(t/) <	• Thus
formulas (27)-(29) provide the bound
llum(*)lll(t/) + A[um(t),Um(t);t]
< c (||5'11я1((у) + 11Л1112(С7) + llf 1112(0,T;L2(17))) 1
Since 0 < t < T was arbitrary, we see from this estimate and (26) that
max (llum(t)|lli(l/) + iKWIll^))
< ^(1191111(17) + ll^lll2(t/) + llf llb(0,T;L2(77))) •
3. Fix any v 6 Hq(U), HvUhi^) < 1, and write v = v1 + v2, where
v1 € spanfwA:}^! and (v2,Wfc) = 0 (k =	Note ||v1||h(‘(c') — 1-
Then (13) and (16) imply
= (u^,v) = (u'^.v1) = (f.n1) -
Thus
|«,n)| < C'(||f||i2(t/) + ||um||Hi(t7)),
since ||v1||#i(t/) < 1. Consequently
llumlll-l(77)^ <	I|f|ll2(t7) + llum|lll((7)dt
- ^(1191111(77) + НЛ'П12(7У) + Hf 111(0,T;L2(t7)))’
384
7. LINEAR EVOLUTION EQUATIONS
c. Existence and uniqueness.
Now we pass to limits in our Galerkin approximations.
THEOREM 3 (Existence of weak solution). There exists a weak solution
Proof. 1. According to the energy estimates (19), we see that the se-
quence {um}“=1 is bounded in Z2(0, T\ Hq(U)), {u'm}m=i is bounded in
L2(0,T;Z2(t/)) and {u"}^=1 is bounded in Z2(0,T; Я-1 (£/)).
As a consequence there exists a subsequence {umJ'^j С {иш}“=1 and
u € L^^.T-HKU)), with u' € Z2(0,T;Z2 (£/)), u" € £2(0,Т;Я-х(Я)),
such that
(30)
um; —> u weakly in Z2(0,T;Hq(U))
< u^ — u' weakly in Z2(0, T; L2(UY)
k u"( — u" weakly in Z2(0, T;
2.	Next fix an integer N and choose a function v € C'1([0,T]; Яд(С7)) of
the form
N
(31)	v(t) = '}Pdk(t)wk,
fc=i
where {dk}^=1 are smooth functions. We select m > N, multiply (16) by
dfc(t), sum к = 1,..., N, and then integrate with respect to t, to discover
(32)	[ (u",v) + B[um,v;t\dt = [ (f,v)dt.
Jo	Jo
We set m = mi and recall (30), to find in the limit that
(33)	i (uz/, v) + 2?[u, v;t] dt = f (f,v)dt.
Jo	Jo
This equality then holds for all functions v € Z2(0,T; Яд (£/)), since
functions of the form (31) are dense in this space. From (33) it follows
further that
(uz/, v) + 2?[u, v, t] = (f, v)
for all v 6 Hq{U} and a.e. 0 < t < T. Furthermore, u € C([0, T]; Z2(t/))
andu'eC'([0,T];ff-1(t7)).
3.	We must now verify
(34)	u(0) = g,
7.2. SECOND-ORDER HYPERBOLIC EQUATIONS
385
(35)	u'(0) = h.
For this, choose any function v € C2([0,T]; Hq(U)), with v(T’) = v'(T) = 0.
Then integrating by parts twice with respect to t in (33), we find
f (v",u) + B\u,v;t\dt = f (f,v)dt
(36)	Jo	Jo
-(u(0),v'(0)) + (u'(0),v(0)).
Similarly from (32) we deduce
T	rT
(v",uTn)4-B[uTn,v;t]dt= / (f,v)dt
Jo
- (um(0),v'(0)) + (14(0), v(0)).
We set m = mi and recall (14), (15) and (30), to deduce
(37)	[ (v",u) + _S[u, v;t] dt = [ (f, v) dt - (g, v'(0)) + (Л, v(0)).
Jo	Jo
Comparing identities (36) and (37), we conclude (34), (35), since v(0), v'(0)
are arbitrary. Hence u is a weak solution of (1).	□
Remark. Recalling the energy estimates from Theorem 2, we observe that
in fact u G L°°(0,T; u' € Z°°(0, T; L2(t/)), u" G Z2(0,T; Я'1^)):
see Theorem 5 below.	□
THEOREM 4 (Uniqueness of weak solution). A weak solution of (1) is
unique.
Remark. The following tricky demonstration would be greatly simplified
if we knew u'(t) itself were smooth enough to insert in place of v in the
definition of weak solution. This is not so, however.	□
Proof. 1. It suffices to show that the only weak solution of (1) with f =
g = h = 0 is
(38)
u = 0.
To verify this, fix 0 < s < T and set
v(t)
ft u(r) dT
0
if 0 < t < s
if s <t <T.
386
7. LINEAR EVOLUTION EQUATIONS
Then v(i) € Hq(U) for each 0 < t < T, and so
[ {u", v) + 2?[u, v; f] dt = 0.
Jo
Since u'(0) = v(s) = 0, we obtain after integrating by parts in the first term
above:
(39)	f-(u',v') Jo Now v' = —u (0 < t < s), and so / (u',u)- Jo	+ 2?[u, v; t] dt = 0. - B[v', v; t] dt = 0.
Thus where C[u,v;t] := and for u,v € Hq(U). Hence i||o(»)lli=(v) + |-B[v(0) Л1	& and consequently l|u(S)||2L2(t/) + ||v(0)||2 <40) 2. Now let us write w(t) :=	r,v;t]\dt = — [ C[u,v;t] + D[v,v;t]dt, /	Jo f ж” л	1 / \2blvXiu+ -bi^uvdx 71 aij,tuXivXi + bi,tuXiv + ctuv dx, =1	i=i ,v(0);t] = — [ C[u,v; t] + £>[v, v; t] dt, Jo W 0 HVHhq((7) + uIl2(^)^ + llv(0)H22(t/)^ • f u(r)dT (0 < t < T);
7.2. SECOND-ORDER HYPERBOLIC EQUATIONS
387
whereupon (40) becomes
llu(s)llb(t/) + llw(s)lltfi(t/)
(41)	<c(J ~ W(S)IIh1(C7) + llu(f)llb(t/) dt + IIw(s)llb2(C7)) •
But ||w(i)-w(s)||2fi(t/) < 2||w(i)||^i(t/)+2||w(s)||^i(t/), and ||w(s)||b2(c/) <
||и(£)||Ь2(^)£/£. Therefore (41) implies
IIu(s)IIl2({/) + (i - 2sCi)||w(s)||^1{[/) < Cl IIw||^1(t/) + ||u||22{U)dt.
Choose Ti so small that
1-2T1C1 > i
Li
Then if 0 < s < Ti, we have
llu(s)llz,2(t/) + l|w(s)||^i(t/) < C ||u||22(l/) + ||w||^i(t/)dt.
Consequently the integral form of Gronwall’s inequality (§B.2) implies u = 0
on [0,Ti],
3.	We apply the same argument on the intervals [7i,27i], [27i,37i],
etc., eventually to deduce (38).	□
7.2.3. Regularity.
As in our earlier treatments of second-order elliptic and parabolic PDE,
the next task is to study the smoothness of our weak solutions.
Motivation: formal derivation of estimates, (i) Suppose for the mo-
ment и = u(x, t) is a smooth solution of this initial-value problem for the
wave equation:
/ иц — Aw = f in Rn x (0, T]
( и — д,щ = h опГ x{< = 0},
and assume also и goes to zero as |x| —> oo sufficiently rapidly to justify the
following calculations. Then as in §2.4.3, we compute
j- ( [ |Z>?z|2 + dx j — 2 [ Du  Dut + щи^ dx
dt	/ Jrh
= 2
388
7. LINEAR EVOLUTION EQUATIONS
Applying Gronwall’s inequality, we deduce
(42)	sup f \Du\2 + u2 dx < c( [ [ ft dxdt + f \Dg\2 + h2 dx
0<t<T JR"	\Jo JR"	JR"
with the constant C depending only on T.
(ii)	Next differentiate the PDE with respect to t and set й := щ. Then
(	— Дй = f in Rn x (0, T]
\ й — g, щ = h on R" x {t = 0},
for f := ft, g := h, h := ии(-,(У) = /(-,0) + Д#. Applying estimate (42) to
u, we discover
(44)
sup I \Dut\2 + u2tdx
0<t<T JR"
<c( [ [ ft dxdt + [ \D2g\2 + \Dh\2 + f(-,0)2dx
\J0 JR"	JR"
Now
(45) шадс ||/(-,4)||1,2(кП) < C'(||/||L2(5{nx(ojT)) + ||/t||L2(R"x(o,7’))),
according to Theorem 2 in §5.9.2. Furthermore, writing —Да = f — utt, we
deduce as in §6.3 that
(46)

for each 0 < t < T. Combining (44) (46), we conclude
(47)
sup /
0<t<T JR’

the constant C depending only on T.
□
This estimate suggests that bounds similar to (42) and (47) should be
valid for our weak solution of a general second-order hyperbolic PDE.
We will calculate using the Galerkin approximations. To simplify the
presentation, we hereafter assume that {wfc}^Sx is the complete collection
of eigenfunctions for — Д on Hq(U), and also that U is bounded, open, with
dU smooth. In addition we suppose
the coefficients aN,Ьг,с (i,j = 1,... ,n) are smooth on
U and do not depend on t.
7.2. SECOND-ORDER HYPERBOLIC EQUATIONS
389
THEOREM 5 (Improved regularity).
(i) Assume
gEH^U), h€L2(U), fEL2(0,T;L2(U)),
and suppose also u G Z2(0, T;	with u' € L2(0, T;L2(Uf), u" G
L2(0, T; H~A(£/)), is the weak solution of the problem
(49)
' utt + Lu = f
и = 0
и = g, щ = h
in Ur
on dU x [0, T]
on U x {t = 0}
Then in fact
u G Z°°(0, T;	u' G L°°(0, T- L2(Uf),
and we have the estimate
ess sup (||u(t)||Я1(1/) + ||u'(t)||L2(t/))
(50)
C(l|f|lL2(0,T;Z.2(t/)) + Нй'Пя^С/) + ll^lk2(t/))-
(ii)	If, in addition,
gEH2(U\ heH^U), ff G Z2(0,T;Z2(C/)),
then
u G Л°°(0,Т;Я2(С/)), u' € Z°°(0, T;
u" G Z°°(0,T;Z2(t/)), u"' G ^(O.TjH-1^)),
with the estimate:
eo<t<r	+	+ 1|ч"(*)11ь2(сп)
(51)
+ lluWПл2(0,Т;Я-1(СГ)) < COIfll#1 (0,Т;Л2(СГ)) + Пй'Ня2^) + НМя1^))-
Remark. Assertions (i), (ii) of this theorem are precise versions of the
formal estimates (42), (47) (for the wave equation in U = R”).	□
390
7. LINEAR EVOLUTION EQUATIONS
Proof. 1. In the proof of Theorem 2, we have already derived the bounds
sup (||um(t)||m(t7) + ||i4(t)||L2(t/))
(52)	Q^T
C(l|fIL2(0,T;L2(t/)) + 1|5|1я1(17) + II^11L2(U))•
Passing to limits as m = mi —► oo, we deduce (50).
2.	Assume now the hypotheses of assertion (ii). Fix a positive integer m,
and next differentiate the identity (16) with respect to t. Writing fim := u'm
we obtain
(53)	(u^,wfc) + B[iim,wfc] = (f',wfc) (k = l,...,m).
Multiplying by dm" (t) and adding for к — 1,..., m, we discover
(54)	(u^, i4) + B[um, u^] = (f', i4).
Arguing as in the proof of the energy estimates, we observe
(55)	^(ll^mllz,2(tz) + -^[йгп) Um])
5= C(ll“inlll,2(i7) + А[йш, um] + llf'll^^)),
the bilinear form A[ , ] defined as before.
3.	Now
(56)	B[um,wfe] = (f- u" ,wk) (k - l,...,m).
Recall we are taking {w)b=i to be the complete collection of eigenfunctions
for —Д on Hq(U). Multiplying (56) by Afcd^(t) and summing к = 1,..., m,
we deduce
(57)	Bfu™, -Дит] = (f - и" , -Aum).
Since Дит = 0 on dU, we have
(58)	ZJ[um, Дит] = (Tum, Дит).
Next we employ the inequality
(59)	/?||u|&2([Z) < (Lu, -Nu) + 7l|u||2L2(tz) (u G H2(U) П Д»);
see Problem 8. We deduce from (56)-(59) that
(60)	llumll#2(t/) — C(ll^llz,2((7) + llum llz,2((7) + IIй™ lli 2(t/))'
7.2. SECOND-ORDER HYPERBOLIC EQUATIONS
391
Using this estimate in (55), recalling iim = u(„, and applying Gronwall’s
inequality, we deduce
oSuH||um(t)||^([;) + lluUO IIh^) +
— С’СН^Ня1 (0,T;L2((7)) + IMI#2(!7) + II^IIhI(IZ))-
Here we estimated ||um(0)||/p(c') < С’||5'||я2(1/), 38 in the proof of Theorem
5 in §7.1.3.
Passing to limits as m = mi —► oo, we derive the same bound for u.
4.	As in the earlier proof of Theorem 5 in §7.1.3, we likewise deduce
uw € T2(0, T; H-1(U)), with the stated estimate.	□
Remark. If L were symmetric, we could alternatively have taken {wk}'^=1
to be a basis of eigenfunctions of L on Hq(U), and so avoided the need for
inequality (59).	□
Now let m be a nonnegative integer.
THEOREM 6 (Higher regularity). Assume
( gEHm+1(U), hEHm(U),
t €Z2(0,T;^m'fe(t/)) (k = Q,...,m).
Suppose also the following mtfl-order compatibility conditions hold:
(62)
' go := g G H'(U), hr := h & H^(U),...,
92! ••= §5T?f(-,0) - L92/-2 € Ho(^) (if m = 21)
. /12/+1 :=fi^(-,0)-Z/l2/_1 6^(1/) (if m = 2Z + l).
Then
(63)
G L°°(0, T- Нт+1~к(иУ) (к = 0,... ,m + 1),
and we have the estimate
(64)
m+l
ess sup
0<t<T ьа
— — fc=0
dku
dtk
dkf
dtk
+ IIsIIh^+h^) +	I •
L2(0,T;H”l-fc((7))	/

392
7. LINEAR EVOLUTION EQUATIONS
Remark. In view of Theorem 2 in §5.9.2, we see that
(65)	f(о) eHm^(u), f'(o) e Hm~2(U),f(m“2)(o) e H\u),
and consequently
go e	hi e нт(и), g2 e Hm~\u), h3 e ят-2(с/),
66)	...,g2ieH\W) (if m = 2Z), h2l+1 € H\U) (ifm = 2Z + l).
The compatibility conditions are consequently the requirements that, in
addition, each of these functions equals 0 on dU, in the trace sense. □
Proof. 1. The proof is by an induction, the case m = 0 following from
Theorem 5,(i) above.
2. Assume next the theorem is valid for some nonnegative integer m,
and suppose
( g e Hm+2(U), h € Hm+1(U),
i eL2(0,T-Hm+1~k(U)) (k = 0,... ,m + 1).
Suppose also the (m + l)tft-order compatibility conditions obtain. Now set
ii := u'. Differentiating the PDE with respect to t, we check that ii is the
unique, weak solution of
(68)
utt + Lu = f
й = 0
й = g, щ = h
in Ut
on dU x [0, T]
on U x {t = 0},
for
(69)
ft, g:=h, h:=f(-,0)-Lg.
In particular, for m = 0 we rely upon Theorem 5,(ii) to be sure that ii €
L2(0,T'Hq(U)), u'eI2(0,T;Z2(Z/)), u" € £2(0,Т;Я-1(С7)).
Since f,g and h satisfy the (m + l)4/i-order compatibility conditions,
f,g and h satisfy the mth-order compatibility conditions. Thus applying
the induction assumption, we see
—r- e Z°°(0, T-	(k = 0,..., m + 1),
dtK
with the estimate
m+l
ess sup 2_.
o<t<T
dku
dtk

m
s
k=0
dkf
dtk
+ Н5||я”‘+1 (U) + НМ1я’п(£/) 1 •
7.2. SECOND-ORDER HYPERBOLIC EQUATIONS
393
Since й = uz, we can rewrite:
m+2
(70) ess sup 2J
o<t<r
dkf
dtk
’m+1
fc=l
dku
dtk
+ ||Л||ят+1(17) + ||1'5'||я’п(С/) + ||^(0)||я™(17)
L2(0,T;Wm+lfc((/))	,
Zm+1
k=0
dkf
dtk
+	+ II^IIh^+vc/) I •
Ят+2- (=([/)
Here we used the inequality
l|f llc([0,T];H’"(t/)) < С'(||Г|к2(0,Т;Ят(£/)) + 11 f' 11L2	,
which follows from Theorem 2 in §5.9.2.
3. Now write for a.e. 0 < t < T: Lu = f — u" =: h. We have
||и||ят+2((7) < ССНЬПя™^) + IIuIIl2((7))
< С'(П^||ят((У) + Пи,/||я™((7) + IIuIIl2((7))’
Taking the essential supremum with respect to t, adding this inequality to
(70) and making standard estimates, we deduce:
m+2
ess sup 2
dku
dtk
Hm+2~k(U)
dkf
dtk
+ ||0||ят+2(17) + ||^11я™+1(£/) I 
L2(0,T;H’"+1-fc(t/))	/
This is the assertion of the theorem for m + 1.
□
THEOREM 7 (Infinite differentiability). Assume
g,heC°°(U), fEC°°(UT),
and the mtfl-order compatibility conditions hold for m = 0,1,... .
Then the hyperbolic initial/boundary-value problem (1) has a unique so-
lution
и € C°°(Ur).
Proof. Apply Theorem 6 for m = 0,1,... .
□
394
7. LINEAR EVOLUTION EQUATIONS
(XQ,tQ)
Domain of dependence
7.2.4. Propagation of disturbances.
Our study of second-order hyperbolic equations has thus far pretty much
paralleled our treatment of second-order parabolic PDE, in §7.1. In the
corresponding earlier section §7.1.4, we discussed maximum principles for
second-order parabolic equations, and noted in particular that the strong
maximum principle implies an “infinite propagation speed” of initial distur-
bances for such PDE. Now strong maximum principles are false for second-
order hyperbolic partial differential equations, and we will instead address
here the opposite phenomenon, namely the “finite propagation speed” of
initial disturbances. This study extends some ideas already introduced in
§2.4.3.
For simplicity we will consider in this section the case U — Rn and L
has the simple nondivergence form
n
(71)	Lu = - ^a^uXiXj,
i,j=l
where the coefficients are smooth, independent of time, and there are no
lower-order terms. We as usual require the uniform hyperbolicity condition.
Let us assume now a is a smooth solution of the PDE
(72)	utt + Lu = 0 in Rn x (0, oo).
We wish to prove a uniqueness/finite propagation speed assertion analogous
to that obtained for the wave equation in §2.4.3. For this, we fix a point
(x0, t0) € Rn x (0, oo), and then try to find some sort of a curved “cone-like”
region C, with vertex (xo,to), such that и = 0 within C if и = ut = 0 on
Co = Cn{t = 0}.
7.2. SECOND-ORDER HYPERBOLIC EQUATIONS
395
Motivated by the geometric optics computation in Example 3 of §4.5.3,
let us guess that the boundary of such a region C is given as a level set
{p = 0}, where p solves the Hamilton-Jacobi PDE
(73)
/ n	X 1/2
Pt ~ ( 52 ^4x)PxtPx,] =0
in Rn x (0, oo).
We will simplify (73) by separating variables, to write
(74)	p(x, t) = g(x) + t — to (x € Rn, 0 < t < to),
where q solves
Г E"j=i а^Чхг<1х3 = 1, q > 0 in Rn — {x0}
l = 0.
We henceforth assume that q is a smooth solution of (75) on Rn — {xq}.
(In fact g(x) = distance of x to xq, in the Riemannian metric determined
by ((av)).) We write
C := {(x, t) | p(x, t) < 0} = {(x, t) | q(x) < t0 - t}.
For each t > 0, we further define
(76)	Ct := {x | g(x) < to — t} = cross section of C at time t.
Since (75) implies Dq 7^ 0 in R" — {xq}, dCt is a smooth, (n— l)-dimensional
surface for 0 < t < to-
THEOREM 8 (Finite propagation speed). Assume и is a smooth solution
of the hyperbolic equation (72). Ifu = ut = 0 on Co, then и = 0 within the
cone C.
Remark. We see in particular that if и is a solution of (72) with the initial
conditions
(77)	и = у, ut = h on Rn x {t = 0},
then u(xq, to) depends only upon the values of g and h within Co- □
Proof. 1. We modify a proof from §2.4.3, and so define the energy

^Xi^Xj dx (0 < t < t0)-
396
7. LINEAR EVOLUTION EQUATIONS
2.	In order to compute e(t), we first note that if f is a continuous
function of x, then
[ fdx] = - [ r^-jdS
di \Jct / Jdct 1-^91
according to the coarea formula from §C.3. Thus
f	"
e(t) — / ututt И- 5 ciN'iLxyiLXjtdx
JCt	i,j=l
1 f ( 2 Л Ц \ 1
---/	гь +	> aJuxux , .	dS
2JdCt\ fa / \Dq\
ilj-1
=: A — B.
Integrating by parts, we calculate
r	/	"
A = / uj utt - 52 (aljuXi}
Jc‘	V	ij=i
n
>2 ах,и^dx+
n
aV uXtiV щ dS
(79)
n
V' a^Ux^utdS,
with p = (i/1,..., z>n) being as usual the outer unit normal to dCt. But
according to the generalized Cauchy-Schwarz inequality (§B.2)
n	\ 1/2 / П	ч 1/2
l/lP |	.
71
(80)
»J=1
In addition, since q = t$ — t on dCt, we have p = Й1 on dCt- Hence
a i j v4 0,1 J4xi4xj	1
by (75). Consequently inequality (80) reads
•• \1/2 1
L alJuxtUxi ) TTTj-
(81)
52 aNuXiij
i,j=l
ij=i
Then returning to (79), we estimate using (81) and Cauchy’s inequality:
f (\ V2 1
|Л| < Ce(t) + / I a4ux^Xj I	|ut| —— dS
JdCt	/	\L>q\
< Ce(t) + | [ (и? + £ alJuxiuxJ'} dS
2 JdCt \	J 1^91
= Ce{t) + B.

7.2. SECOND-ORDER HYPERBOLIC EQUATIONS
397
3.	Therefore inequality (78) gives
e(t) < Ce(t).
Since hypothesis (76) implies e(0) = 0, we deduce using Gronwall’s inequal-
ity that
e(t) =0 for all 0 < t < to-
Hence щ = Du = 0 in C, and consequently и = 0 in C.
□
7.2.5. Equations in two variables.
In this section we briefly consider second-order hyperbolic partial differ-
ential equations involving only two variables, and demonstrate that in this
setting rather more precise information can be obtained. The very rough
idea is that since a function of two variables has “only” three second par-
tial derivatives, algebraic and analytic simplifications in the structure of the
PDE may be possible, which are unavailable for more than two variables.
We begin by considering a general linear second-order PDE in two vari-
ables
2	2
(82)	52 a'iuWj + 52	+cu = 0’
*	i,j=l	i=l
where the coefficients aN,bl,c (i,j = 1,2), with = aJl, and the unknown
и are functions of the two variables xi and x? in some region U C R2. Note
that for the moment, and in contrast to the theory developed above, we do
not identify either xi or x? with the variable t denoting time.
We now pose the following basic question: Is it possible to simplify the
structure of the PDE (82) by introducing new independent variables? In
other words, can we expect to convert the PDE into some “nicer” form by
rewriting in terms of new variables у = Ф(х)?
More precisely, let us set
, .	Г У1 = Ф1(х1,х2)
l У2 = Ф2(Х1,Х2)
for some appropriate function Ф = (Ф\Ф2). To investigate this possibility
let us now write
(84)
u(x) = и(Ф(х)).
398
7. LINEAR EVOLUTION EQUATIONS
That is, we define v(y) := и(Ф(у\), where Ф = Ф \
From (84), we compute
f uXi —	vVk^Xi
I UxiXj	^VkUl^Xi^Xj d" Sfc=l vyk^x,Xj
for i,j — 1,2. Substituting into (82), we discover v solves the PDE
(85)	5 a Vykyi +•••=(),
fc,/=i
for
2
(86)	akl := 52 °Офх.ф^ (k’1 = 1’2)’
i,j=l
where the dots in (85) represent terms of lower order.
We examine the first term in the PDE (85) in the hope we can perhaps
choose the transformation Ф = (Ф^Ф2) so this expression is particularly
simple. Let us try to achieve
(87)	a11 = a22 = 0.
In view of formula (86) this will be possible provided we can choose both
Ф1 and Ф2 to solve the nonlinear first-order PDE
(88)	aU(vxi)2 + 2a12vX1vX2 + a22(yX2)2 = 0 in U.
Observe this is the characteristic equation associated with the partial differ-
ential equation (82), as discussed in §4.6.2.
To proceed further, let us suppose
(89)	det A = апа22 — (a12)2 < 0 in U-,
in which case we say the PDE (82) is hyperbolic.
Utilizing condition (89), we can then factor equation (88) as follows:
(«4, + p2 + ((a12)2 - all«22)1/2] »„)
(»)		+ [a12 - ((a12)2 - ana22)1/2] »12)
= a11 (an(vxl)2 + ZanvnvX2 + a22(vI2)2) = 0.
7.2. SECOND-ORDER HYPERBOLIC EQUATIONS
399
Now the left hand side of (90) is the product of two linear first-order
PDE:
(91i)	a11vXl + [fll2 + ((a12)2 — a11**22)1/2] vX2 = 0 in U
and
(912)	anvX1 + |a12 - ((a12)2 — a11^2)1^2] vX2 —Q in U.
We now assume that we can choose Ф1 to be a smooth solution of the
PDE (91i), with .ОФ1 7^ 0 in U. Then Ф1 is constant along trajectories
x = (ж1, x2) of the ODE
J x1 = a11
(92)	( x2 = [a12 + ((a12)2 - ana22)1/2] .
Similarly, suppose Ф2 is a smooth solution of (91г), with D$2	0 in U;
then Ф2 is constant along trajectories x = (x1,^2) of the ODE
J x1 = a11
(93)	1 x2 = [a12 - ((a12)2 - ana22)1/2] .
Curves which are trajectories of either the ODE (92) or (93) are called
characteristics of the original partial differential equation (82). Returning
now to (83) we see that trajectories of solutions of the characteristic ODE
(92) and (93) provide our new coordinate lines.
Additionally we can verify using (89) that
2
(94)	a12 = 52	° in U-
M=1
Combining then (85), (86), (87) and (94), we see that our PDE (82)
becomes in the у coordinates
(95)	+ • • • = 0,
the dots as before denoting terms of lower order. Let us call equation (95)
the first canonical form for the hyperbolic PDE (82).
If we change variables again by setting zi = yi + 3/2, ^2 = У1 ~ У2, then
(95) becomes
(96)	wZlZ1 - wZ2Z2 +    = 0.
400
7. LINEAR EVOLUTION EQUATIONS
If we then further rename the variables t = zi, x = Z2, then (96) reads
(97)	wtt - wxx + • • • = 0,
the second-order term of which is the one-dimensional wave operator. Equa-
tion (97) is the second canonical form.
Hence any hyperbolic PDE in two variables of the form (82) can be
converted by a change of variables into the wave equation plus a lower order
term, assuming we can find the mapping Ф as above.
7.3. HYPERBOLIC SYSTEMS OF FIRST-ORDER
EQUATIONS
We next broaden our study of hyperbolic PDE (which we may informally
interpret as equations supporting “wave-like” solutions) to the case of first-
order systems. We continue in the manner of §§7.1 and 7.2 by first employing
energy bounds to construct weak solutions for symmetric hyperbolic sys-
tems. For nonsymmetric, constant coefficient hyperbolic systems, however,
we will instead employ Fourier transform methods.
7.3.1.	Definitions.
We investigate in this section systems of linear first-order partial differ-
ential equations having the form
(1)	ut +	= f in Rn x (0, oo),
j=i
subject to the initial condition
(2)	u = g on Rn x {t = 0}.
The unknown is u : Rn x [0, oo) —> Rm, u = (u1 2,..., um), and the functions
Bj : Rn x [0, oo) -> Mmxm (j = 1,..., n), f : Rn x [0, oo) -> Rm, g : Rn ->
Rm are given.
Notation. For each у € Rn, set
B(x,t;y) := j/jBj(x,t) (x E Rn, t > 0).
1=1
□
7.3. HYPERBOLIC SYSTEMS OF FIRST-ORDER EQUATIONS 401
DEFINITION. The system of PDE (1) is called hyperbolic if the mx m
matrix B(x, t; y) is diagonalizable for each x,y £ Rn, t > 0.
In other words, (1) is hyperbolic provided for each x, y,t, the matrix
B(x, t, y) has m real eigenvalues
Ai(x,t;y) < X2(x,t;y) <    _ Xm (x,t;y)
and corresponding eigenvectors {г/Дж, t; y)}™=1 that form a basis of Rm.
There are two important special cases:
DEFINITIONS, (i) We say (1) is a symmetric hyperbolic system if
Bj(x, t) is a symmetric mxm matrix for each x G Rn, t > 0 (y = 1,..., m).
(ii) The system (1) is strictly hyperbolic if for each x,y G Rn, у 0, and
each t > 0, the matrix B(x,t;y) has m distinct real eigenvalues:
Ai(x,t;y) < А2(ж, t;y) <    < Am (x,t;y).
Motivation for the definition of hyperbolicity. We justify the hyper-
bolicity condition as follows. Assume f = 0 and, further, the matrices Bj
are constant (j = 1,..., n). Thus
n
(3)	^yjBj =B(y)
1=1
depends only on у G R".
As in §4.2 let us look for a plane wave solution of (1), (2). That is, we
seek a solution u having the form
(4)	u(x, t) = v(y • x — at) (x € Rn, t > 0)
for some direction у G R", velocity (a G R), and profile v : R —> Rm.
Plugging (4) into (1), we compute:
(n	\
-ст/ + 52 У3В3 v' = 0.
1=1	/
This equality asserts v' is an eigenvector of the matrix B(y) corresponding
to the eigenvalue a.
The hyperbolicity condition requires that there are m distinct plane wave
solutions of (1) for each direction y. These are
(y • x - Afe(y)t)rfe(y) (k = l,...,m),
where
Ai(y) < A2(y) <	< Am(y)
are the eigenvalues of B(y) and {r^(y)}^ the corresponding eigenvectors.
The eigenvalues for |y| = 1 are the wave speeds.	□
402
7. LINEAR EVOLUTION EQUATIONS
(5)
7.3.2. Symmetric hyperbolic systems.
In this section we apply energy methods and the vanishing viscosity
technique to build a solution to the hyperbolic initial-value problem
ut +	= f in Rn x (0, T]
u = g on Rn x {t = 0},
where T > 0, under the fundamental assumption that
(6)	the matrices Bj(x, t) are symmetric {j — 1,..., n),
for x e Rn, 0 < t < T. We will further assume Bj e C2(Rn x [0, T];
with
(7)	sup (|B7|, l^.tBjl, |r>2tBj|) < oo (j =
R"x[0,T]
and
(8)	ge№(Rn;Rm), fG№(Rnx(0,T);Rm).
Remark. More general systems having the form
(9)	Bout +	= f
J=i
are also called symmetric, provided the matrices Bj are symmetric for j =
0,..., n. The theory set forth below easily extends to such systems, provided
Bq is positive definite.
Symmetric hyperbolic systems of the type (9) generalize the second-
order hyperbolic PDE studied in §7.2. For suppose v is a smooth solution
of the scalar equation
(10)	Vtt - ^2	= 0,
i,j=l
where without loss we may take = а?1 (i, j = 1,..., n). Writing
u= (u1,... ,un+1) := (vxl,...,vXn,vt),
we discover u solves a system of the form (9), for m = n + 1, f = 0,
/0	...	0
(j' = 1,..., n),
0	...	0	-a74
\ —a1-7	...	—anJ	0	/	(n+i)x(n+i)
/a11	...	aln	0\
aln	...	ann	0
\ 0	...	0	1/	x(n+i)
Bj =
Bo =
7.3. HYPERBOLIC SYSTEMS OF FIRST-ORDER EQUATIONS 403
Observe that the uniform hyperbolicity condition for (10) implies that the
matrix Bq is positive definite.	□
a.	Weak solutions.
To ease notation, let us define the bilinear form
B[u,v;t] := / Y^(Bj(-,t)ux.) • vdx
Л^=1
for 0 < t < T, u, V G B^R^R™).
DEFINITION. We say
ueL^Tjff^R^R"1)), with uz G L2(0,T;£2(Rn;Rm)),
is a weak solution of the initial-value problem (5) for the symmetric hyper-
bolic system provided
(i)	(uz,v) +B[u,v;t] = (f,v)
for each v G B1(Rn: Rm) and a.e. 0 <t <T, and
(ii)	u(0) = g.
Here and afterwards ( , ) denotes the inner product in L2(Rn;Rm).
Remark. According to Theorem 2 in §5.9.2, u G C*([0, T]; L2(Rn; Rm)) and
so the initial condition (ii) makes sense.	□
b.	Vanishing viscosity method.
We will approximate problem (5) by the parabolic initial-value problem
Г uf — еДие + Bju^. = f inRnx(0,T]
' '	(	u£ = ge on Rn x {t = 0}
for 0 < e < 1, ge := rje * g. The idea is that for each e > 0, problem (11)
has a unique smooth solution u£, which converges to zero as |x| —> oo. The
plan is to show that as e —> 0, the ue converge to a limit function u, which
is a weak solution of (5).
THEOREM 1 (Existence of approximate solutions). For each e > 0, there
exists a unique solution uE of (11), with
(12)	u£GL2(0,T;ff3(Rn;Rm)), ueZ G T2(0,T; Bx(Rn;Rm)).
404
7. LINEAR EVOLUTION EQUATIONS
Proof. 1. Set X = Loo((0,T);#1(IRn;Rm)). For each v e X, consider the
linear system
Г ut - еДи = f -	in Rn x (0, T]
[ и = g£	on Rn x {t = 0}.
As the right hand side is bounded in L2, there exists a unique solution
и e T2(0, T; №(Rn; Rm)), u' e T2(0, T; L2(Rn; Rm)). Indeed, we can utilize
the fundamental solution Ф of the heat equation (§2.3.1) to represent u£ in
terms of g£ and f - £)"=1 Bj vXj •
Similarly, take v G X and let й solve
( ut - еДй = f - BjVx. in Rn x (0, T]
( й = g£	on Rn x {t = 0}.
2. Subtracting, we find й := и — й satisfies
Гщ-€Дй = -Е7=1В^ inRnx(0,T]
( й = 0	on R" x {t = 0},
for v := v — v. From the representation formula of й in terms of the
fundamental solution Ф and Bjv^., we obtain the estimate:
n
ess sup ||u(t)||nl(R";Rm) < C(e)ll ^BjVIJL2(0T;L2(Rr..Rrn))
0<t<T	—[
(14)	< C'(e)||v||L2(0iT.Hl(Rn.Rm})
< C(e)T1/2ess sup ||v(t)||ni(R-;R-)-
0<t<T
Thus
(15)	||й|| < C(e)T1l2||v||.
3.	If T is so small that
(16)	C(e)T1/2 < 1/2,
then (15) reads ||u — й|| < |||v — v||. According to Banach’s fixed point
theorem (§9.2.1) the mapping vmu has a unique fixed point. Then и = u£
solves (11), provided (16) holds.
If (16) fails, we choose 0 < Ti < T so that CTi = 1/2 and repeat the
above argument on the time intervals [0,Ti], [Ti, 2Ti], etc.
Assertion (12) follows from parabolic regularity theory (cf. §7.2.3).	□
7.3. HYPERBOLIC SYSTEMS OF FIRST-ORDER EQUATIONS 405
c. Energy estimates.
We intend to send e —> 0 in (11), and for this as usual need some uniform
estimates.
THEOREM 2 (Energy estimates). There exists a constant C, depending
only on n and the coefficients, such that
(17)	Hu£ HL2(°>T;L2(Rn;Rm))
< C,(lbllH1(Rn;Rm) + l|f||L2(0,T;Wl(R";Rm)) + ||f,||L2(0T.L2(Rn;Rm)))
for each 0 < e < 1.
Proof. 1. We compute
(18)	f|l|u6H12(Kn.Rm)'j = (ue,ue') =	+еДи€\
\	/	\	j=1	/
Now
(19)	|(UE, f)| < ||u£||£2(Rn;Rm) + ||f|lL2(Rn.Rm)
and
(20)	(ие,бДи£) = -6||7>ue||22(Rn;iM^xn) < 0.
2.	Suppose v e C£°(Rn;Rm). Then
1 n r	1 n f
= ((bjv)vkdx-о22 / (Bwv) vdz>
3=1Ju
the last equality following from the symmetry assumption (6). As v has
compact support, we deduce using (7) that
n
(v>52Biv^)
i=i
(B^Xj.v) -vdx < C*||v||L2(]Rn.]Rm).
By approximation therefore:
n
(ut>52Biu^P - ciiu£Hl2(r~;r-)-
406
7. LINEAR EVOLUTION EQUATIONS
Utilizing this bound, (19) and (20) in (18), we obtain the estimate
^(llU 11L2 (Rn ;Rm)) — C(IIU ll£2(Rrl;Rm) + 11^ II.L2(Rn;Rm)) ’
We next apply Gronwall’s inequality, to deduce
(21)	HuE(i)llL2(R";Rm) - (llg|lL2(Rn;Rm) + 11^llL2(0,T;L2(R";Rm))) ’
since ||ge||L2(Rn.Rm) < ||g||L2(R";R’")•
3.	Fix к e {l,...,n} and write vfc := u^. Differentiating (11) with
respect to Xk, we find
Г v4fc - eAvfc + £?=1	= fXk - S7=i 3 in Rn x (0, T]
I	vfc = g,Xk on Rn x {t = 0}.
Reasoning as above, we find
(22)	4(l|V‘ll L2(R";R">)) - C(II-C>u6|Il2(RMM^><’1) + IIIIl2(R";M^x"))•
Sum the previous inequalities for к = 1,..., n, to deduce
^(||^>u£|lL2(Rn;]^nxn)) < C(||^U6||^2(Rn;Mmxn) + ||-E>f|lL2(Rre;Mnxn)).
Gronwall’s inequality now provides the bound
m^Pu'Wll^fRnjM^xn)
(23)	°-*-T
< C'dl-OgHy^Rn jjfnxn) + l|f|lL2(0T;Hl(R>>;Rn>)))>
since ||Dge||L2(Rn.ДопXn) < ||Dg||L2(Rn;]^nxn).
4. Next set v := ue and differentiate (11) with respect to t, to discover
Г vt-eAv + ^B^. =f'-Z;=1Bj,fu^ inR"x(0,T]
( } I	v = f - Bjgx7 + cAge onFx{i = 0}.
Reasoning as before, we compute
max ||ц£ (^Hl^R^R"*) - C(ll£>gllL2(R- ;IM^nXn) "I” 2||Ag£|| L2(R":Rm)
(25)	+ ||f(0)||£2(Rn.Rm) + ||f||L2(0,7;7/1(R";Rm)) + l|f/|lL2(0,T;L2(R";Rm)))-
Now
(J
(26)	||Ag£||
L2(Rn;Rm) < ^2 ll-C>g|lL2(R" .^Jmxn),
since ge = r)e * g. Furthermore
(27)	||f(0) ||£2(Rn.Rm) < C(||f||£2(05T;L2(Rn;Rm)) + 11^ IIL2(0,T;L2(Rn;Rm)))’
This bound, together with (21) and (23), completes the proof.	□
7.3. HYPERBOLIC SYSTEMS OF FIRST-ORDER EQUATIONS 407
d. Existence and uniqueness.
THEOREM 3 (Existence of weak solution). There exists a weak solution
of the initial value problem (5).
Proof. 1. According to the energy estimates (17) there exists a subse-
quence tfc —> 0 and a function u G L2(0,T;//^(R";Rm)), such that u' G
L2(0,T;L2(Rn;Rm)), with
j h' -u weakly in L2(0,T^1(Rn;Rm))
(28)	| u< u' weakly in L2(0,T;L2(Rn;Rm)).
2.	Choose a function v £ C1([0,T]; B'1(Rn; Rm)). Then from (11) we
deduce:
(29)	[ (u£',v)+ e£>u£ : Dv + B[ue,v;t\dt = [ (f,v)dt.
Jo	Jo
Let e = efc —> 0:
(30)	[ (uz, v) + B[u, v;t] dt = /" (f, v)dt.
Jo	Jo
This identity is valid for all v G C([0, T]: B1(Rn; Rm)), and so
(u', v) + B[u, v; t] = (f, v)
for a.e. t and each v G 7f1(Rn;Rm).
3.	Assume now v(T) = 0. Then (29) implies
[ -(u£, v') + e£>u£ : Dv + B[u£, v;t] dt = [ (f, v) dt + (g£, v(0)).
Jo	Jo
Upon sending e = efc —> 0 we obtain:
[ — (u, v) + B[u,v;t]dt = [ (f,v)dt + (g,v(0)).
Jo	Jo
Integrating by parts in (30) gives us the identity
[ — (u,v) + B[u,v;t] dt = [ (f,v) dt + (u(0),v(0)).
Jo	Jo
Consequently u(0) = g, as v(0) is arbitrary.	□
408
7. LINEAR EVOLUTION EQUATIONS
THEOREM 4 (Uniqueness of weak solution). A weak solution of (5) is
unique.
Proof. 1. It suffices to show the only weak solution of (5) with f = g = 0
is u - 0.
To verify this, note
(31)	(uz, u) + B[u, u; t] = 0 for a.e. 0 < t < T.
Since |B[u, u;t]| <	we as usual compute from (31) that
(llU(*)ll L2(R»;R’")) -
whence Gronwall’s inequality forces	= 0 (0 < t < T), since
u(0) = 0.	’	□
7.3	.3. Systems with constant coefficients.
In this section we apply the Fourier transform (§4.3) to solve the constant
coefficient system
(32)	ut + ^2	=0 in Rn x (0, oo),
j=i
with the initial condition
(33)	и = g on Rn x {t = 0}.
We assume that the	are constant m x m matrices and that the
m x m matrix
n
(34)	B(y) :=
1=1
has for each у 6 Rn m real eigenvalues
(35)	Ai(y) < A2(y) <	< Am(y).
There is no hypothesis concerning the eigenvectors, and so we are supposing
only a very weak sort of hyperbolicity here. We also make no assumption
of symmetry for the matrices {Bj}"=1. Consequently the foregoing energy
estimates do not apply. We need a new tool, which we discover in the Fourier
transform.
7.3. HYPERBOLIC SYSTEMS OF FIRST-ORDER EQUATIONS 409
THEOREM 5 (Existence of solution). Assume
geHs(Rn;Rm) (s>^+m).
\	Zi	r
Then there is a unique solution u € C*1([0, oc); Rm) of the initial-value prob-
lem (32), (33).
See §5.8.4 for the definition of the fractional Sobolev spaces Hs.
Proof. 1. We apply the Fourier transform (§4.3.1), as follows. First, tem-
porarily assume u = (ti1,..., um) is a smooth solution. Then set
u = (й1,... ,um),
where ' denotes the Fourier transform in the variable x: we do not transform
with respect to the time variable t. Equation (32) becomes
n
Ut + i^yjBjU = 0;
j=i
that is,
(36)	Ut + iB(y)u = 0 in Rn x (0, oc).
In addition
(37)	u = g on Г x {t = 0}.
For each fixed у e Rn we solve (36), (37) by integrating in time, to find
(38)	u(y, t) = e-itB^g(y) (y £ R",t > 0).
Consequently и — (e-ttB^g)v; so that
(39)	u(x, t) = Д e-'^-tB^g(y) dy (x e R", t > 0).
2.	We have derived formula (39) assuming и to be a smooth solution of
(32), (33). We now verify that the function и defined by (39) is in truth a
solution, and so must first check that the integral in (39) converges.
Since g € 7F(Rn;Rm), we know according to §5.8.4 that there exists
f e L2(Rn;Rm) such that
(40)	|g(y)| < 0(1 +	(yeRn).
410
7. LINEAR EVOLUTION EQUATIONS
So in order to investigate the convergence of the integral (39), we must
estimate ||e-ltB^||.
3.	For a fixed y, let Г denote the path dB(0, r) in the complex plane,
traversed counterclockwise, the radius r selected so large that the eigenvalues
Ai(y),..., Am(y) lie within Г.
We have the formula:
(41)	е-’*В(з/) = J_ f e-itz(zI -B(y))-4z.
2тгг Jr
To verify this, let A(f, y) denote the right hand side of (41) and fix x E Rm.
Then
B(y)A(t,y)x = ~ [ e^B^zI-B^xdz
Ztw Jy
= ~ [ e-itz{z{zI-B^x-x}dz
2тг? Jr
1 d A , 4
= A(t,y)x,
Z Ctt
since Jr e~ttzdz = 0. Consequently
(42)	(-£ + zB(y)Wy)=0.
In addition
A(0,y)x=-^- f (zl - B(y))-1xdz
27TZ Jp
(43)	=2_ Г^ВМ(г1-ВМ)-,<г
2тгг Jr	z
= i + j_ /в^м-вщ)-^
2тгг Jr	z
Now set
(44)	w := (zl — B(y))-1z ;
so that zw — B(y)w = x. Taking the product with w, we deduce |w| < щ
for some constant C. Using this estimate and letting r go to infinity, we
conclude from (43) that A(0, y)x — x. This equality and (42) verify the
representation formula (41).
4.	Define a new path Д in the complex plane as follows. For fixed y,
draw circles Bfc = B(Afc(y), 1) of radius 1, centered at Afc(y) (fc = 1,... , m).
Then take Д to be the boundary of (JZ=i traversed counterclockwise.
7.3.HYPERBOLIC SYSTEMS OF FIRST-ORDER EQUATIONS 411
Deforming the path Г into Д, we deduce from (41) that
(45)	e-^B(j/)= 1 f e-itz^zI -в^))-1^.
2тгг 7д
Now
(46)	\e~itz\ <et (z e Д).
Furthermore
det(zJ - B(y)) = fj(z - Afc(y));
fc=i
whence
(47)	| det(zl — B(y))| > 1 if z € Д.
Now
-! cof(z/-B(y)f
det(zZ — B(y)) ’
where “cof” denotes the cofactor matrix (see §8.1.4). We deduce
||(z7 — B(y))-1|| < || cof(z7 — B(y))||
(48)	^(l + Izr-MBMF-1)
< C(1 + li/p-1) ifzeA.
We have utilized in this calculation the elementary inequality
|Afc(y)| < C|y| (k = l,...,m).
Combining (45)~(48), we derive the estimate
(49)
Це-^в(^)|| < Ce‘(l +	1) (y e Rn).
5. Return now to (37). We deduce using (40), (49) that
f |e^e-^)g(y)|dy<C [ ||e-itB^||(l + |y|s)-1|f(y)|dy
R"	JR^1
<Ce‘ [ If^Kl + lyl—^a + IF)-1^
JRn
-c (X|f|2dy) (X i+iX-m+i0
412
7. LINEAR EVOLUTION EQUATIONS
since s >	+ m — 1. Hence the integral in (39) converges, and it follows
easily that the function
u<*-‘) = 7T^72 [
(2тг)п/2 jRn
is continuous on Rn x [0, oo).
6. To show u is C1, observe for 0 < |Л| < 1 that
u(rr, t + h) — u(rr,t)
h
Since
1	/ pix-y( -i(t+h)B(y)
(2тг)п/2Л JRn 1
e~itB^y)dy.

ft+h,
= -i у B(y)e-isB^ ds,
we can estimate as above that
h
_ e-itB(y)j
<Cet+1(l + \y\m).
Therefore
u(rr, t + h) — u(rr, t)
h
<ce'+i f |ед|(1 + ы”)(1+!№<*»,
JR’1
and the integrand is summable since s >	+ m. Thus u< exists and is
continuous on Rn x [0, oo). A similar argument shows uXi exists and is con-
tinuous (г = 1,..., n). According to the Dominated Convergence Theorem,
we can furthermore differentiate under the integral sign in (39), to confirm
that u solves the system Ut + BjUXj =0.	□
In Chapter 11 we will encounter nonlinear first-order systems of hyper-
bolic equations.
7.4. SEMIGROUP THEORY
Semigroup theory is the abstract study of first-order ordinary differential
equations with values in Banach spaces, driven by linear, but possibly un-
bounded, operators. In this section we outline the basics of the theory, and
present as well two applications to linear PDE. This approach provides an
elegant alternative to some of the existence theory for evolution equations
set forth in §§7.1-7.3.
7.4. SEMIGROUP THEORY
413
7.4.1. Definitions, elementary properties.
We begin in an abstract setting. Let X denote a real Banach space, and
consider then the ordinary differential equation
fu'(t) = Au(t) (i>0)
t u(0) = u,
where ' = и € X is given, and A is a linear operator. More precisely,
suppose £>(A), the domain of A, is a linear subspace of X and we are given
a possibly unbounded linear operator
(2)	A : D(A) X.
We investigate the existence and uniqueness of a solution
u : [0, oo) —> X
of the ODE (1). The key problem is to ascertain reasonable conditions on
the operator A so that (a) the ODE has a unique solution u for each initial
point и € X, and (b) many interesting PDE can be cast into the abstract
form (1). (We have in mind the situation that X is an IP space of functions
and A is a linear partial differential operator involving variables other than
t. In this case A is necessarily an unbounded operator.)
a. Semigroups.
Let us for the moment informally assume u : [0, oo) —> X is a solution
of the differential equation (1), and that (1) in fact has a unique solution
for each initial point и G X.
Notation. We will write
(3)	u(t) := S(t)u (t > 0)
to display explicitly the dependence of u(t) on the initial value и G X. For
each time t > 0 we may therefore regard S(t) as a mapping from X into
X.	□
What properties does the family of operators {S(t)}t>o satisfy? Clearly
S(t) : X —> X is linear. Furthermore
(4)	S(0)u = и (u G X)
and
(5)	S(t + s)u = S(t)S(s)u = S(s)S(t)u (t,s>0, и e X).
Condition (5) is simply our assumption that the ODE (1) has a unique
solution for each initial point. Finally, it seems reasonable to suppose that
for each и E X
(6)	the mapping 11—> S(t)u is continuous from [0, oc) into X.
414
7. LINEAR EVOLUTION EQUATIONS
DEFINITIONS, (i) A family {5(t)}t>o of bounded linear operators map-
ping X into X is called a semigroup if conditions (4)-(6) are satisfied.
(ii) We say {S(t)}t>o is a contraction semigroup if in addition
НЖ1<1 (i>0),
|| || here denoting the operator norm. Thus
l|S(t)«|| < ||u|| (t > о, и e x).
The notion of contraction semigroup captures many properties of a nice
flow on X generated by the ODE (1).
b. Elementary properties, generators.
The real problem now is to determine which operators A generate con-
traction semigroups. We will answer this in §7.4.2, after recording in this
section some further general facts.
Henceforth assume {S(t)}t>o is a contraction semigroup on X.
DEFINITIONS. Write
(8)	D(?l) := G X | lim	—- exists in xj>
and
(9)	Au := tlim 3^~и (u e D(Af).
We call A : D( A) —> X the (infinitesimal) generator of the semigroup
{S(t)}t>o; D(A) is the domain of A.
THEOREM 1 (Differential properties of semigroups). Assume и e D(A).
Then
(i)	S(t)u € D(A) for each t > 0,
(ii)	AS(t)u = Sit) Au for each t > 0,
(iii)	the mapping 11—> S(t)u is differentiable for each t > 0,
and
(iv)	= AS(t)u (t > 0).
7.4. SEMIGROUP THEORY
415
Proof. 1. Let и e D(A). Then
S(s)S(t)u - S(t)u
lim -----------------
s—>0+	S
„ S(t)S(s)u - S(t)u ,	,
= lim ------------------ by the semigroup property (5)
s—>0+	S
S(s)u — u .
= S(t) hm -----------= S(t)Au.
s—+o+	s
Thus S(t)u € D(A) and AS(t)u = S(t}Au. Assertions (i) and (ii) are
proved.
2. Let и € D(A}, h > 0. Then if t > 0,
( S(t)u - S(t - h)u	1
hm < ——-----------------S(t)Au >
h-»o+ ( h	J
. (S(h)u — u\	)
= hm < S(t — h) I ---------- I — S(t)Au >
h—>0+	у h J	J
= lim (S(t -K)(	~ U -	+ (S(t - h) - S(t))Aul = 0,
h—»o+ (	\ h	)	J
since Sl'-l^~u —> Au and \\S(t — h)|| < 1. Consequently
S(t)u - S(t - h)u
hm ——-------H-------— = S(t)Au.
h^o+	h
Similarly
S(t + h)u - S(t)u	S(h)u — u n, . л
lim	= S(t) lim ---------= S(t)Au.
h^0+	h	h->0+ h
Thus exists for each time t > 0, and equals S(t)Au = AS(t)u. □
Remark. Since t i—> AS(t.)u = S(t)Au is continuous, the mapping t
S{t)u is C*1 in (0, oo), if и G D(A).	□
THEOREM 2 (Properties of generators).
(i) The domain D(A) is dense in X,
and
(ii) A is a closed operator.
Remark. To say A is closed means that whenever Uk £ D(A} (fc = 1,...)
and Uk —> u, Auk —> v as к —> oo, then
и G D(A), v = Au.
416
7. LINEAR EVOL UTION EQ UATIONS
Proof. 1. Fix any и 6 X and define then ul := S^uds. In view of (6),
у —► и in X, as t —> 0+.
2.	We claim
(10)	6 D(A) (t > 0).
Indeed if r > 0, we have
where we used the semigroup property (5). Thus
lf‘* ,	If.,..
----------= - / S{s)uds	I S(s)uds
r r	r Jo
—> S(f)u — u, as r —► 0 + .
Hence u* 6 D(A), with Au* = S(fyu — u. This proves (10) and completes
the proof of assertion (i).
3.	To prove A is closed, let Uk E D(A) (к — 1,...) and suppose
(11)	Uk —> u, Auk —> v in X.
We must prove и E D(A), v = Au. According to Theorem 1
S(i)Uk - «к = / S{s)Aukds.
Jo
Let к —> oo and recall (11):
S(t)u — и = [ S(s)vds.
Jo
Hence we have
Sifyu-u 1 Г* . .
hm ---------= hm - / S(s)vds = v.
t-»o+ t t-*o+ t Jo
But then by definition и E D(A), v = Au.
□
7.4. SEMIGROUP THEORY
417
c. Resolvents.
DEFINITIONS, (i) We say a real number Л belongs to p(A), the resolvent
set of A, provided the operator
XI-A : D(A) X
is one-to-one and onto.
(ii)	If X 6 p(A), the resolvent operator R\ : X —> X is defined by
R\u := (AZ — Af^u.
According to the Closed Graph Theorem (§D.3), Rx : X —t D(A) С X
is a bounded linear operator. Furthermore,
AR\u = R\Au if и 6 Z>(A).
THEOREM 3 (Properties of resolvent operators).
(i)	If X,p 6 p(A), we have
(12)	Rx — Rp. = (jz — X)RxRfj. (resolvent identity)
and
(13)	RxRfi = RfiRx-
(ii)	If A > 0, then X € p(A),
(14)	Rxu= f e~xtS(t)udt (izGX),
Jo
and so || Rx || < p
Remark. Thus the resolvent operator is the Laplace transform of the semi-
group (cf. Example 4 in §4.3.2).	□
Proof. 1. Verification of the identities (12), (13) is left to the reader (Prob-
lem 11).
2.	Note first that since A > 0 and ||S(t) || < 1, the integral on the right
hand side of (14) is defined. Let Rxu denote this integral. Then for h > 0
418
7. LINEAR EVOLUTION EQUATIONS
and и 6 X,
S(h)R\u — Rxu If/"00 _\trnz.	o,.4 ,
v 1 -------— = T< e xt[S(t + h)u - S(t)u\ dt !>
h	h I Jo	J
= -| / e-^-^S^udt
h Jo
+ т Г°(е~х^ - e~Xt)S(fi)udt
h Jo
= -eA/l| / e~xtS(t)udt
h Jo
( Xh _ 1 \ f<x>
h J J e~xtS(f)udt.
Hence
S(h)R\u - Rxu	-
lim ——— -----------= -u 4- XR\u.
h—*0-|-	h
Thus AR\u = —u + XR\u-, that is,
(15)	(AZ — A)R\u = и (u 6 X).
On the other hand if и E D(A),
AR\u = A [ e~xtS(t)udt = f e~XtAS(t)udt
/1 zj\	«/0	0
roo
= I e~xtS(f)Audt = R\Au.
Jo
Our passing A under the integral sign is justified since A is a closed operator:
see Problem 12. Thus
R\(XI — A)u = и (и E D(A)).
In view of (15) and the formula above AZ — A is one-to-one and onto. Con-
sequently A E p(A), R\ — (AZ — A)-1 = Rx-	□
7.4.2. Generating contraction semigroups.
We now characterize the generators of contraction semigroups:
THEOREM 4 (Hille-Yosida Theorem). Let A be a closed, densely-defined
linear operator on X. Then A is the generator of a contraction semigroup
{S(t)}t>o if and only if
(17)	(0, oo) C p(A) and ||2?д || < f°r X > Q.
Л
7.4. SEMIGROUP THEORY
419
Proof. 1. If A is a generator, then from Theorem 3,(iii) we immediately
deduce (17).
2.	Conversely, suppose A is closed, densely-defined, and satisfies (17).
We must build a contraction semigroup with A as its generator. For this,
fix Л > 0 and define
(18)	Aa := -AZ + A27?a = XARx.
The operator Ax is a kind of regularized approximation to A.
3.	We first claim
(19)	Axu —► Au as A —> oo (u 6 D(A)).
Indeed, since XRxu — и = ARxu — RxAu, ||XRxu — u|| < ||7?д|| ||Au|| <
| ||Au|| —> 0. Thus XRxu —>uasA—>ooifu£ D(A). But since || XRx || < 1
and D(A) is dense, we deduce then as well
(20)	XRxu —> и as A —> oo, for all и 6 X.
Now if и E O(A), then
Axu = XARxu = XRxAu.
In view of (20), our claim (19) is proved.
4.	Next, define
Sx{f) := etAx = e~Xtex2tR* = e~Xt V ^^Rkx.
t'o k'
Observe that since ||7?д|| < A \
00 \2k.ik
l|S>(t)ll<e-A‘E-y-
k=0
00 \kj.k
V—= 1.
k'.
k=0
Consequently {Sx(t)}t>o is a contraction semigroup, and it is easy to check
its generator is Ax, with D(AX) = X.
5.	Let A, p, > 0. Since = 7?м7?д, we see A>AM = AMA>, and so
Ам5д(£) = Sx(t)Afj, for each t > 0.
Thus if и E D(A), we can compute
/* d
Sx(t)u - 8ц(1)и = / —[SM(t - s)Sa(s)u](/s
Jo as
= [ S^t - s)Sx(s)(Axu - A^u) ds,
Jo
420
7. LINEAR EVOLUTION EQUATIONS
because ^Sx(t)u = A\S\(t)u = S\(t)A\u. Consequently ||S>(t)u — SM(t)u||
< t||A>u — AMu|| —> 0 as А, д —> og. Hence
(21)	S(t)u := lim S>(t)u exists for each t > 0, и G L)(A).
A—>oo
As || (t) || < 1, the limit (21) in fact exists for all и G X, uniformly for t in
compact subsets of [0, oo). It is now straightforward to verify {S(i)}t>o is a
contraction semigroup on X.
6.	It remains to show A is the generator of {S(t)}t>o- Write В to denote
this generator. Now
(22)	Sx(t')u — u= [ Sx(s)Axuds.
Jo
In addition
||Sa(s)Aau - S(s)Au|| < ||Sa(s)|| ||Aau - Au|| + ||(SA(s) - S(s))Au||	0
as A —► oo, if и G L>(A). Passing therefore to limits in (22), we deduce
Stfyu — и — [ S(s)Auds
Jo
if и G D(A). Thus D(A) C D(B} and
Bu = Jim	_ ди D(A)).
Now if A > 0, A G р(А)Пр(В). Also (AZ-B)(D(A)) = (AZ-A)(D(A)) = X,
according to (17). Hence (AZ — В)|д(Л) is one-to-one and onto; whence
D(A) = D(B). Therefore A = B, and so A is indeed the generator of
{S(t)}t>0.	□
Remark. Let w G R. A semigroup {S(i)}t>o is called a>-contractive if
||S(t)|| < ewt (t > 0). An easy variant of Theorem 4 asserts that a closed,
densely-defined linear operator A generates an cu-contractive semigroup if
and only if
(23)	(cu, oo)Cp(A) and ||BA|| < t——for all A > cu.
A — cu
This version of the Hille-Yosida Theorem will be required for our first ex-
ample below.	□
7.4.3. Applications.
We demonstrate in this section that certain second-order parabolic and
hyperbolic PDE can be realized within the semigroup framework.
7.4. SEMIGROUP THEORY
421
a.	Second-order parabolic PDE.
We consider first the parabolic initial/boundary-value problem
(24)
ut + Lu — 0
и = 0
и = g
in Ut
on dU x [0, T]
on U x {t = 0},
a special case (corresponding to f = 0) of (1) in §7.1.1. We assume L has
the divergence structure (2) from §7.1.1, satisfies the usual strong elliptic-
ity condition, and has smooth coefficients, which do not depend on t. We
additionally suppose that the bounded open set U has a smooth boundary.
We propose to reinterpret (24) as the flow determined by a semigroup
on X = T2(t/). For this, we set
(25)
D(A) :=H^U)OH2(U),
and define
(26)	Au := —Lu if и G D(A).
Clearly then A is an unbounded linear operator on X. Recall from §6.2.2
the energy estimate
(27)	0\\u||^i(t/) < B[u, u] + 71|и||22{tz),
for constants /3 > 0, 7 > 0, where , ] is the bilinear form associated with
L.
THEOREM 5 (Second-order parabolic PDE as semigroups). The operator
A generates a 7-contraction semigroup {S(i)}t>o on L2(U\
Proof. 1. We must verify the hypotheses of the variant of the Hille-Yosida
Theorem mentioned in the concluding Remark in §7.4.2, with 7 replacing w.
First, D(A) given by (25) is clearly dense in L2(U).
2.	We claim now that the operator A is closed. Indeed, let	C
D(A) with
(28)	Uk —> u, Auk -+ f in L2(U}.
According to the regularity Theorem 4 in §6.3.2,
||«fc - uiIIh2(c/) < CdlAufc - Au/||L2(tZ) 4- ||ufe - Ui||ь2(С7))
422
7. LINEAR EVOLUTION EQUATIONS
for all к and I. Thus (28) implies	is a Cauchy sequence in H'2(U)
and so
(29)	Uk —> и in H2(U).
Therefore и 6 D(A). Furthermore (29) implies Auk —> Au in L2(U), and
consequently f — Au.
3.	Next we check the resolvent conditions (23), with 7 replacing w. Ac-
cording to Theorem 3 in §6.2.2, for each A > 7 the boundary-value problem
has a unique weak solution и 6 Hq(U) for each f G L2(U\ Owing to elliptic
regularity theory, in fact и G H2(U) Г) Hq(U). Thus и G D(A). We may
now rewrite (30), using (26), and find
(31)	Xu — Au = f.
Thus (XI — A) : D(A) —> X is one-to-one and onto, provided A > 7. Hence
p(A) D [7,00).
4.	Consider the weak form of the boundary-value problem (30):
B[u, v] + X(u, v) = (/, u)
for each v G Hq(C7), where ( , ) is the inner product in L2(U). Set v = и
and recall (27) to compute for A > 7:
(A - 7)||u||^2(t/) < ||/||L2(t/)||u||L2(t/).
Hence, since и = R\f, we have the estimate
This bound is valid for all f G L'2(U) and so
(32)	||Ял|| < (A > -r),
as required.	□
Remark. Semigroup theory provides an elegant method for constructing
a solution to the initial/boundary-value problem (24). It is worth noting
however that this technique requires that the coefficients atJ, Ьг,с (i,j =
l,...,n) of L be independent of t. The Galerkin method in §7.1 works
without this restriction. On the other hand, semigroup theory constructs
at the outset a more regular solution than that produced by the Galerkin
technique, at least until we develop regularity theory.	□
7.4. SEMIGROUP THEORY
423
b.	Second-order hyperbolic PDE.
We turn our attention next to the hyperbolic initial/boundary-value
problem
{utt + Lu = 0 in Ut
и = 0 on dU x [0, T]
и = g, ut = h on U x {t = 0},
for the operator L and open set U as above. We recast (33) as a first-order
system by setting v := щ. Then the foregoing reads
ut = v, vt + Lu = 0 in Ut
и — 0	on dU x [0, T]
и = g, v = h on U x {t = 0}.
We will further assume L has the symmetric form:
OO
Lu = - (atjuXi)Xj -I- cu,
i,j=l
where
(34)	c > 0, at}'= a?1 (i, j = 1,... ,n).
Thus for some constant /3 > 0:
(35)	/3||u||^i^ < B[u,u] for all и e Hq(U).
Now take
X = H^U) x L2(U),
with the norm
(36)	||MI|:=(BM + IH1W1/2-
Define
D(A) := [H2(U)0H^U)]xH^U)
and set
(37)	A(u,v) := (v, — Lu) for (u,v) 6 D(A).
We will show A verifies the hypothesis of the Hille-Yosida Theorem.
424
7. LINEAR EVOLUTION EQUATIONS
THEOREM 6 (Second-order hyperbolic PDE as semigroups). The oper-
ator A generates a contraction semigroup {5(i)}t>o on Hq(U) x U2(U).
Proof. 1. The domain of A is clearly dense in Uq(U) x U2(U).
2. To see A is closed, let {(щ,,C D(A), with
(uk,vk) - (u,v), A(uk,vk) (/,5) in ffJ(U) x U2(U).
Since A(uk, vk) = (vk, —Luk), we conclude f = v and Luk —g in E2(U).
As in the previous proof, it follows that uk —► и in Я2(Я) and g = —Lu.
Thus (u, v) G D(A), A(u, v) = (v, — Lu) = (f,g).
3. Now let Л > 0, (J,g) G X := Hq(U) x L2(U), and consider the
operator equation
(38)	A(u, v) - A(u, v) = (/, g).
This is equivalent to the two scalar equations
(Xu-v = f (иея2(и)пя01(и)),
\ Xu + Lu = g (i'G Hq(U)).
But (39) implies
(40)	X2u + Lu = Xf + g (u G Я2(Я) П H^U)).
Since A2 > 0, estimate (35) and the regularity theory imply there exists
a unique solution и of (40). Defining then v := Xu — f G Hq(U), we have
shown that (38) has a unique solution (u,v). Consequently p(A) D (0, oo).
4. Whenever (39) holds, we write (u, v) = R\(f,g)- Now from the
second equation in (39), we deduce
А|Н1Ь + b[m] = {g,v)L^-
Substituting v = Xu — f, we obtain
+	+ B[“. Я
< (llslly + Bl/Jl)1/2(IMIh + B(u, u])1'2.
Here we used the generalized Cauchy-Schwarz inequality (§B.2), which holds
owing to the symmetry condition (34). In light of our definition (36),
ll(“,»)ll <
A
and so
ЦЯ>11 < | (A > 0),
Л
as required.	□
See Friedman [FR1] or Yosida [Y] for the theory of analytic semigroups.
Aspects of nonlinear semigroup theory will be developed later, in §9.6.
7.5. PROBLEMS
425
7.5. PROBLEMS
In the following exercises we assume U C Rn is an open, bounded set, with
smooth boundary, and T > 0.
1.	Prove there is at most one smooth solution of this initial/boundary-
value problem for the heat equation with Neumann boundary condi-
tions:
{ut — Au = f in Ut
^=0 ondUx[0,T]
и = g on U x {t = 0}.
2.	Assume и is a smooth solution of
Ut — Au = 0
и = 0
и = д
in U х (0, оо)
on dU х [0, оо)
on U х {i = 0}.
Prove the exponential decay estimate:
11и(’>*)11л2(1/) < e XltllffIIl2(77) (< > 0),
where Ai > 0 is the principal eigenvalue of —A (with zero boundary
conditions) on U.
3.	(Galerkin’s method for Poisson’s equation.) Suppose / € L2[U') and
assume that um — ^k=\ dmwk solves
/ Dum  Dwk dx = / f -wk dx
Ju	Ju
for к =	Show that a subsequence of {um}^=1 converges
weakly in Hq(U) to the weak solution и of
—Au = f in U
u = 0 on dU.
4. Assume
Ufc — u
ufc^v
in L2(0, Т-,Н^иУ)
in L2(0,T-H^(U)).
Prove that v = uz. (Hint: Let ф € Cc(0, T), w 6 Hj(t7). Then
[ (u'k^w)dt = - [ (uk,^w)dt.)
0	Jo
5.	Suppose H is a Hilbert space and
ид, —>• u in L2(0, T; H).
426
7. LINEAR EVOLUTION EQUATIONS
Suppose further we have the uniform bounds
esssup ||ufc(t)|| < C (k = 1,...),
0<t<T
for some constant C. Prove
esssup ||u(t)|| < C.
0<t<T
(Hint: If v e H and 0 < a < b < T, we have
v,uk(t))dt < (711^1116-a|.)
6.	Suppose и is a smooth solution of
’ щ — Au + cu = 0
<	и = 0
и = g
in U x (0, oo)
on dU x [0, oo)
on U x {t = 0}
and the function c satisfies
c > 7 > 0.
Prove
|u(a?,t)| < Ce
7.	Suppose и is a smooth solution of the PDE from Problem 6, that
g > 0, and c is bounded (but not necessarily nonnegative). Show that
и > 0. (Hint: What PDE does v := extu solve?)
8.	Prove inequality (54) in §7.1.3, (59) in §7.2.3. (Hints: Assume и is
smooth, и = 0 on dU. Transform the term (Lu, —Au) by integrating
by parts twice. Simplify and then estimate the boundary terms using
trace inequalities.)
9.	Show there exists at most one smooth solution of this initial/boundary-
value problem for the telegraph equation
utt T dut nXx — f
<	и	=	0
и = д,щ = h
in (0,1) x (0, T)
on ({0} x [0,T]) U ({1} x [0,T])
on (0,1) x {t = 0}.
Here d is a constant.
7.6. REFERENCES
427
10.
Prove there exists at most one smooth solution и of this problem for
the beam equation
Utt 4" aXxxx — 0
и = ux = 0
u = g,ut = h
in (0,1) x (0,T)
on ({0} x [0, T]) U ({1} x [0, T])
on [0,1] x {t = 0}.
11.	Prove the resolvent identities (12) and (13) in §7.4.1.
12.	Justify the equality
A [ e~xtS(t)xdt — [ e~xtAS(t)xdt
Jo	Jo
used in (16) of §7.4.1. (Hint: Approximate the integral by a Riemann
sum and recall A is a closed operator.)
13.	Define for t > 0
[S(t)g](x)= [ Ф(х-y,t)g(y)dy (x G Rn),
JR71
where g : R" —> R and Ф is the fundamental solution of the heat
equation. Also set S(0)g = g.
(i)	Prove {S(t)}t>o is a contraction semigroup on £2(R").
(ii)	Show {S(t)}t>o is not a contraction semigroup on L°°(Rn).
14.	Let {S(i)}t>o be a contraction semigroup on X, with generator A.
Inductively define D(Afc) := {x G D(Afc-1) | Ak^1x G D(A)} (k =
2,...). Show that if x G D(Ak) for some k, then S(t)x G D(Ak) for
each t > 0.
15.	Use Problem 14 to prove that if и is the semigroup solution in X =
L2(U) of
щ — Xu = 0 in Uy
и = 0 on dU x [0, T]
и = g on U x {t = 0},
with g G C^°(t7), then u(-,t) G C^(U) for each 0 < t < T.
7.6. REFERENCES
Section 7.1 See Ladyzhenskaya [L, Chapter 3], J.-L. Lions [LI], Lions-
Magenes [L-M], Wloka [WL]. W. Schlag helped me with the
proof of Theorems 5, 6. The proof of the parabolic Harnack
inequality is similar to calculations found in Davies [DA].
428
7. LINEAR EVOLUTION EQUATIONS
	The books of Krylov [KR], Ladyzhenskaya-Solonnikov-Ural- tseva [L-S-U], and Lieberman [LB] have much more on reg- ularity theory.
Section 7.2	See Ladyzhenskaya [L, Chapter 4], J.-L. Lions [Ll], Lions- Magenes [L-M] and Wloka [WL].
Section 7.3	The Fourier transform argument is taken from John [J]; cf. also Treves [T, §15]. D. Serre has shown me a much more precise version of Theorem 5, under the further assumption of strict hyperbolicity.
Section 7.4 Section 7.5	See Friedman [FR1, Part 2, §1] and Yosida [Y, Chapter IX]. Problem 8: see [B-E] for a proof, and see also Ladyzhens- kaya-Uraltseva [L-U, p. 182].
Chapter 8
THE CALCULUS OF
VARIATIONS
8.1	Introduction
8.2	Existence of minimizers
8.3	Regularity
8.4	Constraints
8.5	Critical points
8.6	Problems
8.7	References
8.1. INTRODUCTION
8.1.1. Basic ideas.
We introduce some new ideas by supposing first of all that we wish to
solve some partial differential equation, which for simplicity we write in the
abstract form
(1)
A[u] = 0.
In this formula A[-] denotes a given, possibly nonlinear partial differential
operator and и is the unknown. There is, of course, no general theory for
solving all such PDE.
The calculus of variations identifies an important class of such nonlinear
problems that can be solved using relatively simple techniques from non-
linear functional analysis. This is the class of variational problems, that is,
431
432
8. THE CALCULUS OF VARIATIONS
PDE of the form (1), where the nonlinear operator A[-] is the “derivative”
of an appropriate “energy” functional /[•]. Symbolically we write
(2)
Then problem (1) reads
(3)	Z'[u] = 0.
The advantage of this new formulation is that we now can recognize solutions
of (1) as being critical points of /[•]. These in certain circumstances may be
relatively easy to find: if, for instance, the functional /[•] has a minimum
at u, then presumably (3) is valid and thus и is a solution of the original
PDE (1). The point is that whereas it is usually extremely difficult to solve
(1) directly, it may be much easier to discover minimum (or maximum, or
other critical) points of the functional /[•].
In addition of course, many of the laws of physics and other scientific
disciplines arise directly as variational principles.
8.1.2. First variation, Euler-Lagrange equation.
Suppose now U C R" is a bounded, open set with smooth boundary dU,
and we are given a smooth function
L : R" x R x U R.
We call L the Lagrangian.
Notation. We will write
L = L(p, z,x) = L(pi,... ,pn, z,x-i,... ,Xn)
for p E R", z € R, and x € U. Thus “p” is the name of the variable for which
we substitute Dw(x') below, and “z” is the variable for which we substitute
w(x). We also set
DpL = (LPl,..., LPn)
< DZL = Lz
, DXL = (LX1,..., LXn) .
This notation will clarify the theory to follow.	□
We now make the vague ideas in §8.1.1 more precise by now assuming
/[•] to have the explicit form
(4)	Z[w] = I L(Dw(x),w(x),x) dx,
Ju
8.1. INTRODUCTION
433
for smooth functions w : J7 —► R satisfying, say, the boundary condition
(5)	w = g on dU.
Let us now additionally suppose some particular smooth function u,
satisfying the requisite boundary condition и = g on dU, happens to be
a minimizer of /[•] among all functions w satisfying (5). We will demon-
strate that и is then automatically a solution of a certain nonlinear partial
differential equation.
To confirm this, first choose any smooth function v G C£°(U) and con-
sider the real-valued function
(6)	i(r) := /[u + rv] (г e Ж).
Since и is a minimizer of ![] and и + tv = и = g on dU, we observe that
i(-) has a minimum at т = 0. Therefore
(7)	г'(0) = 0.
We explicitly compute this derivative (called the first variation) by writ-
ing out
(8)	i(r) = / L(Du + rDv, u + tv, x) dx.
Jv
Thus
f n
i'(r)= I (Du + tDv,u + tv,x)vx + Lz(Du + TDv,u + TV,x)vdx.
Let r = 0, to deduce from (7) that
f П
0 = /(0) = I 2~2Lp (Du,u,x)vXi + Lz(Du,u,x)vdx.
Finally, since v has compact support, we can integrate by parts and obtain
0 =
n
- 57 (LPi(Du,u,x))Xi + Lz(Du,u,x)
i=l
v dx.
As this equality holds for all test functions v, we conclude и solves the
434
8. THE CALCULUS OF VARIATIONS
nonlinear PDE
(9)	— У (LPi(Du, u, x))x + Lz(Du,u,x) = 0 in U.
i=l
This is the Euler-Lagrange equation associated with the energy functional
/[•] defined by (4). Observe that (9) is a quasilinear, second-order PDE in
divergence form.
In summary, any smooth minimizer of /[•] is a solution of the Euler-
Lagrange partial differential equation (9), and thus—conversely—we can
try to find a solution of (9) by searching for minimizers of (4).
Example 1	(Dirichlet’s principle). Take
L(p,z,a?) = ||p|2.
Then LPi = pi (г = 1,... ,n), Lz = 0; and so the Euler-Lagrange equation
associated with the functional
Z[w] := | [ \Dw\2 dx
Ju
is
Au — 0 in U.
This fact is Dirichlet’s principle, previously introduced in §2.2.5.	□
Example 2	(Generalized Dirichlet’s principle). Write
n
Up,z,x) = | У al3^PiPj - zf(x^
i,j=l
where a’-7 = a-7’ (i,j = 1,... ,n). Then LPi = Y^j=i alJ(x)Pj (г = 1,... ,n),
Lz — —f(x). Hence the Euler-Lagrange equation associated with the func-
tional
Л777] := / I у aljwXiwXj -wfdx
Ju
is the divergence structure linear equation
= f 111!/
M=1
8.1. INTRODUCTION
435
We will see later (in §8.1.3 and §8.2) that the uniform ellipticity condi-
tion on the а'-7 (г, j = 1,..., ri) is a natural further assumption, required to
prove the existence of a minimizer. Consequently from the nonlinear view-
point of the calculus of variations, the divergence structure form of a linear
second-order elliptic PDE is completely natural. This observation provides
some much belated motivation for the bilinear form techniques utilized in
Chapter 6.	□
Example 3	(Nonlinear Poisson equation). Given a smooth function f :
R —> R, define its antiderivative F(z) = f(y) dy. Then the Euler-Lagrange
equation associated with the functional
I[w] := [
Ju
||Dw|2 — F(w) dx
is the nonlinear Poisson equation
—Au = f(u) in U.
□
Example 4	(Minimal surfaces). Let
L(p,z,a?) = (1 + |p|2)1/2 ;
so that
Z[w] =[ (1 + iDwl2)1/2 dx
Ju
is the area of the graph of the function w : U —> R. The associated Euler-
Lagrange equation is
(10)
^((l + IOuRi/a)^ 0 inU
This partial differential equation is the minimal surfaceriquation. The
expression div Q1+|^|2p/2^ on the left side of (10) is n times the mean cur-
vature of the graph of u. Thus a minimal surface has zero mean curvature.
□
436
8. THE CALCULUS OF VARIATIONS
8.1.3.	Second variation.
We continue in the spirit of the calculations from §8.1.2 by computing
now the second variation of /[•] at the function u. This we find by observing
that since и gives a minimum for /[•], we must have
i"(0) > 0,
i(-) defined as above by (6). In view of (8) we can calculate
f П
i"(r) = У2 LPiPj (Du + tDv, u + tv, x)vXivXj
Juij^
n
+ 2 Тр.г(Т>и + tDv,u + rv,x)vXiv
i=l
+ Lzz(Du + tDv, u + tv, x}v2 dx.
Setting r = 0, we derive the inequality
f П
0 < ?"(0) = / У2 LPiPj(Du,u,x)vXzvXj
Ju
(11)
+ 2 LPiZ(Du, u, x)vXiv + Lzz(Du, u, x)v2 dx,
1=1
holding for all test functions v 6 C'^IO(t7).
We can extract useful information from inequality (11), as follows. First,
note after a routine approximation argument that estimate (11) is valid for
8.1. INTRODUCTION
437
any Lipschitz continuous function v vanishing on dU. We then fix £ G R”
and define
(12)	v(a?) :=	£(a?) (x E U),
where £ G C^(U) and p : Ж —> R is the periodic “zig-zag” function defined
by
f	x	if	0	< x <	~
p(x) = 1	л	-г	i	Z Z	1	’ р(ж + !) = Р(ж) (x 6 R)-
I	1 — x	if	5	< x <	1
Thus
(13)	|p'| = 1 a.e.
Observe further vXi(x) =	£г£+О(е) as e —> 0, and so our substituting
(12)	into (11) yields
0< / LPiPj(Du,u,x')(p,)2€i€jC>2 dx + O(e).
JuC>=i
We recall (13) and then send e —> 0, thereby obtaining the inequality
f П
0< / У2 LPiPj(Du,u,x)£i£jC2 dx.
Ju^=i
Since this estimate holds for all £ E C£°(U), we deduce
(14)	LPiPj (Du>u' x^i 0 (С ё R",	€ tl).
г,1=1
We will see later in §8.2 that this necessary condition contains a clue as
to the basic convexity assumption on the Lagrangian L required for the
existence theory.
8.1.4.	Systems.
a. Euler—Lagrange equations.
The foregoing considerations generalize quite easily to the case of sys-
tems, the only new complications being largely notational. Recall from §A.l
that MmXn is the space of real m x n matrices, and assume the smooth La-
grangian function
L : Mmxn x Rm x U -+ R
is given.
438
8. THE CALCULUS OF VARIATIONS
Notation. We will write
L — L(P, z, x) = L(p],... ,p™, z\..., zm, xi,..., xn)
for P e Mmx", z e Rm, and x e U, where
/ Pi • • • Pn\
P -	.
\ nm	nm /
'Pl • • • Pn 7 mxn
(We are now employing superscripts to denote rows: this notational conven-
tion simplifies the following formulas.)	□
As in §8.1.2 we associate with L the functional
(15)	/[w] := I L(Dw(x),w(x),x) dx,
Ju
defined for smooth functions w : U —> Rm, w = (w1,..., wm), satisfying the
boundary conditions w = g on dU, g : dU —> Rm being given. Here
(wl, ... ж! \
XI	Xn \
j
wm	wm /
is the gradient matrix of w at x.
Let us now show that any smooth minimizer u — (г?,...,ит) of /[],
taken among functions equal to g on dU, must solve a certain system of
nonlinear partial differential equations. We therefore fix v =	e
Cc°°(U;Rm) and write
i(r) := L[u + tv].
As before
г'(0) = 0.
From this we readily deduce as above the equality
0 = г'(0) = / Lvk (Du, u, z)vjj + Lzk (Du, u, x\vk dx .
Ju i=i k=i 1	' fc=i
As this identity is valid for all choices of v1,..., vm, we conclude after inte-
grating by parts:
(16)	— У (l k(Du,u, x)\ + Lzk (Du, u, x) = 0 in U (k = 1,..., m).
\	/ хг
г=1
This coupled, quasilinear system of PDE comprise the Euler-Lagrange equa-
tions for the functional /[•] defined by (15).
8.1. INTRODUCTION
439
b. Null Lagrangians.
Surprisingly, it turns out to be interesting to study certain systems of
nonlinear partial differential equations, for which every smooth function is
a solution.
DEFINITION. The function L is called a null Lagrangian if the system
of Euler -Lagrange equations
(17)	— 52 (Lpk(Du, u, хЙ + Lzk(Du, u, x) = 0 in U (к = 1,... ,m)
1 = 1
is automatically solved by all smooth functions u : U —> Rm.
The importance of null Lagrangians is that the corresponding energy
7[w] = / L(Dw,w, x)dx
Ju
depends only on the boundary conditions:
THEOREM 1 (Null Lagrangians and boundary conditions). Let L be a
null Lagrangian. Assume u,u are two functions in C2(U,Rm) such that
(18)	u = u on dU .
Then
(19)	/[u] = /[й] .
Proof. Define
i(r) := /[tu + (1 — т)й] (0 < т < 1) .
Then
г'(т) = [ 5252Л^(г^и+(1-т)Рй,ти+(1-т)й,х)(г4. -йкх.)
Ju г=1fc=l
+ 52 Lzk (tDu + (1 — r)Du, ти + (1 — т)й, x)(ufe — hfe) dx
fc=i
- 52	(r£)u + (! - т)£)й> TU + (1 - т)й, ж))^
г=1
+ Lzk (tDu + (1 — т)£>й, ти + (1 — т)й, x)
= 0,
(uk — uk) dx
440
8. THE CALCULUS OF VARIATIONS
the last equality holding since the Euler-Lagrange system of PDE is satisfied
by tu + (1 — т)й. The identity (19) follows.	□
In the scalar case that m = 1 the only null Lagrangians are the boring
examples where L is linear in the variable p. For the case of systems (m > 1),
however, there are certain nontrivial examples, which will turn out to be
important for us later.
Notation. If A is an n x n matrix, we denote by
cof A
the cofactor matrix, whose (k, i)th entry is (cof A)^ = (—l)z+fed(A)^, where
d(A)k = determinant of the (n—1) x (n—1) matrix obtained by deleting the
ktfl-row and it/l-column from A.	□
LEMMA (Divergence-free rows). Let и : IR" —> Rn be a smooth function.
Then
n
(20)	J>ofDu)^=0 (fc = l,...,n).
i=i
Proof. 1. From linear algebra we recall the identity
(21)	(det P)I = PT(cof P) (P e Mnxn);
that is,
n
(22)	(detP)^j = J2ftfc(cofP)^ (i,j = l,...,n).
fc=i
Thus in particular
(23)	^ = (cofP)^ (fc,m = l,...,n).
2.	Now set P = Du in (22), differentiate with respect to Xj, and sum
j = 1,..., n, to find
n	n
52 (cofDu)^. = 52 uxiXj(cofL>u)J + ukx.(cofDu)^.
j,k,m=l	k,j=l
for i = 1,..., n. This identity simplifies to read
n	/ n	\
(24)	52 £?cof Du^ = 0 (* = 1, • • •, «)•
fc=l	\j=l	/
8.1. INTRODUCTION
441
3. Now if detDu(zo) 0, we deduce from (24) that
n
22(cof Du)jx. = 0 (fc = 1,..., n)
j=i
at xq. If instead detDu(zo) = 0, we choose a number e > 0 so small that
det(Du(zo) + el) / 0, apply steps 1-3 to й := u+ez, and send e —> 0.	□
THEOREM 2 (Determinants as null Lagrangians). The determinant func-
tion
L(P) = det P (Pg Mnxn)
is a null Lagrangian.
Proof. We must show that for any smooth function и : U —> R",
n
£(b(Du)) =0 (k = l,...,n).
d ' г	'Xi
г=1
According to (23) we have Lpk — (cof P)* (i,k = 1,... ,n). But then em-
ploying the notation and conclusion of the lemma, we see
n	n
E (л*(£)u)) = E(cofDu^ = ° (^ = i, •  •, ^).
i=l	X' i=l
□
Some other interesting null Lagrangians are introduced in the exercises.
c. Application.
A nice application is a quick analytic proof of a topological fixed point
theorem.
THEOREM 3 (Brouwer’s Fixed Point Theorem). Assume
и : B(0,l) -> B(0,l)
is continuous, where B(0,1) denotes the closed unit ball in Rn. Then и has
a fixed point; that is, there exists a point x G B(0,1) with
u(z) = x.
442
8. THE CALCULUS OF VARIATIONS
Proof. 1. Write В = B(0,1). We first of all claim that there does not exist
a smooth function
(25)	w : В dB
such that
(26)	w(z) = x for all x G dB.
2.	Suppose to the contrary that such a function w exists. Let us tem-
porarily write w for the identity function, so that w(x) = x for all x G B.
In view of (26), w = w on dB. Since the determinant is a null Lagrangian,
Theorem 1 implies
(27)	/ det_DwcLr= / det Dwdx = |B|	0.
J в	Jв
On the other hand, (25) implies |w|2 = 1; and so differentiating, we find
(28)	(Dw)Tw = 0.
Since |w| = 1, (28) says 0 is an eigenvalue of DwT for each x € B.
Therefore detDw = 0 in B. This contradicts (27), and thereby proves no
smooth function w satisfying (25), (26) can exist.
2.	Next we show there does not exist any continuous function w verifying
(25), (26). Indeed if w were such a function, we continuously extend w by
setting w(z) = x if x G R" — B. Observe that w(z) 0 (z G R"). Fix e > 0
so small that wi := r]£ * w satisfies wi(i) 0 (r G R"). Note also that since
T]£ is radial, we have wi(ar) = x if x G R" — B(0,2), for e > 0 sufficiently
small. Then
2wi
w2 := I---1
I wi|
would be a smooth mapping satisfying (25), (26) (with the ball B(0,2)
replacing В = B(0,1)), in contradiction to step 1.
3.	Finally suppose u : В —> В is continuous, but has no fixed point.
Define now the mapping w : В —> dB by setting w(x) to be the point
on dB hit by the ray emanating from u(x) and passing through x. This
mapping is well defined since и(ж) x for all x G B. In addition w is
continuous and satisfies (25), (26).
But this in turn is a contradiction to step 2.	□
We will employ Brouwer’s fixed point theorem several times in Chapter
9.
8.2. EXISTENCE OF MINIMIZERS
443
8.2.	EXISTENCE OF MINIMIZERS
In this section we will identify some conditions on the Lagrangian L which
ensure that the functional /[•] does indeed have a minimizer, at least within
an appropriate Sobolev space.
8.2.1.	Coercivity, lower semicontinuity.
Let us start with some largely heuristic insights as to when the functional
(1)	Z[w] := / L(Dw(x), w(z), ж) dx,
Ju
defined for appropriate functions w : U —► IR satisfying
(2)	w = g on dU,
should have a minimizer.
a. Coercivity.
We first of all note that even a smooth function f mapping R to R
and bounded below need not attain its infimum. Consider, for instance,
f = ex or (1 + x2)-1. These examples suggest that we in general will need
some hypothesis controlling Z[w] for “large” functions w. Certainly the most
effective way to ensure this would be to hypothesize that Z[w] “grows rapidly
as |w| —> oo”.
More specifically, let us assume
(3)	1 < q < oo
is fixed. We will then suppose
{there exist constants a > 0, /3 > 0 such that
L(p, z, x) > a|p|9 — (3
for all p 6 IR”, z E IR, x 6 U.
Therefore
(5)	Z[w] > a||Dw\\qLg(u) - 7
for 7 := /3|C|. Thus Z[w] —> oo as ||£>w|fy« —> oo. It is customary to call (5)
a coercivity condition on /[•].
Turning once more to our basic task of finding minimizers for the func-
tional /[•], we observe from inequality (5) that it seems reasonable to define
444
8. THE CALCULUS OF VARIATIONS
I[w] not only for smooth functions w, but also for functions w in the Sobolev
space W1,<7(f7) that satisfy the boundary condition (2) in the trace sense.
After all, the wider the class of functions w for which /[w] is defined, the
more candidates we will have for a minimizer.
We will henceforth write
A := {w 6 W1,<7(f7) | w = g on dU in the trace sense}
to denote this class of admissible functions w. Note in view of (4) that /[w]
is defined (but may equal +oo) for each w G A.
b. Lower semicontinuity.
Next, let us observe that although a continuous function f : R —> R
satisfying a coercivity condition does indeed attain its infimum, our integral
functional /[] in general will not. To understand the problem, set
(6)	m := inf I[w]
w&A
and choose functions uk G A (k = 1,...) so that
(7)	/[ut] —> m as k —> oo.
We call	a minimizing sequence.
We would now like to show that some subsequence of {ufc}^°=1 converges
to an actual minimizer. For this, however, we need some kind of compact-
ness, and this is definitely a problem since the space W1,<7(f7) is infinite
dimensional. Indeed, if we utilize the coercivity inequality (5), it turns out
(cf. §8.2.2) that we can only conclude that the minimizing sequence lies in a
bounded subset of	But this does not imply that there exists any
subsequence which converges in Wl,q(U).
We therefore turn our attention to the weak topology (cf. §D.4). Since
we are assuming 1 < q < oo, so that Lq(U) is reflexive, we conclude that
there exists a subsequence	C	and a function и G W1,9(f7)
so that
.	( ukj —и weakly in L9(f7)
} Dukj — Du weakly in Lq(U-Rn).
We will hereafter abbreviate (8) by saying
(9)	ukj —>• и weakly in W1,<7(f7).
8.2. EXISTENCE OF MINIMIZERS
445
Furthermore, it will be true that и = g on dU in the trace sense, and so
и G A.
Consequently by shifting to the weak topology we have recovered enough
compactness from the coercivity inequality (5) to deduce (9) for an appro-
priate subsequence. But now another difficulty arises, for in essentially all
cases of interest the functional /[•] is not continuous with respect to weak
convergence. In other words, we cannot deduce from (7) and (9) that
(10)	7[u] = lim I[uk.],
j—>oo
and thus и is a minimizer. The problem is that Du^ Du does not imply
Ощ. —> Du a.e.: it is quite possible for instance that the gradients Du^,
although bounded in Lq, are oscillating more and more wildly as >oc.
What saves us is the final, key observation that we do not really need
the full strength of (10). It would suffice instead to know only
(11)	I[u] < liminf 7[m.].
j—'OO
Then from (7) we could deduce 7[u] < m. But owing to (6), m < 7[u].
Consequently и is indeed a minimizer.
DEFINITION. We say that a function /[•] is (sequentially) weakly lower
semicontinuous on W1,<7(?7), provided
Z[u] < liminf/[и/;]
fc—>oo
whenever
Uk -1- и weakly in W1,q(U).
Our goal therefore is now to identify reasonable conditions on the non-
linearity L that ensure /[] is weakly lower semicontinuous.
8.2.2. Convexity.
We next look back to our second variation analysis in §8.1.3 and recall
we derived there the inequality
57 LPiP] (-Dw(z), u(z), rc)CiCj >0 (С € R", x G (7)
i,J=l
holding as a necessary condition, whenever и is a smooth minimizer. This
inequality strongly suggests that it is reasonable to assume that L is convex
in its first argument.
446
8. THE CALCULUS OF VARIATIONS
THEOREM 1 (Weak lower semicontinuity)Assume that L is bounded
below, and in addition
the mapping p^> L(p,z,x) is convex,
for each z gR, x 6 U. Then
![] is weakly lower semicontinuous on W1’<7(U).
Proof. 1. Choose any sequence	with
(12)	Uk и weakly in W1,<7(f7),
and set I := liminf/^^ /[ut]. We must show
(13)	I[u] < I.
2.	Note first from (12) and §D.4 that
(14)	supHufcll^i.,^) < oo.
к
Upon passing to a subsequence if necessary, we may as well also suppose
(15)	I = lim /[m].
k—*oo '
Furthermore we see from the compactness theorem in §5.7 that Uk —> и
strongly in L<7(U); and thus, passing if necessary to yet another subsequence,
we have
(16)	Uk —> и	a.e. in U.
3.	Fix e > 0. Then (16) and Egoroff’s Theorem (§E.2) assert
(17)	Uk —> и	uniformly on Ee,
where Ee is a measurable set with
(18)	\U - Eel < e.
Now write
(19)	Fe :=	6 U||u(z)| + |_Du(z)| < -1.
8.2. EXISTENCE OF MINIMIZERS
447
Then
(20)	|(7 —Fe|—>0	ase^O.
We finally set
(21)	G£:=E£OF£,
and observe from (18), (20): |(7 — Ge| —> 0 as e —> 0.
4.	Now let us observe since L is bounded below, we may as well assume
(22)	L > 0
(for otherwise we could apply the following arguments to L = L + /3 > 0 for
some appropriate constant /3). Consequently
/[ut] = / L(Duk, Uk, x) dx > / L(Duk,Uk,x') dx
(23)	Ju	JGe
> / L(Du,Uk,x)dx + / DpL(Du,Uk,x)  (Duk — Du) dx,
JGe	JGc
the last inequality following from the convexity of L in its first argument;
see §B.l. Now in view of (17), (19) and (21):
(24)	lim / L(Du,Uk,x) dx = / L(Du,u,x) dx.
k~^°° J Ge	J Ge
In addition, since DpL(Du,Uk,x) —> DpL(Du,u,x) uniformly on G£ and
Duk Du weakly in Lq(U;№.n), we have
(25)	lim / DpL(Du,Uk,x)  (Duk — Du) dx = 0.
J Ge
Owing now to (24), (25), we deduce from (23) that
1= lim /[«*;] > / L(Du,u,x~) dx.
k^°° Jgc
This inequality holds for each e > 0. We now let e tend to zero, and recall
(22) and the Monotone Convergence Theorem (§E.3) to conclude
l> / L(Du,u,x) dx = I[u\,
Ju
as required.	□
448
8. THE CALCULUS OF VARIATIONS
Remark. It is very important to understand how the foregoing proof deals
with the weak convergence Duk Du. The key is the convexity inequality
(23), on the right hand side of which Duk appears linearly. Weak conver-
gence is, by its very definition, compatible with linear expressions, and so
the limit (25) holds. Remember that it is not in general true that Du^ —> Du
a.e., even if we pass to a subsequence.
The convergence of Uk to и in Lq is much stronger, and so we do not
need any convexity assumption concerning z i—> L(p, z, ж).	□
We can at last establish that /[•] has a minimizer among the functions
in A.
THEOREM 2 (Existence of minimizer). Assume that L satisfies the co-
ercivity inequality (4) and is convex in the variable p. Suppose also the
admissible set A is nonempty.
Then there exists at least one function и G A solving
/[u] = min I[w] .
w€,4
Proof. 1. Set m := infwe_47[w]. If m = +oo we are done, and so we
henceforth assume m is finite. Select a minimizing sequence	Then
(26)	m.
2.	We may as well take = 0 in inequality (4), since we could otherwise
just as well consider L := L -ИД Thus L > a|p|9, and so
(27)	/[w] > a [ \Dw\q dx.
Ju
Since m is finite, we conclude from (26) and (27) that
(28)	sup 11Dufe 11< oo.
k
3.	Now fix any function w G A. Since щ and w both equal g on dU in
the trace sense, we have Uk — w G Wo1,<7(17). Therefore Poincare’s inequality
implies
IWlL»(iq < ||«fc -
< CIIDu/c — Dw||£o((/) + С < C,
8.2. EXISTENCE OF MINIMIZERS
449
by (28). Hence supfe Цзд||l«([/) < oo. This estimate and (28) imply
is bounded in W1,q(U).
4.	Consequently there exist a subsequence	С	and a
function и G W1,q(U) such that
Uk и weakly in Hzl,<7(U).
We assert next that и G A. To see this, note that for w G A as above,
Uk—w E Wj,<7(U). Now wJ,<7(U) is a closed, linear subspace of W1,q(U), and
so, by Mazur’s Theorem (§D.4), is weakly closed. Hence и — w G wj,<7(?7).
Consequently the trace of и on dU is g.
In view of Theorem 1 then, /[u] < liminf,^^/[ufcj = m. But since
и G A, it follows that
/[u] = m = min I[w].
weA
□
We turn next to the problem of uniqueness. In general there can be many
minimizers, and so to ensure uniqueness we require further assumptions.
Suppose for instance
(29)	L = L(p, x) does not depend on г,
and
( there exists 0 > 0 such that
(30)	(p,CeRn; ®eU).
Condition (30) says the mapping p L(p, x) is uniformly convex for each
x.
THEOREM 3 (Uniqueness of minimizer). Suppose (29), (30) hold. Then
a minimizer и G A of ![•] is unique.
Proof. 1. Assume u,u G A are both minimizers of /[•] over A. Then
v := G A. We claim
(31)	/[v] < 44-+ /|й|,
A
with a strict inequality, unless u = u a.e.
2.	To see this, note from the uniform convexity assumption that we have
(32)	L(p, x) > L(q, x) + DpL(q, x)  (p - q)+ ^-\p - q\2 (z G U, p,qE R").
A
450
8. THE CALCULUS OF VARIATIONS
Set q = Du+Du, p = du, and integrate over U:
(33)
rr i f т (Du + Du \ /Du — Di
/М + JUDPL^----------,xj  (-------
+	[ \Du — Du\2 dx < I[u].
Similarly, set q = Du~Du, p = Di in (32) and integrate:
(34)
rt i f r-> т (Du + Du \ /Du - Du\
Z[v] + / DpLl -----------,x  ------------- dx
Ju \	2	/ \ z }
0 f
+ - / \Du — Dvtf2 dx < I[v\.
8 Ju
Add and divide by 2, to deduce
0 f
Z[v] + - / \Du — Du\2 dx <
Z[u] + Z[€t]
2
This proves (31).
3.	As Z[u] = Z[h] = minw6_4 Z[w] < /[и], we deduce Du = Du a.e. in U.
Since и = й = g on dU in the trace sense, it follows that и — й a.e. □
8.2.3. Weak solutions of Euler—Lagrange equation.
We wish next to demonstrate that any minimizer и G A of /[•] solves the
Euler-Lagrange equation in some suitable sense. This does not follow from
the calculations in §8.1 since we do not know и is smooth, only и G W1,<7(17).
And in fact we will need some growth conditions on L and its derivatives.
Let us hereafter suppose
(35)
|L(p,2,z)|<C(H + H + l),
and also
(36)
r \DpL(p,z,x)\ CCdpH + lz^ + l)
t \DzL(p,z,x)\ < ОД’-1 + IzH + 1)
for some constant C and all p G Rn, z G R, x G U.
Motivation for definition of weak solution. We now turn our attention
to the boundary-value problem for the Euler-Lagrange PDE associated with
our functional L, which for a smooth minimizer и reads
(37)
f ~^2i=1(LPi(Du, u,x))Xi + Lz(Du, u,x) =0 in U
[	и = g on dU.
8.2. EXISTENCE OF MINIMIZERS
451
If we multiply (37) by a test function v G	and integrate by parts,
we arrive at the equality
f A
(38)	/ Lp(Du, u, x)vXi 4- Lz(Du, u, x)vdx — 0.
Of course this is the identity we first obtained in our derivation of (37) in
§8.1.2.
Now assume и G W1,q(U). Then using (36) we see
\DpL(Du,u,x}\ < Cd-Dul9'1 + K”1 + 1) G Lq'(U),
where q' =	|	= 1. Similarly
(39) \DzL(Du,u,x)\ < CQ/M7"1 + M9-1 +1) G Lq'(U).
Consequently we see using a standard approximation argument that the
equality (38) is valid for any v G W(J’9(t/). This motivates the following
DEFINITION. We say и G A is a weak solution of the boundary-value
problem (37) for the Euler-Lagrange equation provided
f П
/ 2 LPi(Du, u,x)vXi + Lz(Du,u,x)vdx = 0
for all v G Rq1,9(Z7).
THEOREM 4 (Solution of Euler-Lagrange equation). Assume L verifies
the growth conditions (35), (36), and и G A satisfies
7[u] = min /[w],
wEA
Then и is a weak solution of (37).
Proof. We proceed as in §8.1.2, taking care about differentiating under the
integrals. Fix any v G Wq’9(U) and set
In view of (35) we see that i(r) is finite for all r.
Let r 0 and write the difference quotient
f L(Du + rDv,u-h rv,x) — L(Du,u,x)
I -------------------------------------dx
и	т
(40)
и
452
8. THE CALCULUS OF VARIATIONS
where
LT(x) := -\L(Du(x) + rDv(x), u(x) + tv(x),x) — L(Du(x), u(x), ж)]
for a.e. x € U. Clearly
n
(41) Lr(x) —> У~^Тр.(Ри,и, x)vXi + Lz(Du, u,x)v a.e.
i=i
as 7" —> 0. Furthermore
1 d
Lr(x) = - / — L(Du + sDv,u + sv, x) ds
T Jo ds
1 fr n
= - y^Lp^DuV sDv,u +sv,x)vXi
T i=i
+ Lz(Du + sDv,u + sv,x)vds.
Next recall from §B.2 Young’s inequality: ab < у + ^r, where |	= 1-
Then since u,v G W1,q(U), inequalities (36) and Young’s inequality imply
after some elementary calculations that
\LT(x)\ < C(\Du\q + |u|9 + \Dv\q + H9 + 1) G L^U)
for each r 0. Consequently we may invoke the Dominated Convergence
Theorem to conclude from (40), (41) that z'(0) exists and equals
и
•u, u, x)vXi + Lz(Du, u, x)v dx.
But then since г(-) has a minimum for т = 0, we know i'(0) = 0; and thus и
is a weak solution.	□
Remark. In general, the Euler-Lagrange equation (37) will have other so-
lutions which do not correspond to minima of /[]; see §8.5. However, in the
special case that the joint mapping (p, z) i—> L(p, z, x) is convex for each x,
then each weak solution is in fact a minimizer.
To see this, suppose и G A solves
f - ^i=1(LPi(Du,u,xy)Xi + L2(Du,u,x) =0	in U
1	и = g	on dU
8.2. EXISTENCE OF MINIMIZERS
453
in the weak sense and select any w 6 A. Utilizing the convexity of the
mapping (p, z) i—> L(p, z, x), we have
L(p, z, x) + DpL(p, z, x)  (q—p) + DzL(p, z, x) • (w — z) < L(q, w, x).
Let p = Du(x), q = Dw(x), z = u(x), w = w(x) and integrate over U:
L[u] + / DpL(Du,u, x) • (Dw — Du) + DzL(Du,u,x)(w — u) dx < /[w],
Ju
In view of (42) the second term on the right is zero, and therefore I[u] < L[w]
for each w 6 A.	□
8.2.4. Systems.
a. Convexity.
We now adopt again the notation for systems set forth in §8.1.4, and
consider the existence question for minimizers of the functional
L[w] := / L(Pw(i),w(i),s) dx,
Ju
defined for appropriate functions w : U —> Rm, where now L : Mmx" x Rm x
U —> IR is given.
It turns out the theory developed in §8.2.2 extends with no difficulty to
the case at hand. Let us therefore assume the coercivity inequality
(43)	L(P,z,x) > a|P|9 -0 (P GMmxn,zGRm,zG U)
for constants a > Q^/3 > 0, and set also
A = {w € U/1',7(U; Rm) | w = g on dU in the trace sense},
where g : dU —> Rm is given.
THEOREM 5	(Existence of minimizer). Assume that L satisfies the co-
ercivity inequality (43) and is convex in the variable P. Suppose also the
admissible set A is nonempty.
Then there exists u 6 A solving
Liu] = min Llw].
wg.4
The proof follows almost exactly the proofs of Theorems 1 and 2 in
§8.2.2. Similarly to Theorem 3 above we have:
454
8. THE CALCULUS OF VARIATIONS
THEOREM 6	(Uniqueness of minimizer). Assume L does not depend on
z and the mapping P i—> L(P, x) is uniformly convex. Then a minimizer
u 6 A <>//[•] is unique.
Now suppose additionally
г |L(P,z,z)| <C(|P|9 + |z|9 + l)
(44)	J \DPL(P,z,x)\ < CCPI9-1 + H’-1 + 1)
I \DzL(P,z,x)\ < CdPl9"1 + H9"1 + 1)
for some constant C and all P 6 Mmx", z G Rm, x G U.
We consider now the system of Euler-Lagrange equations
(...	/-Ег"=1(^(-°и,и,х))1г +Lzk(Du,u,x) = 0 in U
(45)	S	*	к к orr
(	uk = gK on dU
for k = 1,..., m, and define и G A to be a weak solution provided
m p n
_ E..E Lpk (Pu, u, z)w£. + Lzk (Du, u, x)wk dx = 0
k=i Ju i=i
for all w G	w = (w1,..., wm).
THEOREM 7 (Solution of Euler-Lagrange system). Assume L verifies
the growth conditions (44) and u G A satisfies
Pul = min D[w],
wM
Then и is a weak solution of (45).
The proof is almost precisely like that of Theorem 4.
b. Polyconvexity.
It is rather surprising that there are some mathematically and physically
interesting systems which are not covered by Theorem 5 above, but which
can still be studied using the calculus of variations. These include certain
problems where the Lagrangian L is not convex in P, but /[•] is nonetheless
weakly lower semicontinuous.
LEMMA (Weak continuity of determinants). Assume n < q < oo and
Ufc —>• u weakly in Wi,q(U; Rn).
Then
det Du/; —1 det Du	weakly in Lq/n(U).
8.2. EXISTENCE OF MINIMIZERS
455
Proof. 1. First we recall the matrix identity (detP)I = P(cofP)T; conse-
quently
det P = p’ (cofP)j (i = 1,..., n).
j=i
2.	Now let w 6 C'oo(?7;Rn), w = (w1,..., wn). Then
(46)	det Dw =	. (cof Dw)’- (i = l,...,n).
j=i
But the lemma in §8.1.4 asserts J2j=i(c°f = 0- Thus formula (46)
says
det Dw = 57 (w’(cof Dw)’ )Xj.
j=i
Consequently the determinant of the gradient matrix can be written as a
divergence. Therefore if v G	we have
(47)	/ vdetDwdx = — Y^ / vx-wl(cofDw)’dx (i = l,...,n).
Ju	Ju
3.	We have established the identity (47) for a smooth function w; and
so a standard approximation argument yields
(48)	/ v det Du/; dx = — 37 / vx u^cof DuJ)dx
Ju	Ju
for k = 1,2,.... Now since n < q < oo and 1 u in Ил1,<7(Р; Rn), we
know from Morrey’s inequality that {ufc}^ is bounded in C'0,1~r’/<7(D;R").
Thus using the Arzela-Ascoli compactness criterion, §C.7, we deduce —>
u uniformly in U. Returning then to identity (48) we see that we could
conclude
(49)	lim I v det Du/; dx = — Yj /	. u’(cof Du)’dz = / г; det Du dz,
^KJlJ	j=A U	U
if we knew
(50)	lim / ip(cof Duk)} dx = I ^(cof Du)’dz
k-*°° Ju	Ju
for i,j — 1,... ,n and each ip 6	However (cof Du^)’ is the determi-
nant of an (n — 1) x (n — 1) matrix, which can be analyzed as above by being
456
8. THE CALCULUS OF VARIATIONS
written as a sum of determinants of appropriate (n—2) x (n—2) submatrices,
times uniformly convergent factors. We continue and eventually must show
only the obvious fact that the entries of the matrices Du^ converge weakly
to the corresponding entries of Du. In this way we verify (50), and thus
(49).
4.	Finally, since {u/c}^S1 is bounded in Wi,q(U- Rn) and |det£>Ufc| <
C'lZJufcl", we see that {det -Dufe}^ is bounded in Lq/n(U). Hence any sub-
sequence has a weakly convergent subsequence in Lq/n(U), which—owing to
(49)—can only converge to det Du.	□
We next utilize this lemma to establish a weak lower semicontinuity
assertion analogous to Theorem 1, except that we will not assume that the
Lagrangian L is necessarily convex in P. Instead let us suppose that m = n
and L has the form
(51)	L(P,z,x) = F(P, det P,z,x) (P G Mnxn,z G Rn,z G U)
where F : Mnx" x R x Rn xU —> R is smooth. We additionally hypothesize
that
( for each fixed z G Rm, x G R, the joint mapping
( )	[ (P, r) i—> F(P, r, z, x) is convex.
A Lagrangian L of the form (51) is called polyconvex provided (52) holds.
THEOREM 8 (Lower semicontinuity of polyconvex functionals). Suppose
n < q < oo. Assume also L is bounded below and is polyconvex. Then
/[•] is weakly lower semicontinuous on W1,<7(P; R").
Proof. Choose any sequence {uk}^ with
(53)	Ufc —>• u weakly in W1,<7(P; Rn).
According to the lemma,
(54)	det Duk det Du weakly in Lq/n(U).
8.2. EXISTENCE OF MINIMIZERS
457
We can now argue almost exactly as in the proof of Theorem 1. Indeed,
using the notation from that proof, we have
/[ut] = / L(Duk, Ufc.rc) dx > / L(Duk, ufe, x) dx
Ju	JGe
= / F(Duk,deX Duk,uk,x) dx
Jg,
> I F(Du, det Du, uk, x) dx
Jg,
+ / Fp(Du, det Du, u*,, x)  (Duk — Du)
Jg€
+ Fr(Du, det pu, Ufc, z)(det Duk — det Du) dx,
in view of (52). Reasoning as in the proof of Theorem 1 we deduce from
(53), (54) that the limit of the last term is zero as k —> oo.	□
As before, we immediately deduce
THEOREM 9 (Existence of minimizers, polyconvex functionals). Assume
that n < q < oo, and that L satisfies the coercivity inequality (43) and is
polyconvex. Suppose also the admissible set A is nonempty.
Then there exists u G A solving
7[u] = min/[w],
weA
Example. We explain the interest in the hypothesis of polyconvexity with
an application from nonlinear elasticity theory, where n = 3. We consider
an elastic body, which initially has the reference configuration U. We then
displace each point x G dU to a new position g(x) and wish to determine
new displacement u(x) of each internal point x G U.
If the material is hyperelastic, there exists by definition an associated en-
ergy density L such that the physical displacement u minimizes the internal
energy functional
7[w] := / L(Dw, x) dx
Ju
over all admissible displacements w G A. Now it seems reasonable physically
that L, which represents the internal energy density from stretching and
compression, may explicitly depend on the local change in volume, that is,
on det Dw. In other words, it is physically appropriate to suppose that L
has the form L(P, x) — F(P, det P, x). Then F describes in its first argument
changes in internal energy due to changes in line elements, and in its second
argument changes in internal energy due to changes in volume elements. See
Ball [B] for more explanation.	□
458
8. THE CALCULUS OF VARIATIONS
8.3.	REGULARITY
We discuss in this section the smoothness of minimizers to our energy func-
tionals. This is generally a quite difficult topic, and so we will make a number
of strong simplifying assumptions, most notably that L depends only on p.
Thus we henceforth assume our functional /[•] to have the form
(1)	/[w] := I L(Dw) — wf dx,
Ju
for f G L2(U). We will also take q = 2, and suppose as well the growth
condition
(2)	\DpL(p)\ < C(\p\ + 1) (pGK").
Then any minimizer и E A is a weak solution of the Euler-Lagrange PDE
n
(3)	-^(LPt(DU)K = f in U;
i=i
that is,
(4)	/ 57 LPr (Du)vn dx = fv dx
Ju	Ju
for all v E Hq(U).
8.3.1.	Second derivative estimates.
We now intend to show ifuGH1(U)isa weak solution of the nonlinear
PDE (3), then in fact и E H^OC(U). But to establish this we will need to
strengthen our growth conditions on L. Let us first of all suppose
(5)	\D2L{p)\<C (pEV).
In addition let us assume that L is uniformly convex, and so there exists
a constant в > 0 such that
n
(6)	E TpiPj
i,j=l
Clearly this is some sort of nonlinear analogue of our uniform ellipticity
condition for linear PDE in Chapter 6. The idea will therefore be to try to
utilize, or at least mimic, some of the calculations from that chapter.
8.3. REGULARITY
459
THEOREM 1 (Second derivatives for minimizers).
(i) Let и G H^U) be a weak solution of the nonlinear partial differential
equation (3), where L satisfies (5), (6). Then
и G HlfiU).
(ii) If in addition и e Hq(U) and dU is C2, then
иG H\U),
with the estimate
IlulIh2(C7) <
Proof. 1. We will largely follow the proof of Theorem 1 in §6.3.1, the
corresponding assertion of local H2 regularity for solutions of linear second-
order elliptic PDE.
Fix any open set V CC U and choose then an open set W so that
V CC W CC U. Select a smooth cutoff function ( satisfying
( С ее 1 on V, C = 0 in R" - W
t 0 < < < 1.
Let |/z| > 0 be small, choose к G {1,..., n}, and substitute
v := -D'h^2Dhku)
into (4). We are employing here the notation from §5.8.2:
= “(х + Л^-uW
h
Using the identity uD^hv dx = — fL, vDku dx, we deduce
(7)	£ [ Dk(LPi(Duy)(fi2Dku)Xidx = — [ fD^2Dhku)dx.
,=i Ju	Ju
Now
1 r1 d
= T	— LpfisDu(x + hefc) + (1 - s)Du(xy>ds
h Jo ds
1 yi w л
(8)	=T y2LPiPj(sDu(x + hek) + (l-s)Du(x))ds
hJo j=i
(uXj(x + hek) -uXj(x))
= ^aij(X)DkUxj(x),
3=1
460
8. THE CALCULUS OF VARIATIONS
for
(9)	a^(x) := [ LPiPj(sDu(x + hek) + (1 — s)Du(x)) ds (i, j = 1,... ,n).
Jo
We substitute (8) into (7) and perform simple calculations, to arrive at
the identity:
Л1+А2:=^ f ^a^D^uXjDluXidx
i,j=l JU
(10)	+ £ J a^jDkuXjDku2^Xi dx
i,j=l JU
= ~ [ fD^^Dhku)dx=-. B.
Ju
Now the uniform convexity condition (6) implies
(11)	Ai > 0 [ C2\D£Du\2 dx.
Ju
Furthermore we see from (5) that
|A2|<C / C\DhkDu\\Dhku\dx
№	f	C	f
<б/ Q2\DkDu\2 dx + — I \D%u\2dx.
Jw	6	Jw
Furthermore, as in the proof of Theorem 1 in §6.3.1, we have
|B| < e [ C2\DkDu\2dx + — [ f2 + \Du\2dx.
Ju	6 Ju
We select e = | , to deduce from the foregoing bounds on Ai,A2,B, the
estimate
[ C\DkDu\2 dx < C [ f2 + \D%u\2 dx < C [ f2 + \Du\2dx,
Ju	Jw	Ju
the last inequality valid according to Theorem 3,(i) in §5.8.2.
2. Since C = 1 on V, we find
[ \D^Du\2dx<C [ f2 + \Du\2dx
8.3. REGULARITY
461
for к = 1,..., n and all sufficiently small |h\ > 0. Consequently Theorem 3,
(iii) in §5.8.2 implies Du G Нг(У), and so и G H2(V). This is true for each
V CC U- thus и G H2OC(U').
3. If и G Hq(U) is a weak solution of (3) and dU is C2, we can then
mimic the proof of the boundary regularity Theorem 4 in §6.3.2 to prove
и G Я2(17), with estimate
IIuIIh2(c7) < C'(II/IIl2(c7) + IkllHut/));
details are left to the reader. Now from (6) follows the inequality
(DL(p) - DL(0)) • p > в\р]2 (РеП
If we then put v = и in (4), we can employ this estimate to derive the bound
IЫ\hi(LT) < C'll/ll£2([7),
and so finish the proof.
8.3.2. Remarks on higher regularity.
We would next like to show that if L is infinitely ^ifferentiable, then so is
u. By analogy with the regularity theory developed for second-order linear
elliptic PDE in §6.3, it may seem natural to try to extend the H2jC estimate
from the previous section to obtain further estimates in the higher Sobolev
spaces Hioc(U') for к = 3,4,....
This method will not work for the nonlinear partial differential equation
(3) however. The reason is this. For linear equations we could, roughly
speaking, differentiate the equation many times and still obtain a linear
PDE of the same general form as that we began with. See for instance the
proof of Theorem 2 in §6.3.1 Whereas if we differentiate a nonlinear differen-
tial equation many times, the resulting increasingly complicated expressions
quickly become impossible to handle. Much deeper ideas are called for,
the full development of which is beyond the scope of this book. We will
nevertheless at least outline the basic plan.
To start with, choose a test function w G	select к G {1,... , n},
and set v = — wXk in the identity (4), where for simplicity we now take
f = 0. Since we now know и G H2OC(U), we can integrate by parts to find
« n
(13)	/ Lptpj^Du^UxkXjWxt dx = 0.
462
8. THE CALCULUS OF VARIATIONS
Next write
(14)	й := uXk
and
(15)	alJ := LPiPj(Du) (i, j = 1,..., n).
Fix also any V CC U. Then after an approximation we find from (13) (15)
that
« n
(16)	/ (x)uXjwXi dx = 0
U i,j=l
for all w e Hq(V). This is to say that й G Нг(у) is a weak solution of the
linear, second order elliptic PDE
1
(17)
i,J=l
But we cannot just apply our regularity theory from §6.3 to conclude
from (17) that й is smooth, the reason being that we can deduce from (5)
and (15) only that
aVe£OO(y)	= 1,...)П).
However a deep theorem, due independently to DeGiorgi and to Nash, as-
serts that any weak solution of (17) must in fact be locally Holder continuous
for some exponent 7 > 0. (See Gilbarg-Trudinger [G-T, Chapter 8].) Thus
if W CC V we have й G C0,7(W), and so
и e
Return to the definition (15). If L is smooth, we now know aiJ G (U)
(i,j = 1,..., n). Then (3) and an older theorem of Schauder [G-T, Chapter
4] assert that in fact
u G Cf^U).
But then a1-7 G	an<^ so another version of Schauder’s estimate im-
plies
u 6 C?o’cW
We can continue this so-called “bootstrap” argument, eventually to deduce
и is C^(U) for к = 1,..., and so и G C°°(U).
See Giaquinta [GI] for much more about regularity theory in the calculus
of variations.
8.4. CONSTRAINTS
463
8.4.	CONSTRAINTS
In this section we consider applications of the calculus of variations to certain
constrained minimization problems, and, in particular, discuss the role of
Lagrange multipliers in the corresponding Euler-Lagrange PDE.
8.4.1.	Nonlinear eigenvalue problems.
We investigate first problems with integral constraints. To be specific,
let us consider the problem of minimizing the energy functional
(1)	/[w] := | [ \Dw\2 dx
Ju
over all functions w with, say, w = 0 on dill, but subject now also to the
side condition that
(2)	J[w] := [ G(w)dx = 0,
Ju
where G : R —> R is a given, smooth function.
We will henceforth write g = G'. Assume now
(3)	Ш1 < C(\z\ +1),
and so
(4)	|ОД| < C(\z\2 + 1) (z6R)
for some constant C.
Let us introduce as well the appropriate admissible class
A:= {w e Hq(U) \ J[w] = 0}.
We suppose also that the open set U is bounded, connected and has a smooth
boundary.
THEOREM 1 (Existence of constrained minimizer). Assume the admis-
sible set A is nonempty. Then there exists и 6 A satisfying
/[u] - min/[w].
weA
464
8. THE CALCULUS OF VARIATIONS
Proof. Choose, as usual, a minimizing sequence	C A with
/[ufc] —> m = inf /[w],
weA
Then as above we can extract a subsequence
(5)	ukj^u weakly in Hq(U),
with /[u] < m. We will be done once we show
(6)	J[u] = 0,
so that и G A. Utilizing the compactness theory from §5.6, we deduce from
(5) that
(7)	ukj ->u in T2(U).
Consequently
|J(u)| = |J(u)-J(ufe)|< [ \G(u) — G(uk)\dx
Ju
|u - ufe|(l + |u| + |ufc|)dx by (3)
—> 0 as к —> oo .
□
Far more interesting than the mere existence of constrained minimizers
is an examination of the corresponding Euler-Lagrange equation.
THEOREM 2 (Lagrange multiplier). Let и G A satisfy
(9)	/[u] = min/[w],
шеЛ
Then there exists a real number A such that
(10)	I Du-Dvdx = X / g(u)vdx
Ju	Ju
for all v e Hq(U).
Remark. Thus и is a weak solution of the nonlinear boundary-value prob-
lem
, .	f -Au = Xg(u) in U
' '	| и = 0	on dU,
where A is the Lagrange multiplier corresponding to the integral constraint
(12)	J[u] = 0.
A problem of the form (11) for the unknowns (u, A), with и 0, is a non-
linear eigenvalue problem.	□
8.4. CONSTRAINTS
465
Proof. 1. Fix any function v e Hq(U). Assume first
(13)	g(u) is not equal to zero a.e. within U.
Choose then any function w 6 with
(14)	/ g(u)wdx/0;
Ju
this is possible because of (13). Now write
j(r, cr) := J[u + tv + <rw]
(15)	= [ G(u + tv + crw) dx (т, (тЕЙ).
Ju
Clearly
(16)	j (0,0) = [ G(u)dx = 0.
Ju
In addition, j is C1 and
Qj	Г
(17)	— (r, cr) = / g(u + tv + aw)v dx,
от	Ju
Qj	Г
(18)	— (r, cr) = / g(u + tv + (rw)wdx.
<J<r Ju
Consequently (14) implies
(19)	|(0,0) /0.
According to the Implicit Function Theorem (§C.6), there exists a C1
function ф : R —> R such that
(20)	0(0) = 0
and
(21)	Ят,0(т)) = О
for all sufficiently small t, say |r| < tq. Differentiating, we discover
|^(т,^(т)) + ^(t,^(t))^(t-) = 0;
466
8. THE CALCULUS OF VARIATIONS
whence (17) and (18) yield
, >	,, ч Lo(u)udx
22	0' o = -77 ; ,.
\vg(u)wdx
2. Now set
w(r) := tv + </>(t)w (|t| < tq)
and write
i(r) := I[u + w(r)].
Since (21) implies J[u + w(t)] — 0, we see that и + w(r) G A. So the C1
function ?(•) has a minimum at 0. Thus
0 = iz(0) = / (Du + tDu + </>(t)Dw) • (Du + </>z(t)Dw) dx|r=o
(23)	7
= / Du • (Du + </>z(0)Dw) dx.
Ju
Recall now (22) and define
fv Du  Dw dx
ffrg(u)wdx '
to deduce from (23) the desired equality
/ Du-Dvdx = X / g(u)vdx
Ju	Ju
for all и G Hq(U).
3. Suppose now instead of (13) that
g(u) = 0 a.e. in U.
Approximating g by bounded functions, we deduce DG(u) = g(u)Du = 0
a.e. Hence, since U is connected, G(u) is constant a.e. It follows that
G(u) = 0 a.e., because J[u] = G(u) dx = 0. As и — 0 on dU in the trace
sense, it follows that G(0) = 0.
But then и = 0 a.e., as otherwise /[u] > T[0] = 0. Since g = 0 a.e., the
identity (10) is trivially valid in this case, for any A.	□
8.4. CONSTRAINTS
467
8.4.2. Unilateral constraints, variational inequalities.
We study now calculus of variation problems with certain pointwise,
one-sided constraints on the values of u(x) for each x G U. For definiteness
let us consider the problem of minimizing, say, the energy functional
(24)	I[w] := [ ^\Dw\2 — fwdx,
Ju
among all functions w belonging to the set
(25)	A := {w G Hq(U) I w > h a.e. in U},
where h : U —> R is a given smooth function, called the obstacle. The convex
admissible set A thus comprises those functions w E Hq(U) satisfying the
one-sided or unilateral constraint that w > h. We suppose as well that f is
a given, smooth function.
THEOREM 3 (Existence of minimizer). Assume the admissible set A is
nonempty. Then there exists a unique function и E A satisfying
Ify] = minify;].
шеЛ
Proof. 1. The existence of a minimizer follows very easily from the general
ideas discussed before. We need only note explicitly that if {ид,.C A is
a minimizing sequence with и^ и weakly in Hq(U), then by compactness
we have и*,. —> и strongly in L2(U). Since > h a.e., it follows that и > h
a.e. Therefore и E A.
2.	We now prove uniqueness. Assume и and й E A are two minimizers,
with и u. Then w := G A, and
/[w]= f
Ju
= j |(\Du\2 + 2Du  Du + |Z?fi|2) - f (^) dx.
Now 2a • b = |a|2 + |6|2 — |a — 6|2. Thus
I[w]= [ |(2|Z?u|2 + 2|Z?u|2 - \Du — Du\2) - f dx
Ju
< | f ||Z?u|2 — fudx + | f ^\Du\2 — fudx
Ju	Ju
= jlfy] + jlfy],
the strict inequality holding since и й. This is a contradiction, since и
and й are minimizers.	□
We next compute the analogue of the Euler-Lagrange equation, which
for the case at hand turns out to be an inequality.
468
8. THE CALCULUS OF VARIATIONS
THEOREM 4 (Variational characterization of minimizer). Let и G A be
the unique solution of
Z[u] = min Z[w].
weA
Then
(26) I Du  D(w — u)dx > I f(w — u) dx for all weA-
Ju	Ju
We call (26) a variational inequality.
Proof. 1. Fix any element w E A Then for each 0 < r < 1,
и + r(w — u) = (1 — r)u + tw e A,
since A is convex. Thus if we set
г(т) := I[u + r(w — u)],
we see that i(0) < г(т) for all 0 < r < 1. Hence
(27)	i'(0) > 0.
2. Now if 0 < т < 1,
г(т)-г(0)	1 f \Du + rD(w - u)\2 - \Du\2 J1
----------— - I -------------------------------f(u + r(w — u) — u) dx
т т Ju	2
= [ Du • D(w -u) + T\D(W ~	_ f(w _ ц) dx.
Ju	2
Thus (27) implies
0 < iz(0) = / Du ’ D(w ~ u) — f(w — u) dx.
Ju
□
Notice that we obtain the inequality (27), since we can in effect take
only “one-sided” variations, away from the constraint.
Interpretation of the variational inequality. To gain some insight
into the variational inequality (26), let us quote without proof a regularity
assertion (see Kinderlehrer-Stampacchia [K-S]), which states и G W2,oo(t/),
provided dU is smooth. Hence the set
О := {ж G U | u(a?) > h(x)}
8.4. CONSTRAINTS
469
The free boundary for the obstacle problem
is open, and
С := {x G U | u(x) = /i(x)}
is (relatively) closed.
We claim that in fact и G C°°(O) and
(28)
—Au = f in 0.
To see this, fix any test function v G C£°(O). Then if |r| is Sufficiently small,
w := и + tv > h, and so w G A. Thus (26) implies т fo Du  Dv — fvdx > 0.
This inequality is valid for all sufficiently small t, both positive and negative,
and so in fact
/ Du  Dv — fv dx = 0
Jo
for all v G C£°(O). Hence u is a weak solution of (28); whence linear
regularity theory (§6.3) shows и G C°°(O).
Now if v G C^fU) satisfies v > 0 and if 0 < т < 1, then w := u+rv G A,
whence Du  Dv — fvdx > 0. But since и e W2,OO(U), we can integrate
by parts to deduce Jf;(—Au — f)vdx > 0 for all nonnegative functions
v G Cc°°(t7). Thus
—Au > f a.e. in U.
We summarize our conclusions by observing from (28), (29) that
(30)
-Au = /
a.e. in U
on U A {u > h}.
470
8. THE CALCULUS OF VARIATIONS
A harmonic map into a sphere
Remark. The set
F := дО П U
is called the free boundary. Many interesting problems in applied mathemat-
ics involve partial differential equations with free boundaries. Such of these
problems as can be recast as variational inequalities become relatively easy
to study, especially in that there is no explicit mention of the free boundary
in the inequalities (30). Applications arise in stopping time optimal control
problems for Brownian motion, in groundwater hydrology, in plasticity the-
ory, etc. See Kinderlehrer-Stampacchia [K-S].	□
8.4	.3. Harmonic maps.
We consider next the case of pointwise constraints as exemplified by
harmonic maps into spheres. We are interested now in the problem of min-
imizing the energy
(31)	Z[w] := | f |Z?w|2dx
Ju
over all functions belonging to the admissible class
(32)	A := {w e H\U-, Rm) | w = g on dU, |w| = 1 a.e.}.
The idea is that we are trying to minimize the energy over all appropriate
maps from U C R" into the unit sphere = <98(0,1) C Rm. This
problem and its variants arises for instance as a greatly simplified model for
the behavior of liquid crystals.
It is straightforward to verify that there exists at least one minimizer in
A, provided A A 0.
THEOREM 5 (Euler-Lagrange equation for harmonic maps). Let u 6 A
satisfy
Z[u] = minZ[w].
weA
8.4. CONSTRAINTS
471
(34)
Then
(33)	j Du : Dv dx = f |Z)u|2u vdx
Ju	Ju
for eachvt	D L°°(U;Rm).
Remark. We interpret (33) as saying that u = (u1,..., um) is a weak
solution of the boundary-value problem
—Au = |Du|2u in U
u = g on dU.
The function A = |Du|2 is the Lagrange multiplier corresponding to the
pointwise constraint |u| = 1. Note carefully that for a single, integral con-
stant (§8.4.1) the Lagrange multiplier is a number, but for a pointwise con-
straint it is a function.	□
Proof. 1. Fix v e	П L°°(U;Rm). Then since |u| = 1 a.e., we
have
|u + tv| 0 a.e.
for each sufficiently small r. Consequently
Thus
г(т) := I[v(r)]
has a minimum at т = 0, and so, as usual,
(36)	г'(0) = 0.
2. Now	x,
(37)	?'(0) = [ Du : £>v'(0) dx (' = V
Ju	\ dr J
But we compute directly from (35) that
V'M = v _ [(u + tv) • v](u + tv)
|u + tv|	|u + tv|3 ’
whence v'(0) = v — (u • v)u. Inserting this equality into (36), (37), we find
(38)	0= [ Du : Dv — Du : £)((u • v)u) dx.
Ju
However since |u|2 = 1, we have
(fiu)ru = 0.
Using this fact, we then verify
Du : £)((u • v)u) = |£hi|2(u • v) a.e. in U.
This identity employed in (38) gives (33).	□
472
8. THE CALCULUS OF VARIATIONS
8.4.4. Incompressibility.
a. Stokes’ problem.
Suppose U C R3 is open, bounded, simply connected, and set
/[w] := f ||Dw|2 — f • wdx,
Ju
for w belonging to
A = {w G 77j(17;R3) | divw = 0 in U}.
Here f G L2(t7';R3) is given.
There is no problem in showing by customary methods that there exists
a unique minimizer u G A. We interpret u as representing the velocity field
of a steady fluid flow within the region U, subject to the external force f.
The constraint that divu = 0 ensures that the flow is incompressible: see
the Remark at the end of this section.
How does the constraint manifest itself in the Euler-Lagrange equation?
THEOREM 6 (Pressure as Lagrange multiplier). There exists a scalar
function p G L2OC(D) such that
(39)	I Du : Dv dx = I p div v + f • v dx
for all v G 771(D;R3) with compact support within U.
Remark. We interpret (39) as saying that (u,p) form a weak solution of
Stokes ’ problem
(40)
—Au = f — Dp in U
div u = 0 in U
u = 0 on dU.
The function p is the pressure and arises as a Lagrange multiplier cor-
responding to the incompressibility condition divu = 0.
Proof. 1. Assume first v G A. Then for each r G R, u + tv G A. Thus
(41)
0 = i'(0) = / Du : Dv — f • v dx.
Ju
8.4. CONSTRAINTS
473
2. Fix now V CC U, V smooth and simply connected, and select w G
_JZq(V;R3) with divw = 0. Choose 0 < e < dist(V,<9t/) and set v = v£ :=
* w in (41), r]e denoting the usual mollifier and w defined to be zero in
U - V. Then
(42)	0 = [ Du : Dv' — f • v£ dx = [ Du€ : Dw — f£ • w dx
Ju	Ju
for
(43)	u£ := rp * u, f£ :=	* f.
As u£ is smooth, (42) implies
(44)	[ (—Au6 — f£)-wdx = 0
Jv
for each w G 77q(V;R3) with divw = 0.
3.	Fix any smooth vector field £ G C£°(V;R3) and put w = curl£ in
(44). This is legitimate since divw = div(curl£) = 0. Then, temporarily
writing h := Au£ + f£, we find
0= [ h cuACdx= f h1^ -Cx3)+ /i2(Ci3 -Cr1) + /j3(Cr1
for h = (h1, h2, /i3), С = (C1, C2> C3). An integration by parts reveals
0 = / <4h3X2 - /i23) + C2(/>13 - h3X1) + C3(/i^ - h'jdx.
Jv
As C1,^2,^3 C^(V) are arbitrary, we deduce curlh = 0 in V. Since V is
simply connected, there consequently exists a smooth function pe in V such
that
(45)	Dpf = h = Au£ + f£ in V.
4.	If necessary we can add a constant to p£ to ensure fv pc dx = 0.
In view of this normalization, there exists a smooth vector field v£ : V —>
R3 solving
, .	(div v£ =pe in V
[ v£ = 0 on dV.
In addition we have the estimate
(47)	II^IIhTV;®3) C'IIP£llb2(V)>
474
8. THE CALCULUS OF VARIATIONS
the constant C depending only on V. (We omit the proof of the existence of
the vector field v£: the construction both is intricate and requires knowledge
of certain estimates for Laplace’s equation with Neumann-type boundary
conditions beyond the scope of this book. See Dacorogna-Moser [D-M] for
details.)
Now compute
f (pf)2dx = f pedivvedx by (46)
Jv Jv
= — Dpe  v£ dx
Jv
= [ (—Au£ - f£) • v£ dx by (45)
Jv
= [ Dur : Dv£ - f£ • v£ dx
Jv
<	IIVJIH1 (1Z;1R3) (11и<1 II Я1 (V) + l|fe||L2(V))
<	C||p£||L2([7)(||u||hoi([7) + ||f||l2({7)) by (47).
Thus
(48)	IIp£||l2(v) < C (IIuIIh^v) + l|f||£2(v)) •
5.	In view of estimate (48) there exists a subsequence e7 —> 0 so that
(49)	p£j p weakly in L2(V)
for some p e L2(V). Now (45) implies
/ Due : Dv dx = / pe div v + f£ • v dx
Jv	Jv
for all v e #o(V;R3). Sending e = ej —>• 0 we find
(50)	/ Du : Dv dx = p div v + f • v dx
Jv	Jv
as well.
6.	Finally choose a sequence of sets Vk CC U (к = 1,...) as above,
with Vi C V2 C V3 C • • • and U = U/Xi И- Utilizing steps 2-5 we find
Pk^L\Vk} (fc = 1,...) so that
(51)	/ Du:Dvdx = / Pk div v + f  v dx
Jvk	Jvk
for each v G H^Vk; Ж3). Adding constants as necessary to each Pk, we
deduce from (51) that if 1 < I < к, then pk = pi on V/. We finally define
p = Pk on Vk (к = 1,...).	□
8.4. CONSTRAINTS
475
b. Incompressible nonlinear elasticity.
We return now to the model of nonlinear elasticity discussed before in
§8.2.4. Suppose that u represents the displacement of an elastic body which
has the rest configuration U. Let us suppose now that the elastic body is
incompressible, which now means
det Du = 1.
We therefore suppose the energy density function L : M3x3 x U —> R is
given, and consider the problem of minimizing the elastic energy
7[w] := / L(D-w,x}dx
Ju
over all w in the admissible set
A := {w G W1,9(17;R3) | w = g on dU, detDw = 1 a.e.}
for some q > 3.
THEOREM 7 (Minimizers with determinant constraint). Assume the map-
ping
L(P,x)
is convex, and L satisfies the coercivity condition
L(P, x) > a\P\g — (3 (PgM3x3,xg U)
for some a > 0, (3 > 0. Suppose finally A / 0.
Then there exists u G A satisfying
7[u] = min/[w].
weA
Proof. We as usual select a minimizing sequence, with
Ufc5 —- u	weakly in W1,9(P;R3).
Since
7[u] < lim inf I [ufe1 ,
J->OO
we must only show that и G A. However, since in view of the lemma in §8.2.4
we have det k det Du weakly in L?/n(P), we see that det Du = 1 a.e.,
as required.	□
476
8. THE CALCULUS OF VARIATIONS
Remark. It may seem odd that the incompressibility condition in example
(a) is
(52)
div u = 0
and in example (b) is
(53)
det Du = 1.
The explanation is that u represents a velocity in (52) and a displacement
in (53). More generally if w is a velocity field, say of a fluid, we compute
the motion of a particle initially at a point x by solving the ODE
y(i) = w(y(i),f) (teR)
y(0) = x.
Write y(t) = y(t,x) to display the dependence on the initial position x.
Then for each t > 0, the mapping x y(t,x) is volume-preserving if
J(x, t) = det Dxy(t, x) = 1 for all x.
Clearly J(0, ж) = 1, and a calculation verifies Euler’s formula:
Jt(x,t) = (divw)(y(t,x),t)J(x,f),
the divergence taken with respect to the spatial variables. Hence if div w =
0, the flow is volume preserving.	□
8.5. CRITICAL POINTS
Thus far we have studied the problem of locating minimizers of various
energy functionals, subject perhaps to constraints, and of discovering the
appropriate nonlinear Euler-Lagrange equations they satisfy. For this sec-
tion we turn our attention to the problem of finding additional solutions of
the Euler-Lagrange PDE, by looking for other critical points. These critical
points will not in general be minimizers, but rather “saddle points” of ![].
8.5.1. Mountain Pass Theorem.
We develop here some machinery which ensures that an abstract func-
tional ![•] has a critical point.
8.5. CRITICAL POINTS
477
a. Critical points, deformations.
Hereafter H denotes a real Hilbert space, with norm || || and inner
product ( , ). Let I: H —> R be a nonlinear functional on H.
DEFINITION. We say I is differentiable at и 6 H if there exists v 6 H
such that
(1)	7[w] = 7[u] + (v, w — u) + o(||w — u||) (w 6 H).
The element v, if it exists, is unique. We then write T[u] = v.
DEFINITION. We say I belongs to C^TfyR) i/L'M exists for each и e
H, and the mapping I' : H —> H is continuous.
Remark. The theory we will develop below holds if I 6 C1(H; R), but the
proofs will be greatly streamlined provided we additionally assume
(2) I' : H —> H is Lipschitz continuous on bounded subsets of H.
Notation, (i) We denote by C the collection of functions I G C1(H;R)
satisfying (2).
(ii) If c 6 R, we write
Ac := {u 6 H | 7[u] < c}, Kc = {u 6 H | 7[u] = c, /[u] = 0}.
□
DEFINITIONS, (i) We say u 6 H is a critical point if I'\u] = 0.
(ii) The real number c is a critical value if Kc	0.
We now want to prove that if c is not a critical level, we can nicely
deform the set Ac+e into Ac-e for some c > 0. The idea will be to solve
an appropriate ODE in H and to follow the resulting flow “downhill”. As
H is generally infinite dimensional, we will need some kind of compactness
condition.
DEFINITION. A functional I 6 Сг(Н; R) satisfies the Palais-Smale com-
pactness condition if each sequence	С H such that
(i)	is bounded
and
(ii) /'[и*,] —> 0 in H,
is precompact in H.
478
8. THE CALCULUS OF VARIATIONS
THEOREM 1 (Deformation Theorem). Assume I E C satisfies the Palais-
Smale condition. Suppose also
(3)	Kc = 0.
Then for each sufficiently small e > 0, there exists a constant 0 < 6 < e and
a function
7] E C([0,l] x H;H)
such that the mappings
z/t(u) = 7?(t, u) (0 < t < 1, и E H)
satisfy
(i)	T)0(u) = u (и E Hfi
(ii)	z/i(u) = и (и $ /-1[с — e, c + e]),
(iii)	J[z7t(rt)] < I[u] (и E H, 0 < t < 1),
(iv)	тц(Ас+6') C Ac-S.
Proof. 1. We first claim that there exist constants 0 < a, e < 1 such that
(4)	||/'[u]|| > a for each и E Ac+c - Ac-e.
The proof is by contradiction. Were (4) false for all constants a. e > 0, there
would exist sequences ста, —► 0,	—> 0 and elements
(5)	Uk G Ac_f_£k — Ac.,k
with
(6)	||/'Ы|| < ak.
According to the Palais-Smale condition, there is a subsequence
and an element и E H with ukj —> и in H. But then since I E CX(#;R), (5)
and (6) imply I[u] = c, /'[u] = 0. Consequently Kc 0, a contradiction to
our hypothesis (3).
2.	Now fix h to satisfy
2
(7)	0 < <5 < e, 0<6<^-.
Write
A .= {и E H \ /[u] < c - e or 7[u] > c + e},
B:={uEH\c-6< /[u] <c + 6}.
8.5. CRITICAL POINTS
479
Since I' is bounded on bounded sets, we verify that the mapping и i—>
dist(u, A) + dist(u, B) is bounded below by a positive constant on each
bounded subset of H. Consequently, the function
'= dist(u, A) + distfy, B) 6 Я)
0 < g < 1, g = 0 on A, g = 1 on B.
1,	0 < t < 1
h(t) .-
satisfies
(8)
Set
(9)
Finally define the mapping V : H —> H by
(10)	V(u) := -(/(uMH/'HIU'M («ея).
Observe that V is bounded.
3.	Consider now for each и 6 H the ODE
5(0=	(t > 0)
z?(0)= u.
As V is bounded and Lipschitz continuous on bounded sets there is a unique
solution, existing for all times t > 0. We write g = z/(t, u) = z?t(u) (t > 0, u G
H) to display the dependence of the solution on both the time t and the
initial position и G H. Restricting ourselves to times 0 < t < 1, we see that
the mapping rj 6 C([0,1] x H; H) so defined satisfies assertions (i) and (ii).
4.	We now compute
= f/'[ntfy)],-^nt(^)
at	\	at
 l21	= (/'[>)<(«)], V(4,(u)))
= -s(4t(»))A(||/'h,Mill) ll/'h,Mill2.
In particular
-r-HntH] <o (u g н, о < t < i),
at
and so assertion (iii) is valid.
5.	Now fix any point
(13)
и G Ас+&.
480
8. THE CALCULUS OF VARIATIONS
We want to prove
(14)	r?i(u) g Ac_6
and thereby verify assertion (iv). If 7?t(n) В for some 0 < t < 1, we are
done; and so we may as well suppose instead T)t(u) G В (0 < t < 1). Then
<7(z/t(u)) = 1 (0 < t < 1). Consequently, calculation (12) yields
(15)	= -Л(||Г[ч<(«)]||)||Г[к(«)]||2.
at
Now if ||/'[m(^)]|| > 1? then (9) and (4) imply
^I[nt(u)] = -||/'[»7t(u)]|| < -CT.
at
On the other hand, if ||I/[nt(^)]|| < 1, (9) and (4) yield
4Jfat(u)] < -cr2.
dt
Both these inequalities, and (15), then imply
Z[z/t(^)] < I[u\ — a2 <c + 6 — cr2<c — 6 by (7).
This estimate establishes (14) and completes the proof.	□
b. Mountain Pass Theorem.
Next we employ an interesting “min-max” technique, using the defor-
mation T] built above to deduce the existence of a critical point.
THEOREM 2 (Mountain Pass Theorem). Assume I G C satisfies the
Palais-Smale condition. Suppose also
(i)	Z[0] = 0,
(ii)	there exist constants r, a > 0 such that
Z[u] > a if ||u|| = r,
and
(iii)	there exists an element v G H with
||u|| > r, Z[u] < 0.
8.5. CRITICAL POINTS
481
Define
Г := {g € C([0,1]; Я) | g(0) = 0, g(l) = v}.
Then
c = inf max J[g(t)l
ger o<t<i
is a critical value of I.
Think of the graph of /[•] as a landscape with a low spot at 0, surrounded
by a ring of mountains. Beyond these mountains lies another low spot at v.
The idea is to look for a path g connecting 0 to v, which passes through a
mountain pass, that is, a saddle point for /[•]. But note carefully: we are
only asserting the existence of a critical point at the “energy level” c, which
may not necessarily correspond to a true saddle point.
Proof. 1. Clearly
(16)	c > a.
2.	Assume that c is not a critical value of I, so that
(17)	Kc = 0.
Choose then any sufficiently small number
(18)	0<e<^.
According to the Deformation Theorem 1, there exist a constant 0 < <5 < e
and a homeomorphism т] : H —> H with
(19)	п(^с+й) С Ac_5
and
(20)	z?(u) = u if u /-1[c — <5, с + e].
3.	Now select g € Г satisfying
(21)
max /fg(t)] < c + 6.
0<t<l
Then g := 77 о g also belongs to Г, since n(g(0)) = n(0) = 0 and 7?(g(l)) =
7}(u) = v, according to (20). But then (21) implies maxo<t<i J[g(t)] < c — 6-,
whence c = infger maxo<t<i Z[g(£)] < c — 6, a contradiction.	□
482
8. THE CALCULUS OF VARIATIONS
8.5.2. Application to semilinear elliptic PDE.
To illustrate the utility of the Mountain Pass Theorem, let us investigate
now the semilinear boundary-value problem:
f —Nu = f(u) in U
<22>	t и = 0 on dU.
We assume f is smooth, and for some
we have
(23)	|/(г)|<С(1 + И), |m < C(1 + И*-1) (г 6 R),
where C is a constant. We will suppose also
(24)	0 < F(z) <	for some constant 7 <
where F(z) := / f(s)ds and z 6 R. We hypothesize finally for constants
Jo
0 < a < A that
(25)	a\z\p+1 < |F(z)| < А|г|р+1 (г 6 R).
Now (25) implies /(0) = 0, and so obviously и = 0 is a trivial solution
of (22). We want to find another.
Remark. Observe that the PDE
—Nu = |<-1u
falls under the hypotheses above. We will return to this particular nonlin-
earity again in §9.4.2.	□
THEOREM 3 (Existence). The boundary-value problem (22) has at least
one weak solution и 0.
Proof. 1. Define
(26)	/[u] := [ ^\Du\2 - F(u) dx
Ju
for и £ Hq(17). We intend to apply the Mountain Pass Theorem to /[•].
8.5. CRITICAL POINTS
483
We set H = Hq(U), with the norm ||tt|| = (J^ \Du\2 dx)1^2 and inner
product (u, v) = fy Du  Dv dx. Then
I[u] = |||u||2 - [ F(u)dx =t Ii[u] - I2[u].
Ju
2.	We first claim
(27)	I belongs to the class C.
To see this, note first that for each u,w 6 H,
/l[w] = sIMI2 = 5 IIй + w - ^ll2 = 5IMI2 + (u, W - u) + j||w - <
Hence /1 is differentiable at u, with [u] = u. Consequently, /1 6 C.
3.	We must next examine the term /2- Recall from the Lax-Milgram
Theorem (§6.2.1) that for each element v* € H“1(17), the problem
( —Nv = v* in U
I v = 0 on dU
has a unique solution v G Hq(V). We will write v = Kv*; so that
(28)	К : H-^U) —> Hq(U") is an isometry.
Note in particular that if w G L2n/n+2(t/), then the linear functional w*
defined by
(w*,u) := [ wudx (uEHq(U))
Ju
belongs to	(We will misuse notation and say “w 6 H-1(L7)”.)
Observe next that p (7^+2) <	’ 7^+2 = 2*, and so e L2n/n+2(C)
CH~\U) iiuGH^U).
We now demonstrate that if и G Hq(U), then
(29)	I2[u] = K[f(u)].
To see this, note first that
F(a + b) = F(a) + f(a)b + [ (1 — s) f1 (a + sb) ds b2.
Jo
Thus for each w G Hq(U'),
/2[w] = / F(w)dx= / F(u + w — u)dx
Ju	Ju
(30)	= I F(u) + f(u)(w — u) dx + R
Ju
= I2(u)+ [ DK[f(u)]-D(w-u)dx + R,
Ju
484
8. THE CALCULUS OF VARIATIONS
where the remainder term R satisfies, according to (23),
|7?| < C f (1 + |u|p-1 + |w—u|p-1)|w — u|2 dx
Ju
p — 1
< С	\w — u\2 + \w—u\p+1dx^+C^J‘ |u|p+1 dx^
2
|w-u|p+1 dx^ P+ .
Since p + 1 < 2*, the Sobolev inequalities show R = o(||w — u||). Thus we
see from (28) that
/2[w] = /2[u] + (K[f(u)],w) + o(||w - T4||),
as required.
Finally we note that if и, й G Hq(17) with ||u||, ||u|| < L, then
l№] - /2ИII = \\K{f(u)] - K[f(u)]\\Hi{u)
= ll/(u) - /(й)Пя-1({7)
< || f(u) — /(u)|| 2n
But
||/(u) - /(u)|| 2n
<c(J ((1 + |u|p-1 + И”1)|u - й|)^ dx^ 2n
<	C (^(1 + H’1 +	" ||u - u||L2.([/)
<	C(L)||u - u||L2.((7) < C(L)||u - й||,
where we used (23). Thus 1% : Hq(U) —> H^(U) is Lipschitz continuous on
bounded sets. Consequently /2 6 C, and we have established assertion (27).
4. Now we verify the Palais-Smale condition. For this suppose	C
H^(U), with
(31)	bounded
and
(32)	1'Ы^О in Hq1 (17).
485
8.5. CRITICAL POINTS
According to the foregoing
(33)
uk — K(f(uk))—> 0 in H^U).
Thus for each e > 0 we have
I Duk  Dv - f(uk)v dx < e||v||
и

for к large enough. Let v = uk above to find
/ \Duk\2 - f(uk)ukdx
Ju
< е1Ы1
for each e > 0 and for all к sufficiently large. For б = 1 in particular, we see
that
(34)	/ f(uk)ukdx < ||ufc||2 + ||ufc||
Ju
for all к sufficiently large. But since (31) says
flllM2- [ F(uk)dx\ < C < oo
\ Ju /
for all к and some constant C, we deduce
1Ы12 <C + 2 [ F(uk)dx
Ju
< C + 27 (1Ы12 + 1Ы1) by (34), (24).
Since 27 < 1, we discover that	is bounded in Hq(U). Hence there
exists a subsequence	and и G Hq(U), with ukj —> и weakly in
Hq(U) and uk. —♦ и in Lp+1(17), the latter assertion holding since p+1 < 2*.
But then /(lq,) —> /(u) in H-1(17), whence A'[/(u*.)] —> A[/(u)] in Hq(17).
Consequently (33) implies
(35)	ukj->u тЯо({7).
5. We finally verify the remaining hypotheses of the Mountain Pass
Theorem. Clearly /[0] = 0. Suppose now that и G Hq(U), with ||u|| = r, for
r > 0 to be selected below. Then
(36)
/[u] = /1 [u] - /2[u] =	- /2[«].
A
486
8. THE CALCULUS OF VARIATIONS
Now hypothesis (25) implies, since p + 1 < 2*, that
|/2M| < C j \u\p+i dx < C (J |u|2 dx
< C|M|p+1 < Crp+1.
In view of (36), then
2	2
/[tt] > 7-— Crp+l >	= a > 0,
i J - 2	- 4
provided r > 0 is small enough, since p + 1 > 2. Now fix some element
и 6 H, и 0. Write v := tu for t > 0 to be selected. Then
/[v] = /1 [tu] - l2[tu]
= t2/i[u]— I F(tu)dx
Ju
< t2/i[u] — atp+l f \u\p+1dx
Ju
< 0
by (25)
for t > 0 large enough.
6. We have at last checked all the hypotheses of the Mountain Pass
Theorem. There must consequently exist a function и € Hq(1/), и 0, with
/'[u] = u — K[/(u)] = 0.
In particular for each v 6 Hq(U), we have
/ Du -Dvdx= / f(u)vdx,
Ju	Ju
and so и is a weak solution of (22).
See §9.4.2 for further discussion about nonlinear Poisson equations, and
in particular the significance of the critical exponent in hypothesis (23).
8.6. PROBLEMS
In the exercises U always denotes a bounded, open subset of R", with smooth
boundary.
8.6. PROBLEMS
487
1.	This problem illustrates that a weakly convergent sequence can in fact
be rather badly behaved.
(i)	Prove и^(х') — sm(kx') 0 as к —> oo in L2(0,1).
(ii)	Fix a, b G R, 0 < A < 1. Define
( a if j/k < x < X(j + l)/k
“‘W = b if AO + 1)A<X<(J + 1)A 0 = 0' "л-1)'
Prove Uk —‘ Aa + (1 — X)b in L2(0,1).
2.	Find L = L(p, z,x) so that the PDE
—Au + Dtp  Du — f in U
is the Euler-Lagrange equation corresponding to the functional I[w] :=
fu L(Dw, w, x) dx. Here (f),f:U —> R are given smooth functions.
3.	The elliptic regularization of the heat equation is the PDE
(*)	щ — JX.u — eutt = 0 in Ut,
where e > 0 and Ut = U x (0,T]. Show that (*) is the Euler-
Lagrange equation corresponding to an energy functional /e[w] :=
f fu Le(Dw, wt, w, x, t) dxdt.
4.	Assume 77 : R" —> R is C1.
(i)	Show L(P,z,x) = 77(2) det P (P G Mnxn,z G Rn) is a null
Lagrangian.
(ii)	Deduce that if u : Rn —> Rn is C1, then
/ 77 (u) det Du dx
Ju
depends only on u|^j/.
5.	(Continuation). Fix xq u(dU), and choose 77 as above so that
j^gdz = 1, spt?7 C B(xo,r), r chosen so small that B(xo,r) Г)
u(dU) = 0. Define
deg(u,xo) = / 77(11)detDudx,
Ju
the degree of u relative to xq. Prove the degree is an integer.
488
8. THE CALCULUS OF VARIATIONS
6.	Let S C R3 denote the graph of the smooth function и : U —> R,
U C R2. Then
(*)	f (1 + |Du|2)-1 det D2u dx
Ju
represents the integral of the Gauss curvature over E. Prove this
expression depends only upon Du restricted to dU. (The Gauss-
Bonnet Theorem in differential geometry computes (*) in terms of the
geodesic curvature of <ЭГ.)
7.	Let m = n. Prove
L(P) = tr(P2) - tr(P)2 (P e MnXn)
is a null Lagrangian.
8.	Explain why the methods in §8.2 will not work to prove the existence
of a minimizer of the functional
/[w] := [ (1 + fDwf2)1^2 dx
Ju
over A = {w e W1,q(U) | w = g on dU}, for any 1 < q < oo.
9.	(Second variation for systems). Assume u : U —> Rm is a smooth
minimizer of the functional
Z[w] := / L(Dw, w,x) dx.
Ju
(i)	Show
" m d2L
E E	>о
i,j=l k,l=l
for all X e U, £ 6 Rn, r/ e Rm.
(ii)	Give an example of a nonconvex function L : Mmxn —♦ R satis-
fying
1 b, , dPidp\
i,j=l k,l=l г J
for all P e Mmxn, e e Rn, 7? 6 Rm.
10.	Use the methods of §8.4.1 to show the existence of a nontrivial weak
solution и € Hq(U), u 0, of
( ~Ди = |u|9-Iu in U
I и = 0 on dU
8.7. REFERENCES
489
for 1 < q < n > 2.
11.	Assume /3 : R —> R is smooth, with
0 < a < <b (z 6 R)
for constants a, b. Let f G L2(U). Formulate what it means for
и G H1(17) to be a weak solution of the nonlinear boundary-value
problem
( —Ди = f	in U
t	+ /3(u) =0	on dU.
Prove there exists a unique weak solution.
12.	Assume и is a smooth minimizer of the area integral
Z[w] = [ (1 + \Dw\2y/2 dx,
Ju
subject to given boundary conditions w = g on dU and the constraint
J[w] = / wdx = 1.
Ju
Prove the graph of и is a surface of constant mean curvature. (Hint:
Recall Example 4 in §8.1.2.)
13.
(i)	Show there exists a unique minimizer и G A of
Z[w] = I ||Dw|2 — fwdx,
Ju
where f G L2(U) and
A = {w G Hq(U) | |Pw| < 1 a.e.}.
(ii)	Prove
/ Du  D(w — u)dx> / /(w — u) dx
Ju	Ju
for all w G A.
8.7. REFERENCES
Section 8.1 See Giaquinta [GI] for more about the calculus of variations.
Another good source is Giaquinta-Hildebrandt [G-H], and
490
8. THE CALCULUS OF VARIATIONS
	Zeidler [ZD, Vol. 3] is a general reference for variational methods.
Section 8.2	The proof of Theorem 1 in §8.2.2 is from Ladyzhenskaya- Uraltseva [L-U],
Section 8.3	See Giaquinta [GI] for additional information about regular- ity (and partial regularity) of minimizers.
Section 8.4	The book by Kinderlehrer-Stampacchia [K-S] explains much more about variational inequalities.
Section 8.5	The Mountain Pass Theorem is due to Ambrosetti and Ra- binowitz. Consult Rabinowitz [RA] (the source of §8.5) for further results on saddle point methods.
Chapter 9
NONVARIATIONAL
TECHNIQUES
9.1	Monotonicity methods
9.2	Fixed point methods
9.3	Method of subsolutions and supersolutions
9.4	Nonexistence
9.5	Geometric properties of solutions
9.6	Gradient flows
9.7	Problems
9.8	References
We gather in this chapter various techniques for proving the existence,
nonexistence, uniqueness, and various other properties of solutions for non-
linear elliptic and parabolic partial differential equations that are not of
variational form.
9.1.	MONOTONICITY METHODS
Let us look first at this boundary-value problem for a divergence structure
quasilinear PDE:
(1)
— div a(Pu) = f in U
и = 0 on dU,
where f € L2(f/) is given, as is the smooth vector field a : R" —> Rn, a =
(a1,..., a”). As usual the unknown is и : U —> R, и = u(x), where U is a
491
492
9. NON VARIATION AL TECHNIQUES
bounded, open subset of R" with smooth boundary. Now if there exists a
function F : R” —► R such that a is the gradient of F,
(2)	a(p) = DF(p) (p 6 R"),
then (1) is the Euler-Lagrange equation corresponding to the Lagrangian
L(p,z,x) = F(p) — f(x)z. However if there exists no such potential F, the
variational methods from Chapter 8 simply do not apply to the problem (1).
We inquire instead if there is rather some direct method of constructing
a solution of (1), and in particular ask what are reasonable conditions to
place upon the nonlinearity. For motivation let us note that if (2) were valid
and if F were convex (the natural assumption for the variational theory, as
we have seen in §8.2.2), then for each p, q 6 R":
n
(a(p) - a(9)) • (p - q) = ^(FPi (p) - FPi (q))(Pi - qf)
i=l
rl n
= V FpiPj(P + t(q ~ рУ)(Рл-qj)(pi ~qi)dt > 0,
Jo i,j=l
the last inequality following from the convexity of F.
This calculation suggests the following
DEFINITION. A vector field a : R" —> R" is called monotone provided
(3)	(a(p) - a(g)) • (p - q) > 0
for all p,q 6 R”.
We will show below that the quasilinear PDE does indeed possess a
weak solution, under the primary structural assumption that the nonlinear-
ity be monotone. Later we will realize that this condition in effect says that
— div a(Du) = f is a nonlinear elliptic partial differential equation. So let
us henceforth assume that the smooth vector field a is monotone, and that
(4)	|a(p)|<C(l + |p|),
(5)
a(p) • p > a|p|2 - (3
for all p G R" and appropriate constants C, a > 0, (3 > 0. We will see
momentarily that (5) amounts to a coercivity condition on the nonlinearity.
We intend now to build a solution of the boundary-value problem (1) as
9.1. MONOTONICITY METHODS
493
the limit of certain finite dimensional approximations, thereby extending
Galerkin’s method from Chapter 7 to a new class of nonlinear problems.
More precisely, assume that the functions Wk = Wk(x) (к = 1,...) are
smooth and
{wA;}^ is an orthonormal basis of Hq(U),
taken with the inner product (u, fDu  Dv dx. (We could for instance
take	to be the set of appropriately normalized eigenfunctions for
—Д in ^(8).)
We will look for a function um E Hq(U) of the form
(6)	um = ^^dmw/c,
fc=i
where we hope to select the coefficients so that
(7)	/ a(Dum) • Dwk dx = / fwkdx (fc = 1,... ,m).
Ju	Ju
This amounts to our requiring that um solves the “projection.” of the problem
(1) onto the finite dimensional subspace spanned by {w}™=1-
We start with a technical assertion.
LEMMA (Zeros of a vector field). Assume the continuous function v :
Rn —> Rn satisfies
(8)	v(x)  x > 0 if |x| = r,
for some r > 0. Then there exists a point x E 8(0, r) such that
v(x) = 0.
Proof. Suppose the assertion were false; then v(x) ± 0 for all x E 8(0, r).
Define the continuous mapping w : 8(0,r) —> 88(0, r) by setting
w(x)
“Ri)ivW (xeB(0'r))'
According to Brouwer’s fixed point theorem (§8.1.4), there exists a point
z E 8(0, r) with
(9)	w(z) = z.
But then z E 88(0, r), and so (8) and (9) imply the contradiction
r2 = z • z = w(z) • z = — -—-r-r-v(z) • z < 0.
494
9. NONVARIATIONAL TECHNIQUES
THEOREM 1 (Construction of approximate solutions). For each integer
m = 1,..., there exists a function um of the form (6) satisfying the identities
(7)-
Proof. 1. Define the continuous function v : Rm —► Rm, v = (v1,..., vm),
by setting
r	/ m	\
(10)	vk(d) := / a I djDw} 1 • Dw^ — fwk dx —
Ju	J
for each point d = (di,..., dm) € Rm. Now
Now let и £ Hg(U) solve the PDE —Am — f. Then
/ Du-Dwjdx= / fwjdx (j = 1,...),
Ju	Ju
and consequently
m	m
wj)l2(c/) = wi)2 — i iu 11	(u) —
j=i	i=i
Hence v(d) d > ^|d|2 — C, for some constant C, and so v(d) d > 0 if |d| = r,
provided we select r > 0 sufficiently large.
2.	We apply the lemma, to conclude that v(d) = 0 for some point
d € Rm. Then (10) implies um defined by (6) satisfies (7).	□
We want to take the limit as m —♦ oo, and for this will require some
uniform estimates.
9.1. MONOTONICITY METHODS
495
THEOREM 2 (Energy estimates). There exists a constant C, depending
only on U and a, such that
(H)	Нит||н1([/) < C(1 + II/||l2({/))
for m = 1,2,... .
Proof. Multiply equality (7) by dfn and sum for к = 1,..., m:
/ a(Dum) • Dum dx = I fum dx.
Ju	Ju
In view of the coercivity inequality (5), we find
a I \Dum\2 dx < С + I fumdx < C + e [ u2mdx +	[ f2dx.
Ju	Ju	Ju 4e Ju
We recall Poincare’s inequality, and then choose e > 0 small enough to
deduce (11).	□
We wish now to employ the L2 inequalities (11) to pass to limits as
m —> oo, obtaining thereby a weak solution of problem (1), which is to say,
a function и E Hq(D) satisfying the identity
(12)	I a(Dv)-Dvdx = [ fvdx for all v G
Ju	Ju
Employing estimate (11) we can extract a subsequence {umj}j2=1 that con-
verges weakly in H)(U) to a limit u, which we hope to show verifies (12).
However, we encounter a major problem here: we cannot directly conclude
that
a(DumJ) -> a(Pu)
in any sense whatsoever. Take note: nonlinearities are (usually) not contin-
uous with respect to weak convergence. (See Problem 2.)
What saves us is the monotonicity assumption on vector field a.
THEOREM 3 (Existence of weak solution). There exists a weak solution
of the nonlinear boundary-value problem (1).
Proof. 1. As noted in the foregoing discussion, we can extract a subse-
quence {игнДуХ! C {nm}“=1 and a function и E H(j(U) such that
(13)
umj и weakly in H([U)
496
9. NON VARIATION AL TECHNIQUES
and
(14)	umj ->u in L2(U).
We must show и satisfies (12).
2.	In view of the growth condition (4), {a(Dum)}*=1 is bounded in
L2(U; Rn); and so we may as well suppose—upon passing to a further sub-
sequence if necessary—that
(15)	a.(Dumj) —1 £ weakly in L2(17;Rn),
for some £ 6 L2(L/;Rn). Using identity (7), we deduce
/ £ • Dw/- dx = I fwk dx
Ju	Ju
for each к = 1,.... And so
(16)	/ £Dvdx = I fvdx for each v G Hq(U).
Ju	Ju
3.	To proceed further, let us note from the monotonicity condition (3)
that
(17)	j	— a(.Dw)) • (Dum — Dw) dx > 0
Ju
for m = 1,... and all w G H^(U). But as observed before, equation (7)
yields the identity
/ a{Dum) • Dum dx = / fumdx.
Ju	Ju
Substitute into (17), to find:
/ fum — a(Dum) • Dw — a(.Dw) • (Dum — Dw) dx > 0.
Ju
Let m = mj —> oo and recall (13)—(15), to deduce
I fu — £  Dw — a(.Dw) • (Du — Dw) dx > 0.
Ju
We simplify using identity (16) with v = u, and discover
(18)	[ (£ — a(Dw)) • D(u — w) dx > 0 for all w 6 Hq(U).
Ju
9.1. MONOTONICITY METHODS
497
4.	Fix any v E H^(U) and set w := и — Xv (Л > 0) in (18). We obtain
then
I (£ — a(Du — XDvf) • Dv dx > 0.
Ju
Send Л —> 0:
(19)	[ (£ -a(Du))  Dv dx > 0 for all v E
Ju
Replacing v by — v, we deduce that in fact equality holds above. Then (16)
and (19) taken together yield
I a(Du)  Dvdx = [ fvdx for all v E Hq(U).
Ju	Ju
Hence и is indeed a weak solution of (1).	□
Remark. This use of monotonicity is the method of Browder and Minty,
a remarkable technique which somehow employs the inequality condition of
monotonicity to justify passing to weak limits within a nonlinearity. □
Let us assume now the vector field a satisfies the condition of strict
monotonicity, that is,
(20)	(a(p) - a(g)) • (p - q) > 0\p - q|2
for all p,q E Rn and some constant 9 > 0.
THEOREM 4 (Uniqueness of weak solution). Assume the strict mono-
tonicity property (20) holds. Then there exists only one weak solution of
(1).
Proof. Assume that и and й are two weak solutions. Consequently
/ a(Du)  Dv dx= a(Du)  Dv dx= fv dx,
Ju	Ju	Ju
and so
I [a(Pu) — а(-Ой)] • Dv dx — 0
Ju
for each v E	We set v := и — й, and use (20) to deduce
I \Du — Dii\2 dx = 0.
Ju
Thus и = й a.e. in U.	□
498
9. NONVARIATIONAL TECHNIQUES
Remark. Under the strengthened monotonicity assumption (20) our weak
solution и in fact belongs to H2(U), and so satisfies
— div a(Du) = f a.e. in U.
To demonstrate this, we select G R" and set p — q + hfi, h 0, in (20).
We obtain, after dividing by h2, the inequality
[a\q +	- а\д)] >
i=l
Now send h 0:
n
(21)	(9Л6Л-
i,j=l
We can thus interpret the nonlinear PDE — diva(Du) = f as being uni-
formly elliptic. The proof of H2 regularity of the weak solution now follows
almost precisely as in the proof of Theorem 1 in §6.3.1.	□
9.2.	FIXED POINT METHODS
We study next the applicability of topological fixed point theorems to non-
linear partial differential equations. There are at least three distinct classes
of such abstract theorems that are useful. These are:
(a)	fixed point theorems for strict contractions,
(b)	fixed point theorems for compact mappings,
and
(c)	fixed point theorems for order-preserving operators.
We present below applications of types (a) and (b). The utility of order-
preserving properties for nonlinear PDE will be explained later, in §9.3.
9.2.1. Banach’s Fixed Point Theorem.
Hereafter X denotes a Banach space. The simplest fixed point theorem
of all is:
THEOREM 1 (Banach’s Fixed Point Theorem). Assume
A:X X
is a nonlinear mapping, and suppose that
(1)	||A[u]-A[fi]|| <7||u-fi|| (u,uGX)
for some constant 7 < 1. Then A has a unique fixed point.
9.2. FIXED POINT METHODS
499
DEFINITION. We say that A is a strict contraction if (1) holds.
Proof. Fix any point uq & X and thereafter iteratively define Uk+i = А[щ,]
for к = 0,1,... . Then
||A[ufc+i] - ^4[ufc]|| < 7||ufc+i - икII = 71ИЫ - II,
and so
||A[ufc+i] - A[ufc]II < 7fc 1ИЫ - «oil
for к = 1, 2,... . Consequently if к > I,
||Ui-4|| = M(«-i]-A[u,_,]!!< £ ||A[u,+1] - Л(и,Ц|
J=z-1
fc—2
< ||A[u0] — u0||
j=i-i
Hence	is a Cauchy sequence in X, and so there exists a point и G X
with и к —> и in X. Clearly then A[u] = u. Hence и is a fixed point for A,
and hypothesis (1) ensures uniqueness.	□
Applications of Banach’s fixed point theorem to nonlinear PDE usually
involve perturbation arguments of various sorts: given a well-behaved linear
elliptic partial differential equation, it is often straightforward to cast a
small nonlinear modification as a contraction mapping. The hallmark of
such proofs is the occurrence of a parameter which must be taken small
enough to ensure the strict contraction property.
Sometimes however we can eliminate such a smallness hypothesis by an
iteration, as now illustrated.
Example 1 (Reaction-diffusion equations). Let us investigate the solvabil-
ity of the initial/boundary-value problem for the reaction-diffusion system
u O
u = g
(2)
ut — Au = f(u) in Ut
on dU x [0,T]
on U x {t = 0}.
Here u = (u1,... ,um), g = (g1,...^"1), and as usual Ut = U x (0,T],
where U e l" is open and bounded, with smooth boundary. The time
T > 0 is fixed. We assume that the initial function g belongs to (U; Rm).
Concerning the nonlinearity, let us suppose
(3)
f : Rm —> Rm is Lipschitz continuous.
500
9. NONVARIATIONAL TECHNIQUES
This hypothesis in particular implies
(4)	|f(2)| <67(1 + 1^1)
for each z G and some constant C.
Adapting the terminology from §7.1, we say that a function
(5)	ue L2(0,T-H^U;Wn')), with u G L2 (0,T;
is a weak solution of (2) provided
(6)	(u', v) + B[u, v] = (f(u), v) a.e. 0 < t < T
for each v G #d(t7;lRm), and
(7)	u(0) = g.
In (6) ( , ) denotes the pairing of Я-1(17;]йт) and	B[ , ] is the
bilinear form associated with —A in Hq(U; IRm), and ( , ) denotes the inner
product in L2(B;lRm). The norm in HQ(U;Rm) is taken to be
IIuIIh1(C/;R-) = (^|^>u|2dx)2.
Recall from §5.9.2 that (5) implies u G C'([0,T];L2(B;lRm)), after possible
redefinition of u on a set of measure zero.
THEOREM 2 (Existence). There exists a unique weak solution of (2).
Proof. 1. We will apply Banach’s theorem in the space
X = C'([0,T];L2(B;Rm)),
with the norm
llvll =0m^l|v(i)||L2(c/;Rm).
Let the operator A be defined as follows. Given a function u G X, set
h(t) := f(u(t)) (0 < t < Г). In light of the growth estimate (4), we see
h G L2(0, T; L2(B;Rm)). Consequently the theory set forth in §7.1 ensures
that the linear parabolic PDE
(8)
' wt — Aw = h
< w = 0
w - g
in Ut
on dU x [0,T]
on U x {t = 0}
9.2. FIXED POINT METHODS
501
has a unique weak solution
(9)	wG£2(O; T;Ho1(C';r )), with w' G L2(0,T;H-1(lI;'Rm)).
Thus w G X satisfies
(10)	(w', v) + B[w, v] = (h, v) a.e. 0 < t < T
for each v G Bo(B;Rm), and w(0) = g.
Define A : X —» X by setting A[u] = w.
2. We now claim that
if T > 0 is small enough, then
A is a strict contraction.
To prove this, choose u, ii G X, and define w = A[u], w = Л[й] as. above.
Consequently w verifies (10) for h = f(u), and w satisfies a similar identity
for h := f(й).
We calculate as in §7.1
J^llw - w||22([Z;Rm) + 2||w - w|lHl([Z;Rm)
= 2(w - w, h - h)
<	C||W - w||£2([Z;Rm) + | ||f (u) - f(u)||^2([/;1Rm)
<	€C||w -	|||f(u) - f(й)||22([/;Rm),
by Poincare’s inequality. Selecting e > 0 sufficiently small, we deduce
^l|w ~	< C'llf(u) - f(u)||£2([/;Rm) < C||u - u||22([/;1Rm),
since f is Lipschitz. Consequently
||u(t)-u(t)||£2(tZ;Rm)dt
< CT||u- u||2
for each 0 < s < T. Maximizing the left hand side with respect to s, we
discover
||w-w||2 <CT||u —й||2.
Hence
(14)	Ии] - А[й]|| < (СТ)1/2||„-й||,
||w(s) - w(s)||£2([/;Rm) <c Jo
502
9. NON VARIATION AL TECHNIQUES
and thus A is a strict contraction, provided T > 0 is so small that (CT)1/2 =
7 < 1.
3. Given any T > 0 we select Ti > 0 so small that (CTi)1^2 < 1.
We can then apply Banach’s fixed point theorem to find a weak solution
u of the problem (2) existing on the time interval [0,Ti], Since u(t) G
-Hd(17;IRm) for a.e. 0 < t < Ti, we can upon redefining Ti if necessary
assume u(Ti) G Bd(17;]Rm). We can then repeat the argument above to
extend our solution to the time interval [Ti,2Ti]. Continuing, after finitely
many steps we construct a weak solution existing on the full interval [0, Т].
4. To demonstrate uniqueness, suppose both u and u are two weak
solutions of (2). Then we have w — u, w = u in inequality (13); whence
||u(s) - u(s)||22(a;Rm) < C j ||u(t) - fi(t)dt
for 0 < s < T. According to Gronwall’s inequality, u = u.	□
Remark. In common applications problem (2) records the evolution of
the densities u1,..., um of various chemicals, which both diffuse within a
medium and interact with each other. The diffusion term is Au (or more
generally (щДи1,... ,amAum) where the constants a^ > 0 characterize the
diffusion of the ktfl chemical). The reaction term f (u) models the chemistry.
In the foregoing example we made the unreasonable assumption that f is
globally Lipschitz. In more realistic models f is often a polynomial in u
and there are interesting problems as to the global existence or blow-up of
a solution. (A simple such problem is treated in §9.4.1.)	□
9.2.2. Schauder’s, Schaefer’s Fixed Point Theorems.
Next we extend Brouwer’s fixed point theorem (§8.1.4) to Banach spaces.
The key assumption is now compactness. Throughout this section X con-
tinues to denote a real Banach space.
THEOREM 3 (Schauder’s Fixed Point Theorem). Suppose К С X is
compact and convex, and assume also
A-.K^K
is continuous. Then A has a fixed point in К.
Proof. 1. Fix e > 0 and choose finitely many points ui,..., идге G K, so
that the open balls {B°(ui,e)}^1 cover K:
Ne
(15)	tfc|jB0(ube).
i=l
9.2. FIXED POINT METHODS
503
This is possible since К is compact. Let K( denote the closed convex hull
of the points {ui,..., u^e}:
| 0 < Xi <	= 1 > .
1=1
Then Kf С K, since К is convex. Now define P£ : К Kf by writing
(и e K).
Ez^idist(u,A- — B°(ui,e))
The denominator is never zero, because of (15). Now clearly P( is continuous,
and furthermore for each и G K, we have
_	dist(u,/<-B0(ui,€))||ui-u||
'	“ ^rdist^/f-BO^.e))
2. Consider next the operator A£ : Kf —> Ke defined by
Л[и] := P£[A[u] ] (u G Ke).
Now Kf is homeomorphic to the closed unit ball in R'Vf for some Mt < Ne.
Brouwer’s fixed point theorem (§8.1.4) therefore ensures the existence of a
point u£ G Kt with
ЛЫ = ut-
3. As К is compact, there exists a subsequence Cj —> 0 and a point
и G K, such that uej —> и in X. We claim и is a fixed point of A. Indeed,
using estimate (16) we deduce
11% - A[u£j]|| = ||A£>£j] - AK]|| = ||P£j[AK.]] - A[ueJ|| < e7.
Consequently, since A is continuous, we conclude и = A [и].	□
We next transform Schauder’s fixed point theorem into an alternative
form which is often more useful for applications to nonlinear partial differ-
ential equations.
DEFINITION. A nonlinear mapping A : X —> X is called compact pro-
vided for each bounded sequence	the sequence	is pre-
compact; that is, there exists a subsequence	such that
converges in X.
504
9. NONVARIATIONAL TECHNIQUES
THEOREM 4 (Schaefer’s Fixed Point Theorem). Suppose
A.X-+X
is a continuous and compact mapping. Assume further that the set
{u € X | и = АЛ[и] for some 0 < A < 1}
is bounded. Then A has a fixed point.
Remark. The assertion is that if we have a bound on any possible fixed
points of any of the operators XA for 0 < A < 1, then we have the existence
of a fixed point for A. This is in accordance with the remarkable informal
principle that “if we can prove appropriate estimates for solutions of a non-
linear PDE, under the assumption that such solutions exist, then in fact
these solutions do exist”. This is the method of a priori* estimates. □
The advantage of Schaefer’s theorem over Schauder’s for applications is
that we do not have to identify an explicit convex, compact set.
Proof. 1. Choose a constant M so large that
(17)	Ml < M if и = АЛ[и] for some 0 < A < 1.
Define then
(18)
A[u] := <
j4[u] if ||Л[и]|| < M

Observe A : B(0, M) —> B(0, M). Now set К = closed convex hull of
A(B(0,M)). Then since A and thus A are compact mappings, К is a com-
pact, convex subset of X. Furthermore A : К —> К.
2. Invoking Schauder’s fixed point theorem, we infer the existence of a
point и G К with
(19)
A[u] = u.
We now claim additionally that и is a fixed point of A. For if not, then
according to (18) and (19) we would have

* a priori = from before (Latin).
9.2. FIXED POINT METHODS
505
and
(20)
и = АЛ[и] for A =	< 1.
But ||u|| — MM|| = M, a contradiction to (17) and (20).
□
Applications of Schauder’s and Schaefer’s fixed point theorems to PDE
depend upon quite different considerations than applications of Banach’s
theorem. The crucial assumption is now not that some parameter be small,
but rather that we have some sort of compactness. As the inverses of linear
elliptic operators are typically smoothing, compactness is indeed available
for certain nonlinear elliptic equations. Following is a quick, albeit fairly
crude, example:
Example 2 (A quasilinear elliptic PDE). We present now a simple applica-
tion of Schaefer’s theorem by solving the semilinear boundary-value problem
—Au + 6(Du) + pu = 0 in U
u = 0 on dU,
where U is bounded and dU is smooth. We assume b : Rn —> R is smooth,
Lipschitz continuous, and thus satisfies the growth condition
(22)	|6(p)| < C(|p| + 1)
for some constant C and all p G Rn.
THEOREM 5 (Existence). If p > 0 is sufficiently large, there exists a
function и E H2(U) П Hq(U) solving the boundary-value problem (21).
Proof. 1. Given u G Hq(U), set
(23)	f := -6(Du).
Owing to estimate (22), we see that f G T2(t/). Now let w & Hq(U) be the
unique weak solution of the linear problem
—Aw + pw = f in U
w = 0 on dU.
By the regularity theory proved in §6.3, we know additionally that w G
H2(U), with the estimate
(24)
(25)
IMIIh2(p) < c||/IIl2(p)
506
9. NONVARIATIONAL TECHNIQUES
for some constant C.
Let us henceforth write A[u] = w whenever w is derived from и via (23),
(24). In light of (22) and (25), we have the estimate
(26)	1ИМIIh2(c/) < C(llullHi(n) + !)•
2.	We now assert that A : Hq (U) —> Hq (17) is continuous and compact.
Indeed, if
(27)	uk —> и in
then in view of estimate (26) we have
(28)	sup||wfc||H2(n) < oo,
k
for Wk = A[ufc] (к = 1,...). Thus there is a subsequence	and a
function w G Hq(U) with
(29)	wkj->w
Now
/ Dwkj  Dv + pwk vdx = -	b(DukAvdx
Ju	Ju
for each v € Hq(U). Consequently using (22), (27) and (29), we see
/ Dw • Dv + pwv dx = — / b(Du)vdx
Ju	Ju
for each v G Hq(U). Thus w = A[u],
Hence (27) implies А[и*.] —> A[u] in Hq(U), and so A is continuous. A
similar argument shows that A is compact, since if	is bounded in
estimate (22) asserts that	is bounded in Я2(17), and so
possesses a strongly convergent subsequence in Hq(U).
3.	Finally, we must show that if p is large enough, the set
{u G Hq(U') I и = AA[u] for some 0 < A < 1}
is bounded in Lfg(17). So assume и G Hq(U),
и = AA[u] for some 0 < A < 1.
Then = A[u]; or, in other words, и G H2(U) Cl and
—Au + pu = Xb(Du) a.e. in U.
9.3. METHOD OF SUBSOLUTIONS AND SUPERSOLUTIONS
507
Multiply this identity by и and integrate over U, to compute
I \Du\2 + p\u\2dx = — j Xb(Du)udx < f C(|Du| + l)|u| dx
Ju	Ju	Ju
\Du\2dx + C [ |u|2 + 1 dx.
2 Ju	Ju
Thus if p > 0 is sufficiently large, we have ||u||h1(17) < C, for some constant
C that does not depend on 0 < A < 1.
4.	Applying Schaefer’s fixed point theorem in the space X = Hq(U), we
conclude that A has a fixed point и G H^CU) A H2(U), which in turn solves
our semilinear PDE (21).	□
Remark and warning. A plausible plan for constructively solving (21)
would be to select some u° and then iteratively solve the linear boundary-
value problems
( —Aufe+1 + puk+1 = -b(Duk) in U .	.
f	= 0 опЯС/ (fc = 0,l,...).
However, we cannot assert that {nfc}^L0 then converges to a solution of (21).
Schauder’s and Schaefer’s fixed point theorems do not say that any sequence
converges to a fixed point. (But see the proof in §9.3 following.)	□
See Gilbarg-Trudinger [G-T] for much more sophisticated applications
of fixed point theorems to nonlinear elliptic PDE.
9.3. METHOD OF SUBSOLUTIONS AND
SUPERSOLUTIONS
Our application of Schaefer’s theorem above in §9.2.2 depends upon the
regularity estimates for solutions of elliptic equations. We turn attention
now to another basic property of elliptic PDE, namely the maximum prin-
ciple, and demonstrate how various resulting comparison arguments can be
used to solve certain semilinear problems. The idea is to exploit ordering
properties for solutions. More precisely, we will show that if we can find a
subsolution и and a supersolution й of a particular boundary-value problem,
and if furthermore и < й, then there in fact exists a solution satisfying
и < и < й.
We will investigate this boundary-value problem for the nonlinear Pois-
son equation:
(1)
( -Xu = f(u)
( и = 0
in U
on dU,
508
9. NONVARIATION AL TECHNIQUES
where f : R —> R is smooth, with
(2)	\f'\ <C (z G R)
for some constant C.
DEFINITIONS, (i) We say that й G HT(U) is a weak supersolution of
problem (1) if
(3)	/ Du  Dv dx > / f(u)vdx
Ju	Ju
for each v G Hq(U}, v > 0 a.e.
(ii)	Similarly, и G Л1(С7) is a weak subsolution provided
(4)	I DuDvdx< I f(u)vdx
Ju	Ju
for each v G Hq(U), v > 0 a.e.
(iii)	We say и G Hq(U) is a weak solution of (1) if
I Du-Dvdx = I f(u)vdx
Ju	Ju
for each v G Hq (U).
Remark. If й, и G C2(U), then from (3) and (4) it follows that
—Лй > f(u), —Au < /(u) in U.
□
THEOREM 1 (Existence of a solution between sub- and super solutions).
Assume there exist a weak supersolution й and a weak subsolution и of (1),
satisfying
(5) и < 0, й > 0 on dU in the trace sense, u<u a.e. in U.
Then there exists a weak solution и of (1), such that
и < и < u a.e. in U.
9.3. METHOD OF SUBSOLUTIONS AND SUPERSOLUTIONS
509
Proof. 1. Fix a number A > 0 so large that
(6)	the mapping z i—> f(z) + Xz is nondecreasing;
this is possible as a consequence of hypothesis (2).
Now write uo = u, and then given uk (к = 0,1, 2,...) inductively define
Ufc+i G Hq(U) to be the unique weak solution of the linear boundary-value
problem
( -Aufc+i + Aufc+i = /(ufe) + Xuk in U
[	Ufc+i =0	on dU.
2.	We claim
(8)	и = u0 < ui < •  • < uk < • • • a.e. in U.
To confirm this, first note from (7) for к = 0 that
(9)	/ Dui  Dv + Auiu dx = / (/(uo) + Xuo)u dx
Ju	Ju
for each v G	Subtract (9) from (4), recall uq = u, and set
v := (uo — ui)+ G Hq(U), v > 0 a.e.
We find
(10)	/ D(u0 - ui) • D(u0 - ui)+ + A(u0 - ui)(u0 - ui)+ dx < 0.
Ju
But
( D(u0 - Щ) a.e. on {u0 > uj
D(u0 -ui) = <!
I 0	a.e. on {uo < ui).
(See Problem 17 in Chapter 5.) Consequently,
/ |D(uo — ui)|2 + A(uo — ui)2 dx < 0;
J {uo>Ul}
so that uq < ui a.e. in U.
Now assume inductively
(11)	uk-i < uk a.e. in U.
From (7) we find
(12)	/ Duk+i  Dv + Xuk+iv dx = / (f(uk') + Xuk')vdx
Ju	Ju
510
9. NONVARIATIONAL TECHNIQUES
and
I Duk • Dv + Xukv dx = I	+ Xuk-i)v dx
Ju	Ju
for each v E Hq(U). Subtract and set v := (u^ — Ufc+i)+. We deduce
/ \D(uk - ufe+i)|2 + X(uk - uk+1)2dx
J{uk>Vk+l}
= / [(/(«fc-i) + Aufc_i) - (/(ufe) + Xuk)](uk - uk+1)+ dx < 0,
Ju
the last inequality holding in view of (11) and (6). Therefore uk < uk+i a.e.
in U, as asserted.
3.	Next we show
(13)	uk < й a.e. in U (k = 0,1,...).
Statement (13) is valid for к — 0 by hypothesis (5). Assume now for induc-
tion
(14)	uk < й a.e. in U.
Then subtracting (3) from (12) and taking v := (u&+i — й)+, we find
/ |D(ufc+i - S)|2 + A(ufc+i - u)2 dx
J{uk+I>u}
< / [(/(«*) + Aufc) - (/(й) + Au)](ufc+1 - S)+ dx < 0,
Ju
by (14) and (6). Thus Ufc+i < й a.e. in U.
4.	In light of (8) and (13), we have
(15)	u < • • • < uk < uk+i <•••<& a.e. in U.
Therefore
u(x) := lim uk(x)
k—>oo
exists for a.e. x. Furthermore we have
(16)	uk —> и	in 7L2(E7),
as guaranteed by the Dominated Convergence Theorem and (15). Finally,
since we have ||/(«k)||L2(tq < С'(||гх^||ь2(гу) + 1), we deduce from (7) that
9.4. NONEXISTENCE
511
supfe ||ufc||Я1 (ц) < oo. Hence there is a subsequence	which converges
weakly in Hq(U) to и € Hq(U).
5.	We at last verify that и is a weak solution of problem (1). For this,
fix v G Hq(U"). Then from (7) we find
/ Duw • Dv + Xuk+ivdx = / (/(ufe) + Xuk)vdx.
Ju	Ju
Let к —> oo:
/ Du  Dv + Xuv dx = / (f(u) + Xu)vdx.
Ju	Ju
Canceling the term involving A, we at last confirm that
/ Du • Dv dx = / f(u)vdx,
Ju	Ju
as desired.	□
This proof illustrates the use of integration-by-parts, rather than the
maximum principle, to establish comparisons between sub- and supersolu-
tions.
9.4. NONEXISTENCE
We now complement the theory in §§9.1—9.3 with some nonexistence asser-
tions for solutions of various nonlinear partial differential equations. The
overall procedure will be to assume there exists a solution, and then to
obtain certain inequalities, which in turn force a contradiction.
9.4.1. Blow-up.
We begin by considering an initial/boundary-value problem for a para-
bolic equation with a simple quadratic nonlinearity:
(1)
щ — Xu — u2
и — 0
и = g
in Ut
on dU x (0,T)
on U x {t = 0}.
We will show that if T > 0 and g > 0 are large enough in an appropriate
sense, then there does not exist a smooth solution и of (1). We can regard
the nonlinear heat equation in (1) as a simple reaction-diffusion equation (cf.
Example 1 in §9.2.1). The nonlinear term alone corresponds to the ODE
(2)
й = и2
d\
dt) ’
512
9. NON VARIATION AL TECHNIQUES
which certainly blows up in finite time, provided u(0) > 0. The purely
diffusive effects on the other hand yield the heat equation, which tends to
smooth out irregularities. The following analysis must therefore untangle
the competing effects of blow-up from the u2 term and smoothing from the
Au term.
We proceed by choosing wi to be an eigenfunction corresponding to the
principal eigenvalue Ai > 0 of —A in H^U). Then owing to the theory in
§6.5.1, wi is smooth,
( —Awi = Aiw in U
( wi = 0 on dU,
and we may furthermore assume
(3)	wi > 0 in U, / wi dx = 1.
Ju
Suppose u is a smooth solution of (1), with g > Q.g 0; so that u > 0
within Ut by the strong maximum principle. Define
T?(t) := I u(x,t)wi(x)dx (0 < t < T).
Ju
Then
(Au + u2)wi dx
uAwi + u2wi dx = —Ai??(t) + I u2w\ dx.
Ju
I UtWi
и
и
Furthermore
Consequently
r?(t)2 < / u2wi dx.
Ju
Employing this inequality in (4), we find
(6)	7/(t) >-Ai^t) + 77(t)2 (0<t<T).
9.4. NONEXISTENCE
513
Writing £(t) := eAltr?(i) gives
£(*) = ex^t) +	> e^r^tf2 =
for 0 < t < T. Thus
f rlY = M > е-л.«.
\m) W -
so that integrating we find
-1^-1	1 - e“A1<
m - ew + A! •
Rearranging, we deduce
m im > ________________________«^1!_______
provided the denominator is not zero.
But now suppose
(8)
f?(o) = e(o)>A1.
Then
(9)
where
(10)
£(t) —»• +oo as t t*,
C := —log
Ai
T?(0) - Al
7/(0)
The conclusion is that if
= / gwidx > Ai,
Ju
then there cannot exist a smooth solution и of (1). Furthermore, either the
solution is not smooth enough to justify the calculation above, or else
lim / u(x, dx = oo
Ju
for some 0 < t* < t*. In this case we say и “blows up” at time t*.
514
9. NONVARIATIONAL TECHNIQUES
9.4.2. Derrick—Pohozaev identity.
We investigate next a nonlinear elliptic PDE to which different differen-
tial inequality methods apply, namely the nonlinear boundary-value problem
(11)
—Ди = |u|p lu in U
и — 0 on dU.
Now the theory in §8.5.2 applies to (11) provided
and proves the existence of a nontrivial solution и 0. Let us now instead
suppose
Our goal is to demonstrate under a certain geometric condition on U that
(13) implies и = 0 is the only smooth solution of (11). We see therefore
that the restriction to condition (12) in §8.5.2 was in some sense natural,
and consequently say p = is a critical exponent.
DEFINITION. An open set U is called star-shaped with respect to 0 pro-
vided for each x^U, the line segment
{Xz | 0 < A < 1}
lies in U.
A star-shaped domain
9.4. NONEXISTENCE
515
Clearly if U is convex and 0 G U, then U is star-shaped with respect to
0. But a general star-shaped region need not be convex.
LEMMA (Normals to a star-shaped region). Assume dU is C1 and U is
star-shaped with respect to 0. Then
x  v(x) > 0 for all x G dU,
where v denotes the unit outward normal.
Proof. 1. Since dU is C1, if x G dU then for each e > 0 there exists ё > 0
such that \y — x\ < ё andy€U imply v(x) 	< £• In particular
lim sup v(x) 	——y- < 0.
У^х	\y — x\
yeu
Let у = Xx for 0 < A < 1. Then у ей, since U is star-shaped. Thus
, . x	x (Xx — x)
' T = “ lim y^x'> ’ i\------Г - °-
|ж| л—1- |Аж - ж|
□
We next prove that there can exist no nontrivial solution to problem
(11) for supercritical growth, provided U is star-shaped. The proof is a
remarkable calculation initiated by multiplying the PDE —Au = |u|₽-1u by
x  Du and continually integrating by parts.
THEOREM 1 (Nonexistence of nontrivial solution). Assume и G C'2(U)
is a solution of problem (11), and the exponent p satisfies inequality (13).
Suppose further U is star-shaped with respect to 0, and dU is C1. Then
u = 0 within U.
Proof. 1. We multiply the PDE by x  Du and integrate over U, to find
(14)
[ (—Au)(z • Du) dx = j |u|₽ 1u(x-Du)dx.
Ju	Ju
We rewrite this expression as
A = B.
.516
9. NONVARIATIONAL TECHNIQUES
2. The term on the left is
3. Now
On the other hand, since и = 0 on dU, Du(x) is parallel to the normal v(x)
at each point x G dU. Thus Du(x) = ±|Du(a?)|b'(a?). Using this equality we
calculate
(17)	A2 = - [ \Du\2(vx)dS.
JdU
Combine (15)—(17), to deduce
A =	- f iDu^dx — f \Du\2(y  x) dS.
2 Ju	2 JdU
4.	Returning to (14), we compute
n
В :=	/ \u\p~1uxjUx dx
j=iJu
5.	This calculation and (14) yield
(18)	[ \Du\2dx + ^- f \Du\2(y-x)dS = -?— f \u\p+1dx.
\ 2 / Ju	2 JdU	P + 1 Ju
9.5. GEOMETRIC PROPERTIES OF SOLUTIONS
517
In view of the lemma above, we then obtain the inequality
(19)
\ 2 ) Jv	p+1 Ju
But once we multiply the PDE —Au = |u|₽-1u by u and integrate by parts,
we produce the equality
[ \Du\2dx = [ |u|₽+1cLr.
Ju Ju
Substituting into (19), we thus conclude
Hence if и ф 0, it follows that 2^---xr < 0; that is, p < 2-Ц.	□
Remark. Equality (18) is sometimes called the Derrick-Pohozaev identity.
□
9.5. GEOMETRIC PROPERTIES OF SOLUTIONS
9.5.1. Star-shaped level sets.
We explain in this section a simple method that is occasionally useful
for studying the geometric properties of the level sets of solutions to various
PDE. The easiest such case occurs when we look at harmonic functions in
an open set U having the form
U = W - V,
where V CC W, for open sets V, W, each of which is star-shaped with
respect to 0. Write
Го = dW, Г1 = dV.
We consider the problem
(1)
in U
on Г1
on Го.
Physically u corresponds to the electrostatic potential generated within the
region U, once we fix the potential value to be one on Г1 and zero on Fq.
According to the strong maximum principle 0 < u < 1 within U.
518
9. NON VARIATION AL TECHNIQUES
THEOREM 1 (Star-shaped level sets). For each 0 < A < 1 the level set
Г a := {t £ U | u(x) = A}
is a smooth surface and is the boundary of a set star-shaped with respect to
0.
Proof. 1. For each p > 0, the function x ь-> u(px) is harmonic, and thus
so is

-^-(i4(xzrr))|M==i = Du(x)  X
ap
(x £ U).
Now since и = 0 on Го, Du(x) points in the direction of — v(x) at each
point x £ dW. Additionally, we have x  v(x) > 0 on Го, since W is star-
shaped with respect to 0. Consequently v = Du • x < 0 on Tq. Similarly
v < 0 on Г1. According then to the strong maximum principle for harmonic
functions, v < 0 in U. In particular, Du 0 within U. Consequently the
Implicit Function Theorem implies that Гд is a smooth surface for 0 < A < 1.
2. Extend и to equal 1 on all of V and write
Ux := {t e W | и > A}.
Then U\ is an open subset of W and dU\ = Гд. By the strong maximum
principle, U\ is connected.
Now let x £ Гд and let v(x) denote the outer unit normal to Гд at
x. Then Du(x) points in the direction of — is(x). Since v(x) < 0, we have
x  v(x) > 0. This inequality holds for each x £ Гд.
3. It follows that Гд is the boundary of a set star-shaped with respect
to 0. To see this, return to the proof of the lemma in §9.4.2, and notice
that if Гд were not star-shaped with respect to 0, we could then find a point
x £ Гд for which у = px Гд if p is close to 1, p < 1. But then we can
derive the contradiction
/ . X	. (px — x\
v(x) • — = — hm i4ir) • i............. < 0.
ж	/z—i-	\px — т
□
9.5.2. Radial symmetry.
In this section we take U = B°(0,1) to be the open unit ball in Rn and
investigate this boundary-value problem for a semilinear Poisson PDE:
(2)
—Au = /(u) in U
и = 0 on dU.
9.5. GEOMETRIC PROPERTIES OF SOLUTIONS
519
We are interested in positive solutions:
(3)	и > 0 in U,
and will assume f : R —► R is Lipschitz continuous, but is otherwise arbi-
trary. Our intention is to prove that и is necessarily radial, that is, u(x)
depends only on r = |rr|. This is quite an unexpectedly strong conclusion,
since we are making essentially no assumption on the nonlinearity.
a. Maximum principles.
Our proofs will depend upon an extension of the maximum principle for
second-order elliptic PDE.
LEMMA 1 (A refinement of Hopf’s Lemma). Suppose V C Rn is open,
v £ C2(V), and c £ L°°(V). Assume
, .	( — Au + cv > 0 inV
W	I	v > 0 in V.
Suppose also v ф 0.
(i) If x° £ dV, u(rr°) = 0, and V satisfies the interior ball condition at
xq, then
(5)	|(?)<».
(ii)	Furthermore,
(6)	v > 0 in V.
Observe that we are here making no hypothesis concerning the sign of
the zeroth-order coefficient c.
Proof. Let w := e_Allu, where A > 0 will be selected below. Then v =
eXxiw, and so
cv > Au = A(eAllw) = A2u + 2ЛеАх1гсЖ1 + eAllAw.
Thus
—Aw — 2XwX1 > (A2 — c)w >0 in V,
if A = Mll-2.
520
9. NONVARIATIONAL TECHNIQUES
Consequently w is a supersolution for the elliptic operator Kw := —Aw—
2AwX1, which has no zeroth-order term. The strong maximum principle im-
plies w > 0 in V. According to Hopf’s Lemma (§6.4.2) therefore, < 0.
But
— (t°) = Dw(t°)  «/(a;0) = e	^(^°)
since u(rr°) = 0. Assertion (i) therefore holds, and assertion (ii) follows since
w > 0 in V.	□
LEMMA 2 (Boundary estimates). Let и £ C2(U) satisfy (2), (3). Then
for each point x° G dU Г) {xn > 0}, either
(7)	uIn(x°)<0
or else
(8)	г41п(т°) = 0, uXnXJxf)) > 0.
In either case, и is strictly decreasing as a function of xn near xQ.
Proof. 1. Fix any point x° £ dU П {тп > 0} and let u = u(x°) =
(zq,..., un) denote the outer unit normal to dU at t°. Note vn > 0.
2.	We first claim
^„(^°) < 0,
provided
(9)	/(0) > 0.
Indeed
0 = —Au — /(u) = -Au - /(u) + /(0) - /(0)
< —Au + cu,
for c(t) := — £ /'(su(t)) ds. According to Lemma 1, |^(t°) < 0. Since Du
is parallel to v on dU and un > 0, we conclude и1п(т°) < 0.
3.	Now suppose
(10)	/(0) < 0.
If и1п(т°) < 0, we are done. Otherwise, since Du is parallel to v,
(11)	£>u(t°) = 0.
9.5. GEOMETRIC PROPERTIES OF SOLUTIONS
521
As (2) is invariant under a rotation of coordinate axes, we may as well
suppose x° = (0,..., 1), v = (0,..., 1).
4.	We assert
(12)	uXiXj(x0) = -/(0)^ for each = 1,... ,n.
Since и = 0 on dU, we have u(x', 7(x')) = 0 for all x' G Rn-1, \x’| < 1,
where 7(x') = (1 — \x'I2)1/2. Differentiating with respect to Xi and Xj (i,j =
1,..., n — 1) and using (11), we conclude
(13)	uXiXj (x°) = 0 (i,j = 1,... , n - 1).
Since uXn < 0 on dU П {xn > 0} and uXn(x°) = 0, the mapping x' t—>
wIn(x', 7(x')) has a maximum at x' = 0. Thus
(14)	uXnXflxQ} = 0 (г = l,...,n-1).
Finally, (13), (14) and the PDE (2) force uXnXn(x°) = —/(0). This equality
is (12) for v = (0,...,l). Returning to the original coordinate axes, we
obtain (12).
5.	Setting i = j = n in (12), we find using (10) that
Ux„xjx°) = -f(tyvnvn > 0.
□
b. Moving planes.
We introduce next a “moving plane” Px, across which we will reflect our
partial differential equation.
Notation, (i) If 0 < A < 1, define the plane
Px := {x G R" | xn = A}.
(ii)	Write x\ := (xi,..., xn-i, 2A — xn) to denote the reflection of x in
Pa-
(iii)	Ex := {x G U | A < xn < 1}.	□
THEOREM 2 (Radial symmetry). Let и G C2(C7) solve (2), (3). Then и
is radial; that is,
u(x) = u(r) (r = |x|)
for some strictly decreasing function v : [0,1] —► [0, oo).
522
9. NON VARIATION AL TECHNIQUES
Reflection through a plane
Proof. 1. We consider for each 0 < A < 1 the statement
(15д)	-и(ж) < u(x\) for each point x e Ex.
2.	According to Lemma 2, (15д) is valid for each A < 1, A sufficiently
close to 1. Set
(16)	Aq = inf{0 < A < 1 | (15M) holds for each A < /r < 1}.
We will prove:
(17)	Ao = 0.
Assume instead Ao > 0. Write w(t) := u(;ea0) — u(ir) (x £ Ед0). Then
-Aw = f(u(xXo)) - f(u(x)) = —cw in EXo,
for c(t) := — f'(su(xXo) + (1 — s)u(t)) ds. As w > 0 in EXo, we deduce
from Lemma 1 (applied to V = EXo) that w > 0 in EXo, wXn > 0 on PXo DtZ.
Thus
(18)	u(x)<u(xXo) in EXo,
and
(19)	uXn <0 on PXo П U.
Using (18), (19) and Lemma 2, we conclude
(20)	u(t) < -u(tAo_£) in EXo_£ for all 0 < s < e0,
if Eq is small enough. Assertion (20) contradicts our choice (16) of Aq, if
Ao > 0.
3.	Since Ao = 0, we see u(xi,..., xn-i, —xn) > u(xi,..., xn) for all x €
UC]{xn > 0}. A similar argument in C7n{irn < 0} shows u(si,... ,xn_[, —xn)
< u(xi,..., xn) for all x G U П {тп > 0}. Thus и is symmetric in the plane
Pq and uXn = 0 on Po.
This argument applies as well after any rotation of coordinate axes, and
so the theorem follows.	□
9.6. GRADIENT FLOWS
523
9.6. GRADIENT FLOWS
In this section we augment our discussion of abstract semigroup theory for
linear operators (§7.4) by introducing certain nonlinear semigroups, gener-
ated by convex functions. Applications include various nonlinear second-
order parabolic partial differential equations.
9.6.1. Convex functions on Hilbert spaces.
Convexity has been an essential ingredient in much of our analysis of
nonlinear PDE thus far. We now broaden our view by considering convex
functions defined on (possibly infinite dimensional) Hilbert spaces.
Hereafter H will denote a real Hilbert space, with inner product ( , )
and norm || ||.
DEFINITION. A function
I : H —> (—oo, oo]
is convex provided
I[ru + (1 — r)v] < r/[it] + (1 — r)/[v]
for all u,v € H and each 0 < т < 1.
Note carefully that we allow I to take on the value +00 (but not —00).
The function I is called proper if I is not identically equal to +00. The
domain of I is
D(I) := {« £ H I I[u] < +00}.
DEFINITION. We say I : H —► (—00, +00] is lower semicontinuous if
' Uk —> и in H implies
I[u] < lim inf
к—><x>
As in the finite dimensional case (cf. §B.l), it is important to understand
when the graph of I has a supporting hyperplane.
DEFINITIONS. Let I : H —► (—00, +00] be convex and proper.
(i) For each и € H, we write
(1)	d/[u] .= {v E H \ 7[w] > /[it] + (v, w — u) for all w £ H}.
The mapping di : H —► 2H is the subdifferential of I.
(ii) We say и € D(dl), the domain of di, provided 9/[u]	0.
The geometric interpretation of (1) is that v € di [it] if and only if v
is the “slope” of an affine functional touching the graph of I from below
at the point u. Since this graph may have a “corner” at u, cH[u] could be
multivalued.
524
9. NONVARIATIONAL TECHNIQUES
THEOREM 1 (Properties of subdifferentials). Let I : H —► (—oo,+oo] be
convex, proper and lower semicontinuous. Then
(i)	D(dI)Q D(J).
(ii)	If v E dZ[u] and v € then
(v — v,u — u) >0 (monotonicity).
(iii)	Z[u] = minwe// I[w] if and only if 0 £ dZ[u].
(iv)	For each w £ H and A > 0, the problem
и + A9Z[u] Э w
has a unique solution и £ D(dT).
Assertion (iv) means that there exists и £ D(dl) and v £ Э/[и] such that
и + Xv = w.
Proof. 1. Let и £ D(dl), v € dZ[u], Then Z[w] > Z[u] + (v, w — u) for all
w £ H. Since I is proper, there exists a point uq with Z[«o] < +oo. Thus
I[u] < Z[uq] + (y,u — u0) < 00 and so u £ D(F). This proves (i).
2.	Given v € dZ[u], v £ dZ[&], we know
Z[&] > Z[u] + (v,u — u), Z[u] > Z[fi] + (v,u — u).
As (i) implies Z[u], Z[&] < +oo, we may add the foregoing inequalities and
rearrange to obtain (ii).
3.	If Z[u] = min I, then
(2)	Z[w] > L[u] + (0, w — u) for all w £ H.
Hence 0 £ dZ[u]. If conversely 0 £ сЩи], then (2) holds, and so Z[u] = min I.
4.	Given w £ H and A > 0, define
(3)	J[u] := |||u||2 + AZ[u] — (u, w) (u £ H).
We intend to show that J attains its minimum over H.
Let us first claim that
Uk и weakly in H implies
(4)	Z[u] < lim inf Ipu/J.
9.6. GRADIENT FLOWS
525
In other words, we are asserting for a convex function I that lower semicon-
tinuity with respect to strong convergence of sequences implies lower semi-
continuity with respect to weak convergence. To see this, suppose uk и
in H and
lim inf/[iq.] = lim I[ufc7] = I < oo
for some subsequence	C • For each £ > 0 the set K£ = {w £
H | Z[w] < /+c} is closed and convex, and is thus weakly closed according to
Mazur’s Theorem (§D.4). Since all but finitely many of the points {и^}^
lie in K£, и lies in K£, and consequently
Z[u| < I + e = lim inf /[зд] + £.
This is true for each £ > 0 and thus (4) follows.
5.	Next we assert that
(5)	Z[u] > -C - C||u|| (и £ H)
for some constant C. To verify this claim we suppose to the contrary that
for each k = 1,2,... there exists a point uk £ H such that
(6)	I[uk] < -k -&||ufc||.
If the sequence	is bounded in H. there exists according to §D.4 a
weakly convergent subsequence: ukj u. But then (4) and (6) imply the
contradiction /[it] = —oo. Thus we may as well assume, passing if necessary
to a subsequence, that ||iik|| —► oo. Select uq £ H so that I[ito] < oo. Set
zk :=	+ (1 - if) uo (fc = l,2,...).
Ihfcll \	1Ы1/
Then convexity implies
||Дн/|“'=|+
As	is bounded, we can extract a weakly convergent subsequence:
zk. —^z, and again derive the contradiction I[z] = — oo. We thereby estab-
lish the claim (5).
6.	Return now to the function J defined by (3). Choose a minimizing
sequence {'Ufe}^=1 С H so that
—> inf J[w] = m.
wEH
526
9. NON VARIATION AL TECHNIQUES
Owing to (3), (5) m is a finite number. Thus we deduce from (3), (5)
that the sequence	is bounded. We may then extract a weakly
convergent subsequence:	—>• u. As the mapping и i—> ||u||2 is weakly
lower semicontinuous, J has a minimum at u. Then assertion (iii) says
0 £ 9J[u]. A computation verifies that 9J[u] = и — w + Ad/[u], and so
и + A9/[u] Э w.
7.	To confirm uniqueness, suppose as well
й + А9/[й] Э w.
Then и + Xv = w, й + Xv = w for v € 9/[u], v G Э/[й]. Owing to the
monotonicity assertion (ii),
Since A > 0, и = й.	□
We introduce next nonlinear analogues of the operators R\,A\ intro-
duced in §7.4.
DEFINITIONS. (1) For each A > 0 define the nonlinear resolvent J\ :
H —► D(dF) by setting
Ja[w] := u,
where и is the unique solution of
и + Ad/[u] Э w.
(2) For each A > 0 define the Yosida approximation Ад : H —► H by
(7)	АлН := W~fA[w] (w G Я).
A
Think of Ад as a sort of regularization or smoothing of the operator
A = di.
THEOREM 2 (Properties of J\, Ад). For each A > 0 and w,w G H, the
following statements hold:
(i)	||JA[w] - JA[w]|| < ||w-w||,
(ii)	Ma[w] - Aa[w]|| < ^||w - w||,
(iii)	0 < (w — w, Aa[w] — Aa[w]),
9.6. GRADIENT FLOWS
527
(iv)	Aa[w] G S/[JA[w]].
(v)	IfwG D(dl), then
sup||AA[w]|| < |A°[w]|,
A>0
where |A°[w]| := min2e9/[w] ||z||.
(vi)	For each w G D(dl},
lim Jifwl = w.
A^O
Proof. 1. Let и = JA[w], й = JA[w], Then и + Xv = w, й + Xv = w for
some v edl[u],v E di[й]. Therefore
||w — w ||2 = ||u — й + X(y — v)||2
= ||u — й||2 + 2A(u — u, v — v) + A21|v — v||2
> ||u - й||2,
according to Theorem l,(ii). This proves assertion (i), and assertion (ii)
follows at once from the definition (7) of the Yosida approximation A\.
2.	We verify (iii) by using (7) to compute:
(w - w, Aa[w] - Aa[w]) = у (||w - w||2 - (w - w, JA[w] - JA[w]))
> |(||w - w||2 - ||w - w|| ||JA[w] - Jx[w]II) > 0,
A
according to (i).
3.	To prove (iv), note и = JA[w] if and only if и + Xv = w for some
v G d/fy] = dI[JA[w]]. But
w-u w-Jx[w]
u =	=-----------= Aa[w],
4.	Assume next w G D(dl), z G Э/fw]. Let и — JA[w]; so that и + Xv =
w, where v G 9/[u]. By monotonicity
0 < (w — u, z — v) = (w — J\[w],z — ——= (AAa[w], z — Aa[w]).
Consequently
A||Aa>]||2<(AAaM,z)< АЦАлНЦ ||<
528
9. NONVARIATIONAL TECHNIQUES
and so
MaMH < ||z||.
This estimate is valid for all A > 0, z G 9/[w], Assertion (v) follows.
5.	If w G D(dl), then
|| JA[w] - w|| = A||Aa[w]|| < A|A°[w]| ,
and hence JA[w] —> w as A —► 0. Now let w G D(dl) — D(dl). There exists
for each e —► 0 a point w G D(dT) with ||w — w|| < s. Then
II JA[w] - w|| < II Jx[w] - JA[w]II + II JA[w] - w|| + ||w - w||
<	2||w - w|| + II JA[w] - w||
<	2s + || JA[w] - w||.
Since w G D(dl), JA[w] —>• w as A —► 0. Thus
limsup || JA[w] — w|| < 2s
A—>0
for each s > 0.	□
9.6.2. Subdifferentials and nonlinear semigroups.
As above, let H be a real Hilbert space, and take I : H —► (—oo,+oo]
to be convex, proper, lower-semicontinuous. Let us for simplicity assume as
well
(8)	di is densely defined, i.e. D(dl) — H.
By analogy with the theory of linear semigroups set forth in §7.4, we propose
now to study the differential equation
,	J u'(t) + A[u(t)] Э 0 (f > 0)
{	u(0)= u,
where и G H is given and A = di is a nonlinear, discontinuous operator,
which is perhaps multivalued. Assuming for the moment (9) has a unique
solution for each initial point u, we write
(10)	u(t) = S(t)u (t > 0)
and regard S(t) so defined as a mapping from H into H for each time t > 0.
9.6. GRADIENT FLOWS
529
Remark. We will employ the notation (10) to emphasize similarities with
linear semigroup theory, previously introduced in §7.4. But carefully note
here and afterwards that the mapping и S(t)u is in general nonlinear.
□
As in §7.4 it is reasonable to expect further that
(11)
S(0)u = и (и G H),
(12)	S(t + s)u = S(t)S(s)u (t, s > 0, и G Я),
and for each и G H
(13)	the mapping t i—> S(t)u is continuous from [0, oo) into H.
DEFINITIONS, (i) A family {S(t) }t>o of nonlinear operators mapping H
into H is called a nonlinear semigroup if conditions (11)—(13) are satisfied.
(ii)	We say {S(ty}t>o is a contraction semigroup if in addition
(14)	- 5'(<)й|| < ||u — й|| (t > 0, и,й E Hf
Our intention is to show that the operator A = di generates a nonlinear
semigroup of contractions on H. In particular we will prove that the ODE
< u'(f) e -<W[u(f)] (f>0)
(15)	I u(0) = »,
for a given initial point и G H, is well-posed. This is a kind of infinite
dimensional “gradient flow” governed by di. Later in §9.6.3 we will see that
certain quasilinear parabolic PDE can be cast into the abstract from (15).
THEOREM 3 (Solution of gradient flow). For each и G D(dl) there exists
a unique function
(16)	UG C([0,oo);R), with u1 € L°°(f),oo; FT),
such that
(i)	u(0) = u,
(ii)	u(t) G D(dl} for each t > 0,
and
(iii)	u'(t) G — d/[u(t)]	for a.e. t > 0.
530
9. NONVARIATIONAL TECHNIQUES
Proof. 1. We first build approximate solutions by solving for each A > 0
the ODE
z17s	f u'A(t) + AA[uA(f)] = 0 (t > 0)
t	uA(0) = u.
According to Theorem 2,(ii) the Yosida approximation A\ : H —> H is
an everywhere defined, Lipschitz continuous mapping, and thus (17) has a
unique solution ид £ Cx([0, oo); Я).
Our plan is to show that as A —> 0+, the functions ид converge to a
solution of (15). This is subtle, however, as the operator A = dl is in
general nonlinear, multivalued, and not everywhere defined.
2.	First, let us take another point v s H and consider as well the ODE
j V'AW + 4\[vA(f)] = 0 (t > 0)
(18)	t	vA(0) = v.
Then
I “ VA^2 = (UA “ VA,UA ~ VA)
= (—Aa[ua] + AA[vA], UA - VA) < 0,
owing to Theorem 2,(iii). Hence
(19)	||uA(t) - vA(t)|| < ||u-u|| (f>0).
In particular, if h > 0 and v = uA(/i), then by uniqueness vA(t) = uA(t + fi).
Consequently (19) implies
||uA(t + h) - uA(t)|| < ||и(Л.) - u||.
Divide by h and send h —> 0:
(20)	||и'л(0Н < ||и'л(0)|| = Мл(»]|| < M’MI,
the last inequality resulting from Theorem 2,(v).
3.	We next take A, /r > 0 and compute:
(21)	I ^11иА-и^||2 = (и'А-и^,иА-и^)
= (—Ад[ил] + АДиД uA - ид).
Now
UA - UM = (ид - JA[uA]) + (7д[ид] - /д[ид]) + (7д[ид] - ид)
= А Ал [иА] + Тд[иА] - JjuM] - Мм К] •
9.6. GRADIENT FLOWS
531
Consequently
(Aa[ua] - Am[um],ua - ид) = (Aa[ua] - АДиД JA[uA] - 7д[ид])
(22)	+ (Aa[ua] - Ад[ид],ЛАА[иА] -дАДид]).
Since Aa[ua] e 3/[JA[uA]] and АДид] £ д1[7Дид]], the monotonicity prop-
erty implies that the first term on the right hand side of (22) is nonnegative.
Thus
(Aa[ua] - АДид],ил -ид) > A(|Aa[ua]||2 + д||АДид]||2
- (A + m)||Aa[ua]|| ||AUum]||.
Since
(Л + м)||Лл[ил]|| M„[uJ|| < А (||Лл[ил]||2 + IIMUMI2)
+ м(11ЛЫП2 + IIIAWI2).
we deduce
(Ш - /ЦиДид -u„) > - А||Л„[и„]||2 - jll^WII2.
But ||Aa[ua]|| = ||и'л|| < |A°[u]| according to (20); whence
(Aa[ua] - АДид],ил - ид) > -	|A°[u]|2.
Recalling (21), (22), we obtain the inequality
4||ид - u„||2 < 1Ц^|Л“М|2 (i>0);
at	2
and hence
(23)	||uA(i) - u„(<)||2 < <A±rff|4au|2 (s > 0).
In view of estimate (23) there exists a function и € C([0, oo); FT) such that
uA —► и	uniformly in C([0, T], H)
as A —> 0, for each time T > 0. Furthermore estimate (20) implies
(24)	—>• и' weakly in L2(0, T; H)
for each T > 0, and
(25)	||u'(f)|| < |A°[iz]|	for a.e. t.
532
9. NON VARIATION AL TECHNIQUES
4.	We must show u(t) € D(dT) for each t > 0 and
u'(t) + dl[u(t)] Э 0 for a.e. t > 0.
Now
II JA[uA](t) - uA(t)II = AllAA[uA](i)|| = A||u'A(t)II < A|A°[u]|
by (20). Hence
(26)	JA[uA] —> u uniformly in C([0,T]; H)
for each T > 0.
For each time t > 0,
-u'A(i) = AA[uA(f)] e a/[JA[uA(t)]].
Thus given w £ H, we have:
I[w] > Z[JA[uA(t)]] - (uA(t), w - JA[uA(t)]).
Consequently if 0 < s < t,
(t-s)I[w]> f I[Jx[ux(r)]\dr- [ (uA(r),w-JA[uA(r)])dr.
J s	J s
In view of (26), the lower semicontinuity of I, and Fatou’s Lemma (§E.3),
we conclude upon sending A —> 0 that
(t — s)I[w] > [ F[u(r)] dr — [ (u'(r), w — u(r)) dr
Js	J s
for each 0 < s < t. Therefore
/[w] > Z[u(f)] + (-u'(t),w - u(t))
if t is a Lebesgue point of u', Z[u]. Hence for a.e. t > 0,
I[w] > Ли(^)] + (-u'(0,w _ u(0)
for all w £ H. Thus u(f) £ D(dl), with
-u'(t) € 3Z[u(t)]
for a.e. t > 0.
9.6. GRADIENT FLOWS
533
5. Finally we prove u(t) G D(dl) for each t > 0. To see this, fix t > 0
and choose L- —► t such that u(tfe) G D(dl),	G d/[u(tfc)]. In view of
(25) we may assume, upon passing if necessary to a subsequence, that
u'(ifc) —* v weakly in H.
Fix w € H. Then
/[w] > /[u(tfc)] + (-u'(4),w - u(tfe)).
Let tk —► t and recall that u G C([0, oo]; H) and I is lower semicontinuous.
We obtain the inequality
I[w] > Z[u(f)] + (—v,w — u(t)).
Hence u(t) G D(dl) and — v g d/[u(t)].
6. We have shown u satisfies assertions (i)—(iii). To prove uniqueness,
assume ii is another solution and compute
i -у-||и — й||i 2 = (uz — uz,u — u) < 0 for a.e. t > 0,
2 dt
since —u' G <9Z[u], —ii' G <Э/[й].	□
Remarks, (i) The operator A = di in fact generates a nonlinear contrac-
tion semigroup on all of H. If u, v G D(dl), we write as above
lim uA(t) = u(t) = S(t)u
and
lim vA(t) = v(t) = S(t)v.
Owing to (19) we see
\\S(t)u-S(t)v\\ < ||u- v|| (t>0)
if u, v G D(dl). Using this inequality we uniquely extend the semigroup of
nonlinear operators {S(t)}t>o to H = D(dF).
(ii) We have assumed that D(dl) is dense in H purely to streamline the
exposition: in general {S(t)}t>o is a semigroup of contractions on D(dT) C
534
9. NONVARIATIONAL TECHNIQUES
9.6.3. Applications.
We now illustrate how some of the abstract theory set forth in §§9.6.1-2
applies to certain nonlinear parabolic partial differential equations.
Let us hereafter suppose U is a bounded, open subset of Rn, with smooth
boundary dU. We choose H = L2(U), and set
(27)
hUDujdx if и
+oo	otherwise,
where L : Rn —► R is smooth, convex, and satisfies
(28)
|£»2L(p)|<C (peRn),
(29)	52 LpzPj ож,-> 0iei2 (p,esRn)
for constants C, 0 > 0.
THEOREM 4 (Characterization of di).
(i)	I : H —► (—oo,+oo] is convex, proper and lower semicontinuous.
(ii)	D(dl) = H2(U)U\Hq(U).
(iii)	IfuE D(dT), then di is single-valued and
n
dl[u] =-^(LPi(Du))Xi a.e.
i=i
Proof. 1. / is clearly proper and convex. Furthermore, since / is weakly
sequentially lower semicontinuous (cf. Theorem 1 in §8.2.2), I is lower semi-
continuous.
2.	Define the nonlinear operator A by setting
Г D(A) := T/2(C/) P 7/q (C7),
t AM := -E7=i(^P«)k (« e D(A».
We must prove A = di.
3.	First let и £ -D(A), v = A[u], w e L2(U). If w 7/q(C7), then
Z[w] = +oo and so clearly
(30)	I[w] > Z[u] + (u, w — u).
9.6. GRADIENT FLOWS
535
Assume next w G Hq(U). Thus
f П
(v,w-u) = - / V(LPi(Du))Xi(w - u)dx
г=1
Г П
= / \^LPi(Du)(Dw — Du) dx.
Ju i=1
Since L is convex,
L(Dw) > L(Du) + DpL(Du)  (Dw — Du) a.e. in U.
Integrating over U gives (30).
4.	We have thus far shown that A C di, that is, D(A) C D(dl) and
Au G <9/[u] for и € D(A). To conclude, we must prove that A D di.
Select any function f G L2(U). If we minimize the functional
f	w2
J[w] := / L(Dw) + —----fwdx
Ju	2
over the admissible class A = Hq(U), we will find и G H^(U), which is a
weak solution of
(31)	u-±(LPi(Du))Xi=f in U.
i=l
According to calculations similar to those for the proof of Theorem l,(ii) in
§8.3, we see that in fact и G H2(U), with the estimate
(32)	1к11н2([/) < c'II/IIl2([/)-
Thus и G D(A), and и + A[u] = f. Consequently the range of I + A is all
of H. But this implies A — di. For if v G D(dl), w G 9/[u], then there
exists и G D(A) such that u + A[u] = v + w. Since A[u] G d/[u], w G d/[u],
the uniqueness assertion of Theorem l,(iv) implies и — v, w = A[u], Thus
A = di.	□
We can now look at the initial/boundary-value problem:
(33)
<	u = 0
U = g
in U x (0, oo)
on dU x (0, oo)
on U x {t — 0},
536
9. NON VARIATION AL TECHNIQUES
where g € L2(C7). In accordance with Theorem 4, we can recast this problem
into the abstract form
Г u'(f) = -3/[u(t)] (t > 0)
I u(0) = g.
We apply Theorem 3. If g £ №([/) C\Hq(U). there exists a unique function
u £ C([0, oo); L2(C7)), with u' £ L°°((0; oo); L2(C7)),
that is, a weak solution of (33). In view of the estimate
IIu(*)IIh2([/) < C'llu'W,
we see u £ L°°((0, oo), Я2(С7) П ^(Cf)) as well.
9.7. PROBLEMS
In these problems U always denotes a bounded, open subset of Rn, with
smooth boundary.
1.	Assume the vector field v is smooth. Give another proof of the lemma
in §9.1 for this case by solving the ODE
Г x(t) = -v(x(t)) (t > 0)
I x(0) = y.
Let us write the solution as x(t,y) to display the dependence on the
initial point y. For each fixed time t > 0, the map у x.(t,y) is
continuous and so has a fixed point. Conclude v has a zero in the
closed ball B(0, r).
2. Assume a : R —> R is continuous and a(/n) —>• a(/) weakly in L2(0,1)
whenever fn-> f weakly in L2(0,1). Show a is an affine function;
that is, a has the form
a(z) = az + (3 (z £ R)
for constants a, (3.
3. Assume
' ut - Au = f
< и = 0
и = g
in U x (0, oo)
on dU x (0, oo)
on U x {t = 0},
where g £ L2(C7), f £	for each T > 0. Suppose т > 0 and f
is т-periodic in t; that is, f(x,t) = f(x, t + r) (x £ U, t > 0). Prove
9.7. PROBLEMS
537
there exists a unique function g £ L2(U) for which the corresponding
solution и is r-periodic.
4.	Consider the nonlinear boundary-value problem
—Au + b(Du)=f	in U
и = 0	on dU.
Use Banach’s fixed point theorem to show there exists a unique weak
solution и £ H2(U) П Hq(U) provided b : Rn —> R is Lipschitz contin-
uous, with Lip(6) small enough.
5.	Assume f : R —► R is Lipschitz continuous, bounded, with /(0) = 0
and /z(0) > Ai, Ai denoting the principal eigenvalue for — A on Hq (U).
Use the method of sub- and supersolutions to show there exists a weak
solution и of
in U
on dU
in U.
-Au = /(u)
и = 0
и > 0
6.	Assume that и, й are smooth sub- and supersolutions of the boundary-
value problem (1) in §9.3. Use the maximum principle to verify di-
rectly
U = U() < «1 < • • • < Uk < • • • < u,
where the	are defined as in §9.3.
7.	Let e > 0. Define
о / \ _ J ° if > 0
£(*) 	| £ if z < o,
and suppose ue € Hq(U) is the weak solution of
/ X	f -Aue + &(ue) = f in U
'	[	ue = 0 on dU,
where f £ L2(C7). Prove that as e —► 0, ue и weakly in Hq(17), u
being the unique solution of the variational inequality
for all w £ Hq(U) with w > 0 a.e.
Approximating the variational inequality by (*) is the penalty method.
8.	Let К C Rn be a closed, convex, nonempty set. Define
T, ,	f 0 if x £ К
Ж	( +oo if ж K.
538
9. NONVARIATIONAL TECHNIQUES
Explicitly determine A = di, = (I + A A) \ A\ = (A > 0) in
terms of the geometry of К.
9.	Give a simple example showing the flow
u' 6 —<9/[u] (t > 0)
may be irreversible. (That is, find a Hilbert space H and a convex,
proper, lower semicontinuous function I : H —> (—oo, +oo] such that
the semigroup solution of (*) satisfies
S(t)u —
for some t > 0 and и ф й.)
9.8.	REFERENCES
Section 9.	1 See J.-L. Lions [L2] and Zeidler [ZD, Vol. 2]. The booklet
[E] is a survey of methods for confronting weak convergence
problems for nonlinear PDE.
Section 9.	2 Zeidler [ZD, Vol. 1] has more on fixed point techniques.
Consult also Gilbarg-Trudinger [G-T, Chapter 11].
Section 9.	3 See Smoller [S, Chapter 10].
Section 9.	4 Section 9.4.1 is based upon Payne [PA, §9].
Section 9.	5 Section 9.5.2 is due to Gidas-Ni-Nirenberg[G-N-N],
Section 9.	6 This material is from Brezis [BR2], which contains much
more on nonlinear semigroups on Hilbert spaces. See Barbu
[BA] or Zeidler [ZD, Vol. 2] for nonlinear semigroup theory
in Banach spaces.
Chapter 10
HAMILTON-J AC OBI
EQUATIONS
10.1	Introduction, viscosity solutions
10.2	Uniqueness
10.3	Control theory, dynamic programming
10.4	Problems
10.5	References
10.1. INTRODUCTION, VISCOSITY SOLUTIONS
This chapter investigates the existence, uniqueness and other properties of
appropriately defined weak solutions of the initial-value problem for the
Hamilton-Jacobi equation:
, .	( щ + H(Du,x) = 0 in Rn x (0, oo)
[	и = g on Rn x {t = 0}.
Here the Hamiltonian H : Rn x Rn —> R is given, as is the initial function
g : Rn —> R. The unknown is и : Rn x [0, oo) —> R, и = u(x,t), and
Du — Dxu = (uX1,..., uXn). We will write H = H(p,x), so that “p” is the
name of the variable for which we substitute the gradient Du in the PDE.
We recall from our study of characteristics in §3.2 that in general there
can be no smooth solution of (1) lasting for all times t > 0. We recall
further that if H depends only on p and is convex, then the Hopf-Lax
formula (expression (21) in §3.3.2) provides us with a type of generalized
solution.
In this chapter we consider the general case that H depends also on x
and, more importantly, is no longer necessarily convex in the variable p. We
539
540
10. HAMILTON-JACOBI EQUATIONS
will discover in these new circumstances a different way to define a weak
solution of (1).
Our approach is to consider first this approximate problem:
. .	( uet + H(Due, x) — eAu£ = 0 in R” x (0, oo)
' '	[	u( = g on Rn x {t = 0},
for e > 0. The idea is that whereas (1) involves a fully nonlinear first-order
PDE, (2) is an initial-value problem for a quasilinear parabolic PDE, which
turns out to have a smooth solution. The term еД in (2) in effect regularizes
the Hamilton-Jacobi equation. Then of course we hope that as e —> 0 the
solutions ue of (2) will converge to some sort of weak solution of (1). This
technique is the method of vanishing viscosity.
However, as e —> 0 we can expect to lose control over the various esti-
mates of the function ue and its derivatives: these estimates depend strongly
on the regularizing effect of еД and blow up as e —> 0. However, it turns
out that we can often in practice at least be sure that the family {we}c>0
is bounded and equicontinuous on compact subsets of R” x [0, oo). Conse-
quently the Arzela-Ascoli compactness criterion, §C.7, ensures that
(3)	u3 —> и locally uniformly in Rn x [0, oo),
for some subsequence {nN and some limit function
(4)	и G C(Rn x [0, oo)).
Now we can surely expect that и is some kind of solution of our initial-value
problem (1), but as we only know и is continuous, and have absolutely no in-
formation as to whether Du and ut exist in any sense, such an interpretation
is difficult.
Similar problems have arisen before in Chapters 8 and 9, where we had to
deal with the weak convergence of various would-be approximate solutions to
other nonlinear partial differential equations. Remember in particular that
in §9.1 we solved a divergence structure quasilinear elliptic PDE by passing
to limits using the method of Browder and Minty. Roughly speaking, we
there integrated by parts to throw “hard-to-control” derivatives onto a fixed
test function, and only then tried to go to limits to discover a solution. We
will for the Hamilton-Jacobi equation (1) attempt something similar. We
will fix a smooth test function v and will pass from (2) to (1) as e —> 0 by
first “putting the derivatives onto v”.
But since (1) is fully nonlinear, and in particular is not of divergence
structure, we cannot just integrate by parts, as we did in §9.1, to switch
10.1. INTRODUCTION, VISCOSITY SOLUTIONS
541
to differentiations on v. Instead we will exploit the maximum principle to
accomplish this transition, at least at certain points.
We will call the solution we build a viscosity solution, in honor of the
vanishing viscosity technique. Our main goal will then be to discover an
intrinsic characterization of such generalized solutions of (1).
10.1.1	. Definitions.
Motivation for definition of viscosity solution. We henceforth assume
that H, g are continuous and will as necessary later add further hypotheses.
The technique alluded to above works as follows. Fix any smooth test
function v G C°°(lRn x (0, oo)) and suppose
f и — v has a strict local maximum at some point
(5)	<
( (xo,io) £ Rn x (0, oo).
This means
(u - v)(xo, t0) > (u - v)(x, t)
for all points (x,t) sufficiently close to (xo,to), with (x, t) (xo,to).
Now recall (3). We claim for each sufficiently small ej > 0, there exists
a point (xe., t£j) such that
(6)	— v has a local maximum at (x£j,t£j)
and
(7)	(xt7,ttj) -> (xo,to) as j -> oo.
To confirm this, note that for each sufficiently small r > 0, (5) implies
тахдв(и — v) < (u — v)(a;o,to)> В denoting the closed ball in Rn+1 with
center (rro,to) and radius r. In view of (3), u£j —> и uniformly on B, and so
max0B(u£J— v) <	— v)(xo,to) provided €j is small enough. Consequently
ue — v attains a local maximum at some point in the interior of B. We can
next replace r by a sequence of radii tending to zero to obtain (6), (7).
Now owing to (6) we see that the equations
(8)	Duej (x£., t£j) = Dv(x, ., t£}),
(9)	ut3(x£],t£.) = vt(x£j,t£j),
and the inequality
(10)	-Nuej(x£j,t£j) > -Nv(x£j,t£j)
542
10. HAMILTON JACOBI EQUATIONS
hold. We consequently can calculate
Vt(Xe} , tej ) + H(Dv(xtj Xt. )
= utj(xt tt ) + H(Du^(xt tt ),xe ) by (8),(9)
(11)
= €jAu^(x£.,f£j) by (2)
< 6jAv(x£j,f£J by (10).
Now let €j —> 0 and remember (7). Since v is smooth and H is continuous,
we deduce
(12)	vt(xQ,to) + H(Dv(xo,to),xo) <0.
We have established this inequality assuming (5). Suppose now instead
that
(13)	и — v has a local maximum at (xo,to),
but that this maximum is not necessarily strict. Then u — v has a strict local
maximum at (xo, to), for v(x, t) := v(x, i)+^(|x—rro|2 + (i — io)2) (<$ >0). We
thus conclude as above that vt(xo,io) + H(Dv(xo,to),xo) < 0; whereupon
(12) again follows.
Consequently (13) implies inequality (12). Similarly, we deduce the re-
verse inequality
(14)	vt(xo,to) + H(Dv(xo,to),xo) > 0,
provided
(15)	u — v has a local minimum at (a?o, io)-
The proof is exactly like that above, except that the inequalities in (10),
and thus in (11), are reversed.
In summary, we have discovered for any smooth function v that inequal-
ity (12) follows from (13), and (14) from (15). We have in effect put the
derivatives onto v, at the expense of certain inequalities holding.	□
Our intention now is to define a weak solution of (1) in terms of (12),
(13) and (14), (15).
DEFINITION. A bounded, uniformly continuous function и is called a
viscosity solution of the initial-value problem (1) for the Hamilton-Jacobi
equation provided:
(i)	и = g on Rn x {t = 0},
10.1. INTRODUCTION, VISCOSITY SOLUTIONS
543
and
(ii)	for each v G C°°(lRn x (0, oo)),
(16)
' if и — v has a local maximum at a point (a?o, io) € Rn x (0, oo),
< then
, vt(xo,to) + H(Dv(xo,to),xo) <0,
and
(17)
' if и — v has a local minimum at a point (xq, to) G Rn x (0, oo),
< then
- vt(xo,to) + H(Dv(xo,to),xo) >0.
Remark. Note carefully that by definition a viscosity solution satisfies (16),
(17), and so all subsequent deductions must be based on these inequalities.
The previous discussion was purely motivational.
For emphasis, we repeat the same point, which has caused some confu-
sion among students. To verify that a given function и is a viscosity solution
of the Hamilton-Jacobi equation ut + 77(Du,x) = 0, we must confirm that
(16), (17) hold for all smooth functions v. Now the argument above shows
that if и is constructed using the vanishing viscosity method, it is indeed a
viscosity solution. But we will also see later in §10.3 that viscosity solutions
can be built in entirely different ways, which have nothing whatsoever to do
with vanishing viscosity.
The point is that the inequalities (16), (17) provide an intrinsic charac-
terization, and indeed the very definition, of our generalized solutions.
□
We devote the rest of this chapter to demonstrating that viscosity so-
lutions provide an appropriate and useful notion of weak solutions for our
Hamilton-Jacobi PDE.
10.1.2. Consistency.
Let us begin by checking that the notion of viscosity solution is consistent
with that of a classical solution. First of all, note that if и G Сг(Кп x [0, oo))
solves (1) and if и is bounded and uniformly continuous, then и is a viscosity
solution. That is, we assert that any classical solution ofut + H(Du,x) = 0
is also a viscosity solution. The proof is easy. If v is smooth and и — v
obtains a local maximum at (xo,to), then
Du(xo,to) = Dv(xo,to)
Ut(xo,to) = Vt(xo,to).
544
10. HAMILTON-JACOBI EQUATIONS
Consequently
vt(xo,to) + H(Dv(xo,to),xo)
= ut(xQ, to) + H(Du(x0, io), ^o) = 0,
since и solves (1). A similar equality holds at any point (a?o> io) where u — v
has a local minimum.
Next we assert that any sufficiently smooth viscosity solution is a clas-
sical solution, and, even more, that if a viscosity solution is differentiable at
some point, then it solves the Hamilton-Jacobi PDE there. We will need
the following calculus fact:
LEMMA (Touching by a C1 function). Assume и : Rn —> R is continuous
and is also differentiable at some point xq. Then there exists a function
v € C1(Rn) such that
(18)	u(x0) = n(rr0)
and
(19)	u — v has a strict local maximum at xq.
Proof. 1. We may as well assume
(20)	a?o = 0, u(0) = Du(0) = 0;
for otherwise we could consider й(ж) := u(x + a?o) — u(xo) — Du(xo) • x in
place of u.
2.	In view of (20) and our hypothesis, we have
(21)	и(ж) = Hpi(z),
where
(22)	pi : Rn —> R is continuous, pi(0) = 0.
Set
(23)	p2(r) := max {|pi(x)|} (r > 0).
xeB(r)
Then
(24)	p2 : [0, oo) —> [0, oo) is continuous, p2(0) = 0,
10.1. INTRODUCTION, VISCOSITY SOLUTIONS
545
and
(25)	p2 is nondecreasing.
3.	Now write
/•2|x|
v(x) := / Pz(r) dr + |x|2 (x 6 Rn).
J\x\
Since |n(x)| < |ж|р2(2|х|) + |x|2, we observe
(26)	v(0) = Pv(0) = 0.
Furthermore if x 0, we have
2'7*	'T
1)ф) = --р2(2И)--р2(И) + 2т,
la'l	1^1
and so v e C^R").
4.	Finally note that if x ± 0,
/•2|x|
u(x) - v(x) = |x|pi(x) - /	P2(r)dr-\x\2
J\x\
/•2|x|
<	|x|p2(|x|) - /	p2(r)dr-|x|2
J|x|
<	-И2 by (25)
<	0 = u(0) — v(0).
Thus и — v has a strict local maximum at 0, as required.
THEOREM 1 (Consistency of viscosity solutions). Let и be a viscosity
solution o/(l), and suppose и is differentiable at some point (xo,to) € Rn x
(0,oo). Then
Ut(xo,to) + Я(Т>и(хо,^о),жо) = 0.
Proof. 1. Applying the lemma above to u, with R”+1 replacing Rn and
(a?o, io) replacing xq, we deduce there exists a C1 function v such that
(27)	и — v has a strict maximum at (xo,io)-
2. Now set ve := гр * v, rp denoting the usual mollifier in the n + 1
variables (x,i). Then
(28)
' ve —> v
< Dv€ —> Dv uniformly near (a?o,io)
. vf vt;
546
10. HAMILTON-JACOBI EQUATIONS
and so (27) implies
(29)	и — ve has a maximum at some point (xe,tQ,
with
(30)	(xe, fe) —> (a?o,to) as e —> 0.
Applying then the definition of viscosity solution, we see
vl(xe,te) + Я(Рие(жеЛ),я:е) <0.
Let e —> 0 and use (28), (30) to deduce
(31)	vt(xo,to) + Я(£>г|(хо,^о),жо) < 0.
But in view of (27), we see that since и is differentiable at (a?o, to)>
Du(x0,t0) = Dv(xo,to), ut(xo,to) = vt(xo,to).
Substitute above, to conclude from (31) that
(32)	ut(xo,tQ) + Я(£)и(жо,*о),яо) < 0.
3.	Now apply the lemma above to — и in Rn+1, to find a C1 function v
such that u — v has a strict minimum at (a?o,io)- Then, arguing as above,
we likewise deduce
wt(®o,<o) + H(Du(xo,to),a;o) > 0.
This inequality and (32) complete the proof.	□
10.2. UNIQUENESS
Our goal now is to establish the uniqueness of a viscosity solution of our
initial-value problem for Hamilton-Jacobi PDE. To be slightly more general,
let us fix a time T > 0 and consider the problem
, .	( ut + H(Du, x) = 0 in Rn x (0, T]
|	и = g on Rn x {t = 0}.
We say that a bounded, uniformly continuous function и is a viscosity
solution of (1) provided и = g on Rn x {t = 0}, and the inequalities in (16)
(or (17)) from §10.1.1 hold if и — v has a local maximum (or minimum) at
a point (xq,<o) CKnx (0, T).
LEMMA (Extrema at a terminal time). Assume и is a viscosity solution
of (1) and u — v has a local maximum (minimum) at a point (xo,to) 6
Rnx(0,T]. Then
(2)	«tfcojo) + H(Dv(xo,to),xo) < 0 (> 0).
The point is that we are now allowing for to = T.
10.2. UNIQUENESS
547
Proof. Assume и — v has a local maximum at the point (xo, T); as before
we may assume that this is a strict local maximum. Write
v(x, t) := v(x, t) + e — (x 6 Rn, 0 < t < T).
Then for e > 0 small enough, u — v has a local maximum at a point (x£, te),
where 0 < te < T and (x£,t£) —> (xo,T). Consequently
«t(x£, tQ + H(Dv(xe, tQ, x£) < 0,
and so
Vt(x£,t£) + 7- J 2 + H(Dv(Xe,tQ,Xe) < 0.
U te)
Letting e —> 0, we find
nt(x0,T) + H{Dv{xq,T},xq) < 0.
This proves (2) if и — v has a maximum at (xq, T). A similar proof gives the
reverse inequality should и — v have a minimum at (xo,T).	□
To go further, let us hereafter suppose the Hamiltonian H to satisfy
these conditions of Lipschitz continuity:
(3)
\H(p, x) - H(q, x)| < C|p - g|
\H(p,x) -H(p,y)\ < C|x — 2/|(l + |p|)
for x, y,p, q G Rn and some constant C > 0.
We come next to the central fact concerning viscosity solutions of the
initial-value problem (1), namely uniqueness. This important assertion jus-
tifies our taking the inequalities (16) and (17) from §10.1.1 as the foundation
of our theory.
THEOREM 1 (Uniqueness of viscosity solution). Under assumption (3)
there exists at most one viscosity solution o/(l).
Remark. The following proof is based upon an unusual idea of “doubling
the number of variables”. See the proof of Theorem 3 in §11.4.3 for a related
technique.	□
548
10. HAMILTON-JACOBI EQUATIONS
Proof*. 1. Assume и and й are both viscosity solutions with the same
initial conditions, but
(4)	sup (u — u) =: a > 0.
R" x [0,T]
Choose 0 < e, A < 1 and set
Ф(ж, у, t, s) -=u(x, t) — u(y, s) — A(t + s)
(5)	- ^(|x - y|2 + (i - s)2) - e(|rr|2 + |y|2),
for x, у 6 Rn, t, s > 0. Then there exists a point (xq, yo, to, so) G R2n x [0, T]2
such that
(6)	Ф(жо, Уо, to, so) = ^max Ф(ж, у, t, s).
2.	We may fix 0 < e, A < 1 so small that (4) implies
(7)	$(xo,yo,to,$o} > sup $(x,x,t,t) >
Rnx[0,T]	2
In addition, Ф(хо,Уо,^о>5о) > Ф(0,0,0,0); and therefore
(8)	A^° + S°) +	- y°|2 + “ S°)2) + е(1ж°|2 + ly°|2)
< u(x0, to) - u(yo, s0) - u(0,0) + fi(0,0).
Since и and й are bounded, we deduce
(9)	|®o - Уо|, |<o - s0| = O(e) as c —> 0.
Furthermore (8) implies e(|rro|2 + |j/o|2) — 0(1), and consequently
e(|®o| + |z/o|) = e1/4e3/4(|rr0| + |y0|)
<	e1/2 + Ce3/2(|x0|2 + |yo|2)
<	Ce1/2.
Thus
(10)	edxol + |yo|) = O(e1/2).
*Omit on first reading.
10.2. UNIQUENESS
549
3.	Since Ф(хо, yo,to, «о) > Ф(жо, xq, to, to), we also have
u(x0, to) - й(уо, «о) - A(t0 + s0) - (ко - yo|2 + (t0 - s0)2)
- e(|rr0|2 + |y012) > u(xo,to) - й(жоЛо) - 2Ai0 - 2e|rr0|2.
Hence
^(ko ~ Уо|2 + (to - so)2) < u(x0,t0) -tz(y0,s0) + A(t0 - s0)
+ е(жо + Уо) • (xo-yoY
In view of (9), (10) and the uniform continuity of u, we deduce
(11)	ко — Z/o|, |io - Sol = o(e).
4.	Now write w(-) to denote the modulus of continuity of u; that is,
|u(rr, t) - u(y, s)| < w(k - y\ + |t - s|)
for all x, у 6 Rn, 0 < t, s < T, and w(r) —> 0 as r —> 0. Similarly, o>(-) will
denote the modulus of continuity of u.
Then (7) implies
< u(xq, to) - u(yo, so) = u(x0, to) - u(x0,0) + u(x0,0) - u(xq, 0)
+ й(ж0,0) - й(хо, t0) + й(хо, io) - й(у0, s0)
< w(to) + w(to) + £(°(6))>
by (9), (11) and the initial condition. We can now take e > 0 to be so
small that the foregoing implies < u(to) + a) (to); and this in turn implies
io > // > 0 for some constant p > 0. Similarly we have sq> p> 0.
5.	Now observe in light of (6) that the mapping (x,t) f-> <b(x,yo,t,so)
has a maximum at the point (a?o,io)- In view of (5) then,
и — v has a maximum at (a?o, io)
for
v(x, t) := u(y0, s0) + A(i + s0) + (k ~ Уо|2 + (i - s0)2) + e(|ж|2 + |z/o12)•
Since и is a viscosity solution of (1) we conclude, using the lemma if neces-
sary, that
vt(xo, io) + H(Dxv(x0,t0), x0) < 0.
550
10. HAMILTON-JACOBI EQUATIONS
Therefore
(12)	A + 2^°e2 S°) + H ^(x0 - yo) + 2€x0,x0^ < 0-
We further observe that since the mapping (y,s) i—> — Ф(я?о,у, to,s) has
a minimum at the point (yo, so),
й — v has a minimum at (yo, so)
for
v(y,s) := u(xo,*o) - A(t0 + s) - ^(|rr - y0|2 + (to ~ s)2) - e(|rr0|2 + |y|2).
As й is a viscosity solution of (1), we know then that
vs(yo,so) + H(Dyv(yo,so'),yo') > 0.
Consequently
(13)	-A + 2^°^2 S°) + H [^(xo ~ Уо) ~ 2еуо, Уо^ > 0.
6.	Next, subtract (13) from (12):
(2	A /2	A
^2 (жо - Уо) - ^tyo, Уо\ - H f (x0 - yo) + 2ex0, z0 I •
In view of hypothesis (3) therefore,
(15)	A < Ced^ol + |yo|) + Cl^o — yoI	+ е(1жо| + |?/o 1)^ •
We employ estimates (10), (11) in (15), and then let e —> 0, to discover
0 < A < 0. This contradiction completes the proof.	□
10.3.	CONTROL THEORY, DYNAMIC
PROGRAMMING
It remains for us to establish the existence of a viscosity solution to our
initial-value problem for the Hamilton-Jacobi partial differential equation.
One method would be now to prove the existence of a smooth solution ue of
the regularized equation (2) in §10.1 and then to make good enough uniform
10.3. CONTROL THEORY, DYNAMIC PROGRAMMING
551
estimates. This technique in fact works, but requires knowledge of certain
bounds for the heat equation beyond the scope of this book.
In this section we provide an alternative approach of independent inter-
est, which is suitable for Hamiltonians which are convex in p.
We will first of all introduce some of the basic issues concerning control
theory for ordinary differential equations and the connection with Hamilton-
Jacobi PDE afforded by the method of dynamic programming. This discus-
sion will make clearer the connections of the theory developed above in
§§10.1-2 with that set forth earlier in §3.3.1. The remarkable fact is that
the defining viscosity solution inequalities (16), (17) in §10.1.1 are a conse-
quence of the optimality conditions of control theory.
10.3.1.	Introduction to control theory.
We will now study the possibility of optimally controlling the solution
x(-) of the ordinary differential equation
fx(s)=f(x(s),a(s)) (t < s < T)
t x(t) = X.
Here ' = T > 0 is a fixed terminal time, and x 6 Rn is a given initial
point, taken on by our solution x(-) at the starting time t > 0. At later
times t < s < T, x( ) evolves according to the ODE, where
f : Rn x A -+ Rn
is a given bounded, Lipschitz continuous function, and A is some given
compact subset of, say, Rm. The function «(•) appearing in (1) is a control,
that is, some appropriate scheme for adjusting parameters from the set A as
time evolves, thereby affecting the dynamics of the system modeled by (1).
Let us write
(2)	A = {a : [0, T] —> A | a(-) is measurable}
to denote the set of admissible controls. Then since
(3)	|f(x, a)| < C, |f(rr, a) — f(y, a)| < C\x — y| (x,y 6 Rn, a 6 A)
for some constant C, we see that for each control a(-) 6 A, the ODE (1)
has a unique, Lipschitz continuous solution x(-) = xQ( )(•), existing on the
time interval [i,T] and solving the ODE for a.e. time t < s < T. We call
x(-) the response of the system to the control «(•), and x(s) the state of the
system at time s.
552
10. HAMILTON-JACOBI EQUATIONS
x(a)=f(x(s),a(s))
Response of system to the control a(-)
Our goal is to find a control «*(•) which optimally steers the system. In
order to define what “optimal” means however, we must first introduce a
cost criterion. Given x G R” and 0 < t < T, let us define for each admissible
control a(-) G A the corresponding cost
(4)	C^,<[«(•)]:=	/r(x(s),a(s))ds + 5(x(T)),
where x(-) = x°4 \-) solves the ODE (1) and
h : Rn x A -+ R, g : Rn -> R
are given functions. We call h the running cost per unit time and g the
terminal cost, and will henceforth assume
J |/r(rr,a)|, \g(x)\ <C
I \h(x, a) - hfy, a)|, |р(я:) - #(у)| < C|x - y\ V
for some constant C.
Given now x G R” and 0 < t < T, we would like to find if possible
a control «*(•) which minimizes the cost functional (4) among all other
admissible controls.
10.3.2.	Dynamic programming.
The method of dynamic programming investigates the above problem by
turning attention to the value function
(6)	u(x,t) := inf	(x E Rn, 0 < t < T).
а(-)еЛ
10.3. CONTROL THEORY, DYNAMIC PROGRAMMING
553
The plan is this: having defined u(x, t) as the least cost given we start at
the position x at time t, we want to study и as a function of x and t. We are
therefore embedding our given control problem (1), (4) into the larger class
of all such problems, as x and t vary. The idea then is to show that и solves
a certain Hamilton-Jacobi type PDE, and finally to show conversely that a
solution of this PDE helps us to synthesize an optimal feedback control.
Hereafter, we fix x € Rn, 0 < t < T.
THEOREM 1 (Optimality conditions). For each h > 0 so small that
t + h <T, we have
C f-t+h
(7) u(x,t) = inf < I h(x(s),a(s)) ds + u(x(t + h),t + h)
«()еЛ [Jt
where x(-) = x“0)(.) solves the ODE (1) for the control of-).
Proof. 1. Choose any control ai(-) € A and solve the ODE
(8)	( Xi(s) = f(xi(s),oti(s)) (t<s<t + h) [ Xi(t) = X.
Fix e >	> 0 and choose then аг(-) G A so that
(9)	u(xi(i + h),t + h) + e > ( /г(хг(в), a2(s)) ds + ^(x2(T)), J t+h
where
(10)	( x-z(s) = f(x2(s),»2(s)) ft + h < s < T) t x2(t + h) = xi(t + h).
Now define the control
(И)	( <*i(s) if t < s <t + h a3^( a2(s) it t + h<s<T,
and let
(12)	Г x3(s) = f(x3(s),a3(s)) (t < s < T) ( X3(t) = X.
By uniqueness of solutions to the differential equation (1), we have
(13)	( xi(s) if t < s < t + h 3( t x2(s) if t + h < s < T.
554
10. HAMILTON-JACOBI EQUATIONS
Thus the definition (6) implies
u(x,t) < CI>f[a3(-)]
= У h(x3(s),a3(s))ds+ 5(x3(T))
/•t+Zi	rT
= / h(xi(s),ai(s))ds + /	h(x2(s), a2(s)) ds + ^(x2(T))
J t	J t-\~h
< J /i(xi(s),ai(s)) ds + u(xi(t + h),t + h) + e,
the last inequality resulting from (9). As ai(-) G A was arbitrary, we con-
clude
f	'I
(14) u(x,t) < inf < /	h(x(s),a(s)) ds + u(x(t + h),t + h) > + e,
«(•)e-4 Ut	J
x(-) = xQ()(-) solving (1).
2. Fixing again e > 0, select now a4(-) 6 A so that
(15)	u(x, t) + e > У h(x4(s), a4(s))ds + p(x4(T)),
where	t x4(s) = f(x4(s), o4(s)) (t < s < T) ( x4(t) = X.
Observe then from (6) that
(16) u(xi{t + h),t + h) < i h(x4(s),a4(s))ds + g(x4(T)).
Jt+h
Therefore
u(x,t) + e > inf < / h(x(s),a(s))ds + u(x(t + h),t + h)
a( )e-4 Ut
x(-) = xQ( )(•) solving (1). This inequality and (14) complete the proof of
(7).	□
10.3.3. Hamilton—Jacobi—Bellman equation.
Our eventual goal is writing down as a PDE an “infinitesimal version”
of the optimality conditions (7). But first we must check that the value
function и is bounded and Lipschitz continuous.
10.3. CONTROL THEORY, DYNAMIC PROGRAMMING
555
LEMMA (Estimates for value function). There exists a constant C such
that
|и(ж, i)| < C,
|u(rr,f) — u(x,t)| < C(|rr — ж| + \t — t|)
for all x,x G Rn, 0 < t, t < T.
Proof. 1. Clearly hypothesis (5) implies и is bounded on R” x [О, Т].
2. Fix x, x 6 Rn, 0 < t < T. Let e > 0 and then choose «(•) G A so that
(17)	u(x,t)+e>f h(x(s),a(s)) ds + 5(x(T)),
where x(-) solves the ODE
J x(s) = f(x(s),d(s)) (t < s < T)
[ x(f) = x.
Then
u(x, t) — u(x, t) < f h(x(s),a(s)) ds + g(x(Ty)
(19)	J‘ T
— J h(x(s),a(s)) ds — д(х(ТУ) + e,
where x(-) solves
Г x(s) = f(x(s),d(s)) (t<s<T)
( }	t x(i) = x.
Since f is Lipschitz continuous, (18), (20) and Gronwall’s inequality (§B.2)
imply |x(s) — x(s)| < C|x — (t < s <T). Hence we deduce from (5) and
(19) that u(x, t) — u(x, t) < C\x —	+ e. The same argument with the roles
of x and x reversed implies
|u(x, f) — u(x, t)\ < C[x — ж| (ж, x G Rn, 0 < t < T).
3. Now let x G Rn, 0 < t < t < T. Take e > 0 and choose a(-) G A so
that	т
u(x, i) + e > £ h(x(s), a(s)) ds + <7(x(T)),
x(-) solving the ODE (1). Define
d(s) := a(s + t — t) for t < s < T
556
10. HAMILTON JACOBI EQUATIONS
and let x(-) solve
( x(s) = f(x(s), d(s)) (i < s < T)
X x(t) = x.
Then x(s) = x(s +1 — t). Hence
u(x, t) — u(x, i) <	h(x(s), d(s))ds + д(х(ТУ)
(21)	h(x(s),a(s))ds — g(x(T)) + e
= - [	h(x(s), a(s))ds + g(x(T + t - t)) - g(x(T}) + e
JT+t-t
< C\t — t\ + e.
Next pick «(•) so that
u(x, t) + e >	h(x(s), d(s))ds + g(x(T)),
where
I x(s) = f(x(s), d(s)) (t < s < T)
( x(t) = X.
Define
f a(s + t — t) if t<s<T + t — t
(x(• — s
v '  I d(T) if T + t - t < s < T,
and let x(-) solve (1). Then a(s) = a(s + t — t), x(s) = x(s + t — t) for
t < s <T + t — t. Consequently
u(x, t) — u(x, t) < £ h(x(s), a(s))ds + д(х(ТУ)
-	К{х{8),а{8)^8-д(х{ТУ) + €
= [ fi(x(s),a(s))ds+ 5(x(T)) -g(x(T + t-i)) + e
JT+t-t
< C\t — t\ + €.
This inequality and (21) prove
ju(x, t) — u(x, t)| < C\t —	(0 < t < t < T, x E Жп).
□
We prove next that the value function solves a Hamilton-Jacobi type
partial differential equation.
10.3. CONTROL THEORY, DYNAMIC PROGRAMMING
557
THEOREM 2 (A PDE for the value function). The value function и is the
unique viscosity solution of this terminal-value problem for the Hamilton-
Jacobi-Bellman equation:
( щ + min {f(x, a)  Du + h(x, a)} = 0 in R” x (0, T)
(22)	< aeA
I	и = g on Rn x {t = T}.
Remarks, (i) The Hamilton-Jacobi Bellman PDE has the form
щ + H(Du, x) = 0 in Rn x (0, T),
for the Hamiltonian
(23) H(p, x) := min{f(x, a) -p + h(x,a)} (p, x 6 Rn).
From the inequalities (5), we deduce that H satisfies the estimates (3) in
§10.2.
(ii) Since (22) is a terminal-value problem, we must specify what we mean
by a solution. Let us say that a bounded, uniformly continuous function и
is a viscosity solution of (22) provided:
(i)	и = g on Rn x {£ = T},
and
(ii)	for each v 6 C°°(Rn x (0,T))
(24)
' if и — v has a local maximum at a point (a?o, to) £ Kn x (0, T),
< then
vt(x0, to) + H(Dv(x0, to), xo) > 0,
and
(25)
' if и — v has a local minimum at a point (a?o, to) € Rn x (0, T),
< then
. vt(xo,to) + H(Dv(xo,to),xo) < 0.
Observe that for our terminal-value problem (22) we reverse the sense of the
inequalities from those for the initial-value problem.
(iii)	The reader should check that if и is the viscosity solution of (22),
then w(x,t) := u(x,T — i) (ж € Rn,0 < t < T) is the viscosity solution of
the initial-value problem
wt — H(Dw, x) = 0 in Rn x (0, T)
w = g on Rn x {t = 0}.
□
558
10. HAMILTON-JACOBI EQUATIONS
Proof. 1. In view of the lemma, и is bounded and Lipschitz continuous. In
addition, we see directly from (4) and (6) that
u(s,T) = inf С'1,т[а(-)] = g(x) (x & Rn).
a(-)e4
2.	Now let v G C*00^” x (0,T)), and assume
и — v has a local maximum at a point (so, to) 6 Rn x (0, T).
We must prove
(26)	vt(xo, to) + min{f (so, a) • Dv{xq, to) + h(so, a)} > 0.
aeA
Suppose not. Then there exist a G A and 0 > 0 such that
(27)	vt(x, t) + f(s, a)  Dv(x, t) + h(x, a) < —0 < 0
for all points (s,t) sufficiently close to (so, to), say
(28)	|s - s0| + |t - to| < <5-
Since и — v has a local maximum at (so, to), we may as well also suppose
(29)
(u - u)(s, t) < (u - u)(s0, to)
for all (s,t) satisfying (28).
Consider now the constant control a(s) = a (to < s < T) and the corre-
sponding dynamics
Г x(s) = f(x(s), a) (t0 < s < T)
t x(to) = s.
Choose 0 < h < 6 so small that |x(s) — so| < 6 for to < s < to + h. Then
(31)	ut(x(s), s)+f(x(s), a)-Du(x(s), s)+h(x(s), a) < —0 (to < s < to+h),
according to (27), (28). But utilizing (29) we find
u(x(to + h), to + h) - u(so, to) < w(x(to + h), to + h) - u(so, to)
= f y-^(x(s),s)ds = [ ut(x(s),s) + Du(x(s),s)-x(s)ds
(32)	Jto ds	Jto
= / ut(x(s),s)+f(x(s),a) • Du(x(s),s)ds.
Jto
10.3. CONTROL THEORY, DYNAMIC PROGRAMMING
559
In addition, the optimality condition (7) provides us with the inequality
(33)	u(xo,to) < / h(x(s),a)ds+ u(x(to + h),to + h).
Jt0
Combining (32) and (33), we discover
rto+h
0< I ut(x(s),s)+ f(x(s),a) • Du(x(s),s)+ h(x(s),a)ds <-0h,
J to
according to (31). This contradiction establishes (26).
3.	Now suppose
и — v has a local minimum at a point (so,to) Gl"'x (0,T);
we must prove
(34)	vt(xQ, to) + min{f(sq, a) • Dv(xq, to) + h(xo, a)} < 0.
aEA
Suppose not. Then there exists в > 0 such that
(35)	vt(x, t) + f (x, a)  Dv(x, t) + h(x, a) >0 > 0
for all a G A and all (s, t) sufficiently close to (so, to), say
(36)	|s - s0| + |t - to| < <$•
Since и — v has a local minimum at (sq, to), we may as well also suppose
Г (u - u)(s, t) > (u - u)(s0, to)
t for all (s, t) satisfying (36).
Choose 0 < h < 6 so small that |x(s) — sq| < 6 for to < s < to + h, where
x(-) solves
Г x(s) = f(x(s), a(s)) (t0 < s < T)
W	1	<4. \
I x(to) = So
for some control a(-) G A- This is possible owing to hypothesis (3).
Then utilizing (37), we find for any control a(-) that
u(x(t0 + h), to + h) - u(so, t0)
> v(x(t0 + h), t0 + h) - u(s0, t0)
rto+h j
(39)	= /	—u(x(s),s)ds
Jto	ds
/•to+h
— /	Vt(x.(s), s') + f(x(s), a(s)) • _Du(x(s), s) ds,
Jto
560
10. HAMILTON-JACOBI EQUATIONS
by (38). On the other hand, according to the optimality condition (7) we
can select a control a(-) £ A so that
fto+h	Oh
(40) n(so,to) > / h(x(s),a(s))ds+ n(x(t0 + h),to + h)——.
Jto	2
Combining (39) and (40), we discover
Oh fto+h
-z- >	t>t(x(s),s)+ f(x(s),a(s)) • Du(x(s),s)
2 Jto
+ h(x(s), a(s)) ds > Oh,
according to (35). This contradiction proves (34).	□
Remark (Design of optimal controls). We have now shown that the value
function u, defined by (6), is the unique viscosity solution of the terminal-
value problem (22) for the Hamilton-Jacobi-Bellman equation. How does
this PDE help us solve the problem of synthesizing an optimal control? In
informal terms, the method is this. Given an initial time 0 < t < T and an
initial state x E Rn, we consider the optimal ODE
Г x*(s) = f(x*(s),a*(s)) (t<s<T)
t x*(t) = X,
where at each time s, a*(s) G A is selected so that
f(x*(s),a*(s)) • Du(x*(s),s) + h(x*(s),a*(s))
= H'(Du(x*(s), s),x*(s)).
In other words, given the system is at the point x*(s) at time s, we adjust
the optimal control value a*(s) so as to attain the minimum in the definition
(23) of the Hamiltonian H. We call a*(-) so defined a feedback control.
It is fairly easy to check that this prescription does in fact generate a
minimum cost trajectory, at least in regions where и and a*(-) are smooth
(so that (42) makes sense). There are however problems in interpreting (42)
at points where the gradient Du does not exist.	□
10.3.4. Hopf-Lax formula revisited.
Remember that earlier in §3.3 we investigated this initial-value problem
for the Hamilton-Jacobi equation:
z .	( ut + H(Du) = 0 inRnx(0,T]
1	[	и = g on Rn x {t = 0},
10.3. CONTROL THEORY, DYNAMIC PROGRAMMING
561
under the assumptions that
p i—> H(p) is convex,
r Я(р)
hm ———
IpHoo |p|
and
g : Rn —> R is Lipschitz continuous.
Notice that we are now taking 0 < t < T, to be consistent with §10.2. We
introduced as well the Hopf-Lax formula for a solution:
(44) u(x, t) = min ItL (X + <?(?/) j (® 6 Rn,t > 0),
where L is the Legendre transform of H:
(45)
L(q) = swp{p-q-H(p)} (qGR").
In order to tie together the theory set forth here and in §3.3, let us
now check that the Hopf-Lax formula gives the correct viscosity solution,
as defined in §10.1.1. (The proof is really just a special case of that for
Theorem 2.)
THEOREM 3 (Hopf-Lax formula as viscosity solution). Assume in addi-
tion that g is bounded. Then the unique viscosity solution of the initial-value
problem (43) is given by the formula (44).
Proof. 1. As shown in §3.3 the function и defined by (44) is Lipschitz
continuous and takes on the initial function g at time t = 0. It is easy to
verify as well that и is also bounded on Rn x (0,T], since g is bounded.
2.	Now let v G C^R” x (0, oo)) and assume и — v has a local maximum
at (a?o,to) G Rn x (0, oo). According to Lemma 1 in §3.3.2,
(46)	u(xq, to) = min < (to — t)L (	—- ] + u(x, t) >
zeRn [	\ to — t J	J
for each 0 < t < to- Thus for each 0 < t < to, x G Rn
(47)	u(x0,t0) < (to - t)L	—^+u(s,t).
\ to — t /
But since и — v has a local maximum at (xo,to),
u(xq, to) — v(xq, to) > u(x, t) — v(x, t)
562
10. HAMILTON-JACOBI EQUATIONS
for (x, t) close to (a?o,to). Combining this estimate with (47), we find
(48)	v(so,to) - v(x,t) < (t0 - t)L	—r)
\ to — t /
for t < to, (x, t) close to (а?о, to)- Now write h = to — t and set x = xq — hq,
where q G Rn is given. Inequality (48) becomes
v(xq, to) — v(xo — hq, to — h) < hL(q).
Divide by h > 0 and send h —> 0:
vt(x0,to) + Dv{xo,to)-q- L(q) < 0.
This is true for all q G Rn and so
(49)	vt(xo,to) + H(Dv(x0,t0)) < 0,
since
(50)	H(p) = sup {p • q - L(q)},
qgR"
by the convex duality of H and L. We have, as desired, established the
inequality (49) whenever и — v has a local maximum at (xo,to)-
3.	Now suppose instead и — v has a local minimum at a point (xq, to) G
Rn x (0,T). We must prove
(51)	vt(x0, to) + H(Dv(x0, to)) > 0.
Suppose to the contrary estimate (51) fails; in which case
vt(x, t) + H{Dv{x,t)') < —в < 0
for some 0 > 0 and all points (a?,t) close enough to (so,to). In view of (50)
(52)	Vt(x,t) + Dv(x,t) • q — L(q) <—0
for all (<r,t) near (a?o,to) and all q G Rn.
Now from (46) we see that if h > 0 is small enough
/	<	, v	/ Xd — X] \
(53)	tz(xo^o) = hL I ----------- I +u(xi,to — h)
\ h J
10.4. PROBLEMS
563
for some point Xi close to xq- We then compute
f1 d
u(xo,to) — v(xi,to — h) = / — v(sxo + (1 — s)a?i, to + (s — l)h)ds
Jo “s
= / Dv(sxq + (1 - s)si, to + (s - l)h) • (a?o - ah)
Jo
+ vt(sxo + (1 — s)si, to + (s — l)h)hds
= h (J Dv(- • •) • q + vt(- • •) ds^ ,
where q = XQ^X1  Now if h > 0 is sufficiently small, we may apply (52), to
find
v(xo, to) — v(xi, to — h) < hL ( Ж° X1 ) — Oh.
\ h )
But then (53) forces
u(a?o, to) — v(xi, to — h) < u(xo,to) — u(xi,to — h) — Oh,
a contradiction, since и — v has a local minimum at (a?o,to)- Consequently
the desired inequality (51) is indeed valid.	□
10.4.	PROBLEMS
1.	Let	be viscosity solutions of the Hamilton-Jacobi equations
+ H(Duk, x) = 0 in Rn x (0, oo)
(fc = 1,...), and suppose uk —» и uniformly. Assume as well that H
is continuous. Show и is a viscosity solution of
Ut + H(Du, x) = 0	in Rn x (0, oo).
Hence the uniform limits of viscosity solutions is a viscosity solution.
2.	Let и1 (г = 1,2) be viscosity solutions of
( и[ + H(Dul, x) = 0 in Rn x (0, oo)
[	и1 = дг on Rn x {t = 0}.
Assume H satisfies condition (3) in §10.2. Prove the contraction prop-
erty
sup|t?(-,t) - u2(-, t)| < sup Iff1 -ff2| (t > 0).
R"	R"
564
10. HAMILTON-JACOBI EQ UATIONS
3.	Suppose for each e > 0 that u!- is a smooth solution of the parabolic
equation
n
uf + H(Due, x) - € 22 aiJu€XtXi = 0
«,j=i
in Rn x (0, oo), where the smooth coefficients a’J (i,j = 1,
satisfy the uniform ellipticity condition from Chapter 6. Suppose also
that H is continuous, and that ue —> и uniformly as e —> 0.
Prove that и is a viscosity solution of щ + H(Du,x) = 0. (This
exercise shows that viscosity solutions do not depend upon the precise
structure of the parabolic smoothing.)
4.	(i) Show that u(x) := 1 — |x| is a viscosity solution of
Г M = 1 in (-1,1)
1	>	(n(-l) = u(l) = 0.
This means that for each v E C°°(—1,1), if и — v has a maximum
(minimum) at a point xq E (—1,1), then |v'(xq)| < 1 (> 1).
(ii)	Show that й(х) := |ж| — 1 is not a viscosity solution of (*).
(iii)	Show that й is a viscosity solution of
( —|й'| = —1 in (-1,1)
k J	\й(-1) =й(1) = 0.
(Hint: What is the meaning of a viscosity solution of (**)?)
(iv)	Why do problems (*),(**) have different viscosity solutions?
5.	Let U C Rn be open, bounded. Set u(x) := distfy, dU) (x E U).
Prove и is Lipschitz continuous and is a viscosity solution of the eikonal
equation
|Du| — 1 in U.
This means that for each v E C°°(U), if и — v has a maximum (mini-
mum) at a point xq E U, then |Du(xo)| < 1 (> 1).
10.5.	REFERENCES
Section 10	.1 The definition of viscosity solutions presented here is due to
[C-E-L], who recast an earlier definition set forth in the basic
paper [C-L] of Crandall-Lions.
Section 10	.2 The uniqueness proof follows [C-E-L], with some recent im-
provements I learned from M. Crandall.
10.5. REFERENCES
565
Section 10	.3	P.-L. Lions [LI] observed the connection between the def-
inition of viscosity solution and the optimality conditions
of control theory. The books of Fleming-Soner [F-S] and
Bardi-Capuzzo Dolcetta [В-CD] provide much more infor-
mation about viscosity solutions and the connections with
deterministic and stochastic optimal control.
Chapter 11
SYSTEMS OF
CONSERVATION
LAWS
11.1	Introduction
11.2	Riemann’s problem
11.3	Systems of two conservation laws
11.4	Entropy criteria
11.5	Problems
11.6	References
11.1. INTRODUCTION
In this final chapter we study systems of nonlinear, divergence structure
first-order hyperbolic PDE, which arise as models of conservation laws.
Physical interpretation. In the most general circumstance we would like
to investigate a vector function
u = u(x,t) = (u1^, t),... ,um(x, t)) (x G Rn, t > 0),
the components of which are the densities of various conserved quantities in
some physical system under investigation. Given then any smooth, bounded
region U cRn, we note that the integral
(1)
f u(x,t)dx
и
567
568
11. SYSTEMS OF CONSERVATION LAWS
represents the total amount of these quantities within U at time t. Now
conservation laws typically assert that the rate of change within U is gov-
erned by a flux function F : Rm —> Mmxn, which controls the rate of loss or
increase of u through dU. Otherwise stated, it is appropriate to assume for
each time t
(2)
iz denoting as usual the outward unit normal along U. Rewriting (2), we
deduce
(3)
As the region U C R" was arbitrary, we derive from (3) this initial-value
problem for a general system of conservation laws:
(4)
Ut + divF(u) = 0
u = g
in R" x (0, oo)
on Rn x {t = 0},
the given function g = (g1,..., </m) describing the initial distribution of
u = (u\...,um).	□
At present a good mathematical understanding of problem (4) is largely
unavailable (but see Zheng [ZH]). For this reason we shall henceforth con-
sider instead the initial-value problem for a system of conservation laws in
one space dimension:
(5)
ut + F(u)z = 0
u = g
in R x (0, oo)
on R x {t = 0},
where F : Rm —> Rm and g : R —> Rm are given and u : R x [0, oo) —> Rm is
the unknown, u = u(x,t). We call Rm the state space, and write
F = F(2) = (F1^),...,^^)) (^6Rm)
for the smooth flux function.
We intend to study the solvability of problem (5), properties of its solu-
tions, etc.
Example 1.	The p-system is this collection of two conservation laws:
(6)
Ut ~u2x = 0
u2 -p(u^x = 0
(compatibility condition)
(Newton’s law),
11.1. INTRODUCTION
569
in R x (0, oo), where p : R —> R is given. Here
(7)	F(z) = (-z2,-p(zi))
for z = (zi,Z2). The p-system arises as a rewritten form of the scalar
nonlinear wave equation
(8)	utt ~ (p(ux))x = 0 in R x (0, oo).
Taking и1 := ux, и2 := щ, we obtain the system (6), with the stated inter-
pretations.	□
Example 2.	Euler’s equations for compressible gas flow in one dimension
are
{Pt + (pv)x = 0 (conservation of mass)
(pv)t + (pv2 +P}x — 0 (conservation of momentum)
(pE')t + (pEv +pv)x = 0 (conservation of energy)
in R x (0, oo). Here p is the mass density, v the velocity, and E the energy
density per unit mass. We assume
where e is the internal energy per unit mass and the term corresponds to
the kinetic energy per unit mass. The letter p in (9) denotes the pressure.
We assume p is a known function
(10)
p = p(p, e)
of p and e; formula (10) is a constitutive relation. Writing u = (u1, u2, u3) =
(p, pv,pE}, we check (9) is a system of conservation laws of the requisite
form
ut + F(u)x = 0 in R x (0, oo)
for F = (F^F^F3),
(11)
' F1!::) = 22
 ^W = ¥+P(21,g-^g)2)
where z = (<21,22,23), z\ > 0.
570
11. SYSTEMS OF CONSERVATION LAWS
Example 3.	The one-dimensional shallow water equations are
<f>t + (иф)х = 0 (conservation of mass)
vt + (u2/2 + ф}х = 0 (conservation of momentum)
in R x [0, oo). In these equations v is the horizontal velocity, ф denotes gh,
where h is the height and g the gravitational constant, and
F(z) = (ziz2,^- +z-t
\ £
□
(12)
11.1.1. Integral solutions.
The great difficulty in this subject is discovering a proper notion of weak
solution for the initial-value problem (5). We have already encountered
similar issues in §3.4 in our study of the much simpler case of a single or
scalar conservation law (i.e., m = 1 above).
Following then the development in §3.4.1, let us suppose
v : R x [0, oo) —> Rm is smooth,
with compact support, v = (u1,..., um).
We temporarily assume u is a smooth solution of our problem (5), take
the dot product of the PDE u< +	= 0 with the test function v, and
integrate by parts, to obtain the equality
(13)	f f u • vt + F(u) • vz dxdt + [ g • v dx\t=o = 0.
JO J —oo	J—oo
This identity, which we derived supposing u to be a smooth solution, makes
sense if u is merely bounded.
DEFINITION. We say that u G L°°(Rn x (0, oo);Rm) is an integral so-
lution of the initial-value problem (5) provided the equality (13) holds for all
test functions v satisfying (12).
Continuing now to parallel the development in §3.4.1 for a single conser-
vation law, let us now consider the situation that we have an integral solution
u of (5) which is smooth on either side of a curve C, along which u has sim-
ple jump discontinuities. More precisely, let us assume that V C R x (0, oo)
is some region cut by a smooth curve C into a left hand part Vi and a right
hand part Vr-
11.1. INTRODUCTION
571
Rankine-Hugoniot condition
Assuming that u is smooth in Vi, we select the test function v with
compact support in Vi and deduce from (13) that
(14)	ut + F(u)a; = 0 in Vi-
Similarly, we have
(15)	ut + F(u)a; = 0 in Vr,
provided u is smooth in Vr. Now choose a test function v with compact
support in V, but which does not necessarily vanish along the curve C.
Then utilizing the identity (13) we find
ПОО
u  vt + F(u) • vx dxdt
°°
= // u • Vt + F(u) • vx dxdt + // u • Vt + F(u) • vx dxdt.
J Jv,	J Jvr
As v has compact support within V, we deduce
/ / U • Vt + F(u) • vx dxdt = - [ut + F(u)x] • v dxdt
J JVi	J JVi
(17)	+ [ (u;i/2 + F(uz)i/1) • vdl
Jc
= [ (u/i/2 + F(u;)i/1) • vdl,
Jc
572
11. SYSTEMS OF CONSERVATION LAWS
owing to (14). Here v — (i/1,!/2) is the unit normal to the curve C point-
ing from Vi into Vr, and the subscript “1” denotes the limit from the left.
Likewise (16) implies
II u • vj + F(u) • vx dxdt = — I (urz/2 + F(ur)p1)  v dl,
J Jvr	Jc
“r” denoting the limit from the right. Adding this identity to (17) and
remembering (16), we deduce
[ [(F(u/) - F(ur))z/1 + (uz - ur)z/2] • vdl = 0.
Jc
This identity obtains for all smooth functions v as above; whence
(18)	(F(uz) - F(ur))z/X + (uz - ur)z/2 = 0 along C.
Suppose now the curve C is represented parametrically as {(x, t) | x =
s(t)} for some smooth function s(-) : [0, oo) —> R. Then v = (z/1,!/2) =
(1 + s2)-1/2(l, —s). Consequently (18) reads
(19)	F(uz) - F(ur) = s(u( - ur)
in V along the curve C.
Notation.
{[[u]] = uz — ur = jump in u across the curve C
[[F(u)]] = F(uz) - F(ur) = jump in F(u)
a = s = speed of the curve C.
□
Let us then rewrite (19) as the identity
(20)	[[F(u)]] = a[[u]]
along the discontinuity curve, the Rankine-Hugoniot jump condition. Note
carefully this is a vector equality.
11.1.2.	Traveling waves, hyperbolic systems.
We have seen in §3.4 that the notion of an integral solution for conser-
vation laws is not adequate: such solutions need not be unique. We are
therefore intent upon discovering some additional requirements for a good
definition of a generalized solution. This will presumably entail as in §3.4
11.1. INTRODUCTION
573
an entropy criterion based upon an analysis of shock waves. This expecta-
tion, now as carried over to systems, is largely correct, but first of all we
must study more carefully the nonlinearity F in the hopes of discovering
mathematically appropriate and physically correct structural conditions to
impose.
Let us start by first considering the wider class of semilinear systems
having the nondivergence form:
(21)	ut + B(u)uI = 0 in R x (0, oo),
where В : Rm —» Mmxm. This system is for smooth functions equivalent to
the conservation law in (5), provided
/ F1 F1 \
/ Z1 •	Zm \
В = BF = |
\ /7 771	/
'^21	••• rzm ' mxm
We consider now the possibility of finding particular solutions which
have the form of a traveling wave:
(22)	u(s, t) = v(s — at) (r6Rn,t>0),
where the profile v : R —> Rm and the velocity ст G R are to be found.
We substitute expression (22) into the PDE (21), and thereby obtain the
equality
(23)	— av'(x — at) + B(v(s — at))v'(x — at) = 0.
Observe (23) says that a is an eigenvalue of the matrix B(v) corresponding
to the eigenvector v'.
This conclusion suggests (exactly as for the linear theory in §7.3) that
if we wish to find traveling waves or, more generally, wavelike solutions of
our system of PDE, we should make some sort of hyperbolicity hypothesis
concerning the eigenvalues of B.
DEFINITION. If for each z E Rm the eigenvalues ofB(z) are real and
distinct, we call the system (21) strictly hyperbolic.
We henceforth assume the system of partial differential equations (21)
(and the special case В = DF of conservation laws ) to be strictly hyperbolic.
574
11. SYSTEMS OF CONSERVATION LAWS
Notation, (i) We will write
(24)	Ai(z) < Л2(г) < • • • < Am(z) (zeRm)
to denote the real and distinct eigenvalues of В (г), in increasing order.
(ii)	Then for each к = 1,..., m, we let
Tfc(^)
denote a corresponding nonzero eigenvector, so that
(25)	B(z)rfe(z) =	(fc = 1,..., m, z G Rm).
Since we are always assuming the strict hyperbolicity condition, the vectors
{rZc(z)}™=1 span Rm for each z G Rm.
(iii)	Next, since a matrix and its transpose have the same spectrum, we
can introduce for each к = 1,... , m a nonzero eigenvector
kfc)
for the matrix B(z)T, corresponding to the eigenvalue Afc(z). Thus
(26)	B(z)Tlfc(z) = Afc(z)lfc(z) (fc = 1,..., m, z G Rm).
This equality is usually written
(27)	lfc(z)B(z) = A(z)lfc(z) (k = l,...,m, z E Rm).
Thus {Ifc(z) jfcLj can be regarded as left eigenvectors of B(z), and {rZc(z)}^l1
are right eigenvectors.	□
Remark. Additionally, we observe
(28)	lz(z)-rfc(z) =0 iffc// (zGRm).
To confirm this, we compute using (25) and (26) that
Afc(z)(lz(z) • rfc(z)) = lz(z)  (B(z)rfc(z)) = (B(z)Tlz(z)) • rfc(z)
= Az(z)(lz(z) • rfc(z));
whence (28) follows since Afc(z) / Az(z) if к / I.	□
Let us first show that the notion of strict hyperbolicity is independent
of coordinates.
11.1. INTRODUCTION
575
THEOREM 1 (Invariance of hyperbolicity under change of coordinates).
Let и be a smooth solution of the strictly hyperbolic system (21). Assume
also Ф : Rm —> Rm is a smooth diffeomorphism, with inverse Ф. Then
(29)	й := Ф(и)
solves the strictly hyperbolic system
(30)	Ut + B(u)£Lr = 0 in R x (0, oo),
for
(31)	B(z) := Г»Ф(Ф(г))В(Ф(г))Г>Ф(г) (z e Rm).
Proof. 1. We compute fit = ИФ(и)и/, = РФ(и)и1, and so equation
(30) is valid for B(z) = ПФ(г)В(г)1)ф_1(г), where z = Ф(г). Substituting
z = Ф(5), we obtain (31).
2.	We must prove the system (30) is strictly hyperbolic. If АЦг) is an
eigenvalue of В (г), with corresponding right eigenvector rfc(zr), we have
B(z)rfc(z) = Afe(z)rfe(z).
Setting
(32)	rfe(z) := ГФ(Ф(г))гк(Ф(г)),
(33)	Xk(z) := Ал(Ф(г)),
we compute
(34)	B(z)rfc(z) = Afc(z)rfe(z).
Similarly if lfc(z) is a left eigenvector, we write
(35)	Ifc(z) := к(Ф(г))РФ(г),
and calculate
(36)	ife(z)B(2) = А*,(г)й(г).
In view of (32)-(36), we conclude that the system (30) is strictly hyperbolic.
□
Next we study how Afc(z), rk(z) and lfc(z) change as z varies:
576
11. SYSTEMS OF CONSERVATION LAWS
THEOREM 2 (Dependence of eigenvalues and eigenvectors on parame-
ters). Assume the matrix function В is smooth, strictly hyperbolic. Then
(i) the eigenvalues Xk(z) depend smoothly on z G Rm (fc = 1,... ,m).
(ii) Furthermore, we can select the right eigenvectors г&(г) and left eigen-
vectors lfc(z) to depend smoothly on z e Rm and satisfy the normalization
|rfc(z)|, |U(z)| = 1 (fc = 1,... ,m).
Proof. 1. Since B(z) is strictly hyperbolic, for each zq G Rm we have
(37)
Ai(г0) < Л2(г0) <	< Am (•Zq)-
Fix к G {1,... ,m} and any point zq G Rm, and let гд,(го) satisfy
B(^o)rfc(^o)= А/г(го)г/с(го)
|rfcUo)|= 1.
Upon rotating coordinates if necessary, we may assume
(38)
rfc(^o) —	— (0,..., 1).
We first show that near zq, there exist smooth functions А/с(г),г^(г) such
that
( B(z)rk(z)= Xk(z)rk(z)
I lrfc(u)|= 1-
2.	We will apply the Implicit Function Theorem to the smooth function
Ф : Rm x R x Rm —> Rm+1 defined by
Ф(г, А, г) = (В(г)г - Ar, |r|2) (r, z G Rm, A G R).
Now
/	-ri	\
ЭФ(г, X, г) = В(г)	-	XI	:
д(г, А)	-Гт
\2ri...2rm	0 /	(TO+I)x(m+1)
and so, according to (38), it suffices to check that
(39)
det В(г0) - Afc(^o)/
\	0......2
-1
0 /
/0.
11.1. INTRODUCTION
577
3.	Note that for e > 0 sufficiently small, the matrix
(40)
B£ = B(z0) - (Xk(zo) +
is invertible. In light of (38),

Therefore
/
Be
\0...2
\o...o
/
Be
\0...2
0
0
2(-e)-1
Consequently, since the determinant of the second matrix before the equals
sign is one, we have
det
\0...2
° \
= 2(det Be)(-e)-1
0 /
— 2 U(AjUo) — (Afc(zo) + e))(—e)(-e) 1
-> 2 H(Aj(z0) - Afc(zo))	as e —> 0.
As B(zq) is strictly hyperbolic, the last expression is nonzero. Condition
(39) is verified. We may thus invoke the Implicit Function Theorem to find
near zq smooth functions Afc(-z) and rfc(z), satisfying the conclusion of the
theorem.
4.	It remains to show that we can define A/cfy) and r/c(z) for all z G Rm,
and not just near any particular point zq. To do so, let us write
R := sup{r > 0 | Afc(z), rfc(z) as above exist and are smooth on B(0, r)}.
If R — oo, we are done. Otherwise, we cover dB(O. R) with finitely many
open balls into which we can smoothly extend Afc(-) and !>(•), using steps
1-3 above. This yields a contradiction to the definition of R.
A similar proof works for the left eigenvectors.	□
578
11. SYSTEMS OF CONSERVATION LAWS
Remark. Observe that we are not only globally and smoothly defining
the eigenvalues and eigenspaces of B, but are also globally providing the
eigenspaces with an orientation.
This proof depends fundamentally upon the one-dimensionality of the
eigenspaces. See Problem 2 for an example of what could go wrong in the
hyperbolic, but not strictly hyperbolic, setting.	□
Example 1 (continued). For the p-system (6), we have
ut + B(u)uz = 0,
for
B(z) = DF(z') =
-1\
0 / '
The eigenvalues are Ai = —а, A2 = <7, for a := p'(zi)1/2. These are real and
distinct provided we hereafter suppose the strict hyperbolicity condition
(41)
p’ > 0.
For the nonlinear wave equation (8) this is the physical assumption that the
stress p(ux) is a strictly increasing function of the strain ux.	□
Example 2 (continued). Euler’s equations (9) comprise a strictly hyper-
bolic system provided we assume p > 0 and
(42)
dp
dp
dp
de
where p = p(p, e) is the constitutive relation between the mass density, the
internal energy density and the pressure. This assertion is however difficult
to verify directly, as the flux function F defined by (11) is complicated.
Let us rather change variables and regard the density p, velocity v and
internal energy e as the unknowns. We can then rewrite Euler’s equations
(9) in terms of these quantities, and, so doing, obtain after some calculations
the system
(43)
' Pt + vpx + pvx = 0
< Vt + vvx + lpx = 0
. et + vex +	= 0,
provided p > 0. These equations are not in conservation form. Setting now
u = (u1,^2,^3) = (p, u,e), we rewrite (43) as
(44)
u< + B(u)u! = 0 in R x (0,00),
0
11.2. RIEMANN’S PROBLEM
579
for
(45)	В(г) = z2I + B(z),
where
/	0	z\	0
B(z) :=	I	7Г§;(г1’гз)	о	^-^(21,г3)
\	0	^р(г1,г3)	0
The characteristic polynomial of В is — A(A2 — cr2), for a2 =
Recalling (45) and reverting to physical notation, we see that the eigenvalues
of В are
(46)	Ai = v — сг, A2 = v, A3 = v + a,
where
p_dp d?\1/2 n
p2 de dp J
is the local sound speed. We therefore see that the system (44) is strictly hy-
perbolic, provided assumption (42) is valid. Remembering now Theorem 1,
we deduce that Euler’s equations (9) are also strictly hyperbolic, with eigen-
values given by (46).	□
11.2. RIEMANN’S PROBLEM
In this section we investigate in detail the system of conservation laws
(1)
и* + F(u)a; = 0 in R x (0,00),
with the piecewise-constant initial data
(2)
if x
if x
< 0
> 0.
This is Riemann’s problem. We call the given vectors щ and ur the left and
right initial states.
11.2.1. Simple waves.
We commence our study of (1), (2) in very much the same spirit as in
§11.1.2, in that we look for solutions of (1) having a special form. Before we
searched for traveling waves, that is, solutions of the type u(x, t) = v(x—at).
We now seek simple waves. These are solutions of (1) having the structure
(3)
u(x, t) = v(w(x, t)) (x G R, t > 0),
580
11. SYSTEMS OF CONSERVATION LAWS
where v : R —> Rm, v = (v1,..., vm), and w : R x [0, oo) —> R are to be
found. To discover the requisite properties of v and w, let us substitute (3)
into (1) and obtain the equality
(4)	v(w) + DF(y(wy)v(w)wx = 0.
Now in view of equation (25) from §11.1.2, with В = DF, we see (4)
will be valid if for some к G {1,..., m} w solves the PDE
(5)	wt + Xk(v(wy)wx = 0
and v solves the ODE
(6)	v(s) = rfc(v(s)) 6 =
\ ds /
If (5) and (6) hold, we call the function u defined by (3) a к-simple wave.
The point of all this is that we can regard (6) as an ODE for the vector
function v, and then—once v has been found by solving (6)—can interpret
equation (5) as a scalar conservation law for w.
Let us next identify circumstances under which we can employ the con-
struction (3)-(6) to build a continuous solution u of (1). We must examine
first the ODE (6).
DEFINITION. Given a fixed state zq G Rm, we define the kth-rarefaction
curve
Rktzo')
Rarefaction curve
Given then the solution v of (6), we turn to the PDE (5), which we
rewrite as the scalar conservation law
(7)	wt + Ffc(w)a; = 0
11.2. RIEMANN’S PROBLEM
581
for
(8)	Ffc(s) := [ Xk(y(tf)dt (sGR).
Jo
The PDE (7) will fall under the general theory developed in §3.4 provided
Fk is strictly convex (or else strictly concave). Let us therefore compute
(9)	F'fc(s) = Afc(v(s)),
(10)	Fk (5) = £>Afc(v(s))  v(s) = DAfc(v(s)) • rfc(v(s)).
Owing to (10), the function Fk will be convex if
DAfc(z) • rfc(z) > 0 (2GRm),
and concave if
DAfc(z)-rfc(z)<0 (zGRm).
The function Fk is linear provided
DXk(F)  rfe(z) = 0 (zeF).
These possibilities motivate the following
DEFINITIONS, (i) The pair (Afc(z),rfc(z)) is called genuinely nonlinear
provided
(11)	£>Afc(z) • Tfc(z) / 0 for all z G Rm.
(ii)	We say (Xk(z),rk(zf) is linearly degenerate if
(12)	DAfc(z) • Tfc(z) = 0 for all z G Rm.
Notation. If the pair (Хк,гк) is genuinely nonlinear, write
Rk (Zo) = {z G Rk(z0) I Afc(z) > Afc(zo)}
and
Rk (z0) := {z G Rk(z0) | Afc(z) < Afc(z0)}.
Then
Rfc(zo) = R%(z0) U {zq} U Rf (z0).
Two parts of the rarefaction curve
582
11. SYSTEMS OF CONSERVATION LAWS
11.2.2. Rarefaction waves.
We turn our attention again to Riemann’s problem (1), (2).
THEOREM 1 (Existence of fc-rarefaction waves). Suppose that for some
к G {1,..., m},
(i) the pair (А&,г&) is genuinely nonlinear, and
(ii) ur G Rfc (иг).
Then there exists a continuous integral solution u of Riemann’s problem (1),
(2), which is a к-simple wave constant along lines through the origin.
Remark. We call u a (centered) k-rarefaction wave.
fc-rarefaction wave
Proof. 1. We first choose wi and wr G R so that щ = v(w;), ur = v(wr).
Suppose for the moment
(13)
Wl < Wr
2.	Consider then the scalar Riemann problem consisting of the PDE (7)
together with the initial condition
wi if x < 0
wr ii x > 0.
Now in view of hypothesis (ii) we have Afe(ur) > A^(u/); that is, according
to (9), F'k(wr) > Fk(wi). But then it follows from (i) that the function Fk
defined by (8) is strictly convex. Accordingly we can apply Theorem 4 in
§3.4.4 to the scalar Riemann problem (7), (14), whose unique weak solution
is a continuous rarefaction wave connecting the states wi and wr. More
specifically,
' Wl
w(x,t) = Gk(f~)
wr
if f <
if F'kM < f < F'kM (z G R, t > 0)
if > F'k(wr),
11.2. RIEMANN’S PROBLEM
583
where Gk =	1. Thus u(x, t) = v(w(x, t)), where v solves the ODE (6)
and passes through щ, is a continuous integral solution of (1), (2).
3.	The case wi > wr is treated similarly, since Fk is then concave. □
11.2.3. Shock waves, contact discontinuities.
We consider next the possibility that the states ui and ur may be joined
not by a rarefaction wave as above, but rather by a shock.
fc-shock wave
a. The shock set.
Recalling the Rankine-Hugoniot condition from §11.1.1, we see that nec-
essarily we must have the equality F(iq) — F(ur) = <t(ui — ur), where a G R,
for such a shock wave to exist. This observation motivates the following
DEFINITION. Given a fixed state zq G Rm, we define the shock set
S(zq) := {z G Rm | F(z) - F(zq) = cr(z — zq) for a constant a = a(z, zq)}.
THEOREM 2 (Structure of the shock set). Fix zq G Rm. In some
neighborhood of zq, S(zq) consists of the union of m smooth curves Sk(zo)
(к — 1,..., m), with the following properties:
(i)	The curve Sfc(zo) passes through zq, with tangent г^(го).
(ii)	Л™ <t(z,z0) = Afe(z0).
Z 'ZQ
zesk(z0)
(iii)	cr(z, z0) = AtH+Mzo) +	_ z0|2), as z —> zq with z G Sfc(z0).
Proof. 1. Define
B(z) := [ DF(z0 + t(z-z0))dt (z G Rm).
Jo
584
11. SYSTEMS OF CONSERVATION LAWS
Contact between and St
Then
(15)	В(г)(г — г0) = F(z) — F(z0).
In particular z G S(zq) if and only if
(16)	(В(г) - a/)(z - г0) = 0
for some scalar a = cr(z, го).
2.	We study equation (16) by first of all noting
(17)	В(го) = ОТ(г0).
Now in view of the strict hyperbolicity, the characteristic polynomial A i—>
det(AZ — В(го)) has m distinct, real roots, and hence the polynomial A i—>
det(AZ — В(г)) likewise has m distinct roots if г is close to го- Recall-
ing Theorem 2 in §11.1.2 we see that near го there exist smooth functions
Ai(г) < • • • < Ат(г) and unit vectors {fyfy), к(г)})’к1 satisfying
Afc(^o) = Afcfyo), ffcfy0) = rfcfy0), Ifcfyo) = к(го) (k = 1,..., m),
and
( В(г)Ь-(г) = Xk(z)rk(z)
(18)	' (k = l,...,m).
I lk(z)B(z) = Afcfy)lfcfy)
Note that {г&(г)}, {tfc(-2;)}fc^1 are bases of Rm, and also
(19)	1/(г) • гй(г) = 0 (к / /).
3.	Equation (16) will hold provided a = А&(г) for some к G {1,..., m},
and (г — го) is parallel to ik(z). In light of (19), these conditions are equiv-
alent to asking
(20)
1г(г) • (г - г0) = 0 (I / к).
11.2. RIEMANN’S PROBLEM
585
These equalities amount to m — 1 equations for the m unknown components
of z, which we intend to solve using the Implicit Function Theorem. So
define Фk  Rm Rm-1 by setting
:= (... ,ifc-i(z) • (z - zo),ife+i(z) • (z - z0),.. .).
Now ФЦго) = 0 and
	/ li(^o) \	
ЖЫ =	Ifc—i(*o) lfc4-l(^o)	1
	к Imfyo) )	(m—l)xm
the entries of this matrix being regarded as row vectors. Since the vectors
{k(-2o)}fcLi f°rm a basis of Rm, we see
rank D$fc(zo) — m — 1.
Accordingly, there exists a smooth curve </>*,: R —> Rm such that
(21)	= z0
and
(22)
Ф&(</>&(£)) = 0 for all t close to 0.
The path of the curve </>&(•) for t near zero defines Sk(zo). We may repara-
meterize as necessary to ensure
(23)
£ \
dt )
|<£fc(*)| = 1
4.	Now (20)-(22) imply
(24)
Фк(£) = Zo +^(«)rfc(0fc(«))
for all t near zero, where p: R —> R is a smooth function satisfying /r(0) =
0, /t(0) = 1. Differentiating (24) with respect to t and setting t = 0, we thus
find
(25)
0fe(O) = rfc(z0).
Hence the curve Sfc(zo) has tangent r/,.(zo) at zq. Assertion (i) is proved.
586
11. SYSTEMS OF CONSERVATION LAWS
5.	In light of the foregoing analysis, there exists a smooth function
cr : Rm x Rm —> R such that
(26)	F(0fc(<)) - F(z0) = <r(0fc(t), zo)(0fc(t) - z0)
for all t close to zero. Differentiating with respect to t and setting t = 0, we
deduce from (21) that
DF(zo)0fc(O) = a(zo,zo)0fc(O).
In light of (25), we see that ct(zo,zq) = Afc(zo). This establishes assertion
(ii).
6.	Now write cr(t) := <r(</>fc(t), zq); so that (26) reads
F(0fc(t)) - F(z0) = o-(«)(0fe(t) - z0).
Differentiate twice with respect to t:
(D2F(</>fc(t))0fc(t))0fc(t) + DF(<fe(i))<fe(i)
= cr(t)(</>fc(t) - z0) + 2d(t)0fc(t) + a(t)^(t).
Evaluate this expression at t = 0 and recall cr(0) = Afc(zg), </>fc(0) = zq,
</>fc(0) = rfc(z0):
(27)	(2<r(0)Z - D2F(zo)rfc(zo))rfc(zo) = (DF(z0) - Afc(zo)/)^(O).
7.	Let ^fc(t) = v(t) be a unit speed parameterization of the rarefaction
curve .Rfc(zo) near zq (as in (6) above). Then
(28)	^fc(O) = z0,	= ric(^k(t))-
Thus
T)F(^fe(«))rfc(t) = Afe(t)rfc(t),
for
Afc(«) := Afc(^’fc(t)), rfc(t) := rfc(^fc(«)).
Next differentiate with respect to t and set t = 0:
(29)	(D2F(z0)rfc(z0) - Afe(O)Z)rfc(zo) = ~(DF(z0) - Afc(zo)Z)rfc(O).
Add (27) and (29), to obtain
(30)	(2d(0) - Afc(O))rfc(zo) = (DF(z0) - Afc(zo)Wfc(O) - rfc(0)).
11.2. RIEMANN’S PROBLEM
587
Take the dot product with ifc(zo) and observe U •	± 0, to conclude
(31)	2<r(0) - Afc(0).
We deduce from (31) that
2a(t) - Afc(z0) - Afc(i) = O(£2) as t -> 0.
Assertion (iii) follows.	□
We see from Theorem 2,(iii) that the curves Rk(zo') and Sfc(zg) agree at
least to first order at zq. Next is the assertion that in the linearly degenerate
case these curves in fact coincide.
THEOREM 3 (Linear degeneracy). Suppose for some he {1,..., m} that
the pair (Afc,rj.) is linearly degenerate. Then for each Zq G Rm,
(i)	7?fc(z0) = Sfe(zo)
and
(ii)	cr(z, z0) = Afc(z) = Afc(zo) for all z G Sfc(z0).
Proof. Let v = v(s) solve the ODE
Г v(s) = rfc(v(s)) (s G R)
I v(0) = z0.
Then the mapping s i—> A^(v(s)) is constant, and so
F(v(s))-F(zo)= f DF(v(t))v(t) dt = Г DF(v(t))rfc(v(t)) dt
Jo	Jo
= [ Afc(v(t))rfc(v(t))dt = Afc(z0) [ v(t)dt
Jo	Jo
= Afc(z0)(v(s) - z0).
□
b. Contact discontinuities, shock waves.
We next undertake to analyze in light of Theorems 2 and 3 the possibility
of solving Riemann’s problem by joining two given states щ and ur by some
kind of shock wave.
Contact discontinuities. Suppose first that (Afc,rfc) is linearly degenerate
and
(32)
ur G Sfc(uf).

11.2. RIEMANN’S PROBLEM
589
or else
(39)	Afc(uz) < Afc(ur)-
Now in view of assertion (iii) from Theorem 2, we have then either
(40)	Afc(ur) < a(ur,u0 < Afc(uz)
or
(41)	Afc(uz) < (T(ur,ui) < Afc(ur),
provided ur is close enough to ui.
This dichotomy is reminiscent of a corresponding situation in §3.4, for a
scalar conservation law. By analogy with the entropy conditions introduced
there, let us hereafter agree to reject the inequalities (41) as allowing for
“nonphysical shocks” from which characteristics emanate as we move for-
ward in time. We rather take (40) as being physically correct. The informal
viewpoint is that then the characteristics from the left and right run into
the line of discontinuity, whereupon “information is lost” and so “entropy
increases”. This interpretation was largely justified mathematically in §3.4
with our uniqueness theorem for weak solutions that satisfied this sort of
entropy condition.
Refocusing our attention again to systems, we therefore agree to regard
(40) as the correct inequalities to be satisfied:
DEFINITION. Assume the pair (Afc, r^) is genuinely nonlinear at щ. We
say that the pair (ui,ur) is admissible provided
ur € >S'fc('U/)
and
(42)	Afc(ur) < cr(ur,iq) < Afc(uz).
We refer to (42) as the Lax entropy condition. If (ui,ur) is admissible,
we call our solution u defined by (36), (37) a k-shock wave.
By analogy with our decomposition of -Rfc(zo) into R^(zq), let us intro-
duce this
DEFINITION. If the pair (Afc,rfe) is genuinely nonlinear, we write
'= {z G sk(zo) | Afc(z0) < o-(z,z0) < Afc(z)}
590
11. SYSTEMS OF CONSERVATION LAWS
and
Sk(z0) :== {z G sk(zo} I Afc(z) < a(z,z0) < Afc(zo)}.
Then
Я(г0) = Sf(г0) U {г0} U S~(г0)
near го. Note then that the pair (iq,ur) is admissible if and only if
ur G S^(ui).
11.2.4.	Local solution of Riemann’s problem.
Next we glue together the physically relevant parts of the rarefaction
and shock curves.
DEFINITIONS, (i) If the pair (А&,г&) is genuinely nonlinear, write
Tk(z0) := Д^(г0) U {г0} U S^(z0).
(ii)	If the pair (Afc,rjt) is linearly degenerate, we set
Tk(zo') := 74(г0) = Sfc(z0).
Owing to Theorem 2,(ii) the curve 7/(го) is C1. Employing this notation
we see that nearby states ui and ur can be joined by a fc-rarefaction wave,
a shock wave or a contact discontinuity provided
(43)	ui G Tfc(ur).
We now at last ask if we can find a solution to Riemann’s problem
provided only that ur is close to ui (but (43) may fail for each k = 1,... , rn).
The hope is that by moving along various paths Tk for different values of
к, we may be able to connect щ to ur, utilizing a sequence of rarefaction
waves, shock waves, and/or contact discontinuities.
THEOREM 4 (Local solution of Riemann’s problem). Assume for each
к =	that the pair (А^,г^) is either genuinely nonlinear or else
linearly degenerate. Suppose further the left state ui is given. Then for each
right state ur sufficiently close to щ there exists an integral solution u of
Riemann’s problem, which is constant on lines through the origin.
11.2. RIEMANN’S PROBLEM
591
Structure of the T-curve
Proof. 1. We intend to apply the Inverse Function Theorem to a mapping
Ф : Rm —» Rm, defined near 0 as follows.
First, for each family of curves Tk (k = 1,..., m) choose the nonsingular
parameter Tk to measure arc length; that is, if z, z G Rm with z G Tk(z),
then
Tk(z) — Tk(z) — (signed) distance from z to z along the curve Tk(г).
We take the plus sign for Tk(z) if z G (-г), the minus sign if z G S^(z).
2. Given then t = (ti,..., tm) G Rm, with |t| small, we define Ф(£) = z
as follows. First, temporarily write
(44)	ui = z0.
Then choose states zi,..., zm to satisfy
G Ti(z0), n(zi) - n(z0) = ti,
z2 G T2(21), t2(z2) - r2fyi) = t2,
.
. Zm € Tm(zm-1), Tm(<Zm')
Now write
(46)	z = zm,
and define
(47)	Ф(г) = z.
Note Ф is C1, and
(48)	Ф(0) = z0.
3. We claim
(49)
ДФ(О) is nonsingular.
592
11. SYSTEMS OF CONSERVATION LAWS
To see this, observe
Ф(0,..., tk, •. •, 0) - Ф(0,..., 0) = tfcrfc(z0) + o(tfc) as tk -> 0.
Thus
and so
(0) = rk(zo)
(k = 1,..., m),
2?Ф(0) — (ri(zo)> • •  , Гт(^о)) mxm
the entries regarded as column vectors. This matrix is nonsingular, since
{rfc(z0)}™=1 is a basis.
4. In light of (49), the Inverse Function Theorem applies: for each state
ur sufficiently close to ui there exists a unique parameter t = (ti,... ,tm)
close to zero such that Ф(£) — ur.
Recall next that if zk-i and zk are joined by a fc-rarefaction wave, this
wave is
' zk_L	if f <
<	if	< f < Afc(zfc), forGfc-CF'J-1
. zk	if Afc(zfc) < f.
Moreover if zk-i, zk are joined by a fc-shock, it has the form
Zk-l
Zk
if f < <y(zk,Zk--i)
if <r(zfc,zfc_i) < f,
where Afc(zfc) < ct(z/.:, z^-i) < Afe(zfc-i). In both cases the waves are constant
outside the regions Afc(zg) — e < у < Afc(zg) + e, for small e > 0, provided
Zk, Zk-i are close enough to zq. This is true for к = 1,..., m.
Since Ai(zq) < • • • < Am(zo), we see then that the rarefactions, shock
waves and/or contact discontinuities connecting ui = zq to zi, z\ to Z2, z^
to Z3, ..., zm_i to zm = ur do not intersect.	□
11.3. SYSTEMS OF TWO CONSERVATION LAWS
In this section we more deeply analyze the initial-value problem for m = 2,
which is to say, for a pair of conservation laws:
(1)
' Uf + F1(u1,u2)x = 0
< u2 + F2(u1,u2)x = 0
1	12	2
i? = g , i/ = gz
in R x (0, oo)
on R x {t = 0}.
11.3. SYSTEMS OF TWO CONSERVATION LAWS
593
Here F = (F1, F2),g = (дг,д2), u = (u1,u2).
11.3.1. Riemann invariants.
Our intention is first to demonstrate that we can transform (1) into a
much simpler form by performing an appropriate nonlinear change of de-
pendent variables. The idea is to find two functions w\w2 : R2 —> R with
nice properties along the rarefaction curves 7?i, R2-.
DEFINITION. We say
wl : R2 —► R
is an ith -Riemann invariant provided
(2)	Dw1(г) is parallel to lj(z) (zGR2,z^j).
We will see momentarily how useful condition (2) is, but let us first pause
to ask whether Riemann invariants exist. It turns out that since we are now
taking m — 2, this is easy. Indeed, because L,(z) • гДг) = 0 (i j), we see
(2) is equivalent in R2 to the statement
(2')	£W(z) • rj(z) = 0 (i = 1,2, z G R2);
which is to say
(3)	wz is constant along the rarefaction curve Ri (z = 1,2).
In particular, any smooth function w‘ satisfying (3) satisfies also (2'), (2)
and so is an i^'-Riemann invariant.
Remark. In the case that m > 2, Riemann invariants do not in general
exist.	□
Now we can regard w = (wi,W2) = (wfyzi, z2), w2(zi, Z2)) as being new
coordinates on the state space R2, replacing z = (21,22). More precisely, we
define w : R2 —► R2 by setting
(4)	w(z) = w(zb z2) = (w1(z1,z2),w2(z1,z2)).
The inverse mapping is z(w) = z(wi,w2) = (z1 (wi, w2), z2(wi, w2)).
Let us now utilize the transformation (4) to simplify our system of two
conservation laws (1). For this, let us suppose henceforth u — (zz1,^2) is a
smooth solution of (1). We now change dependent variables by setting
(5)	v(z, t) := w(u(x, t)) (x G R, t > 0).
What system of PDE does v = (v1, v2) satisfy?
594
11. SYSTEMS OF CONSERVATION LAWS
THEOREM 1 (Conservation laws and Riemann invariants). The func-
tions v^v2 solve the system
Vf + A2(u)?4 — 0
2 л / \ 2 za m R x (0, oo).
v2 + Ai(u)v2 = 0	1
(6)
The point is that the system (6), although not in conservation law form,
is in many ways rather simpler than (1). Note in particular that whereas
the PDE for u1 involves the term u2, the PDE for v1 does not entail v2.
Similarly, the PDE for v2 does not involve v%.
Proof. According to (5), we see that for i = 1,2, i j,
vf + Aj(u)?4 — Dw'(u) • ut + Aj(u)Dwl(u) • иг
= Dw’(u)  (-F(u)I + A/uJuJ
= TW(u)  (-DF(u) + Aj(u)/)Ua; = 0,
since, by definition, Dw1 is parallel to lj.	□
Remarks, (i) We can interpret the system of PDE (6) by introducing the
ODE
(7)	Xi(s) = Aj(u(xi(s),s)) (s > 0)
for i = 1,2, j; / i. Then we see from (6) that
(8)	vl is constant along the curve (xj(s), s) (s > 0)
for г = 1,2.
(ii)	Recall from §11.2 our condition of genuine nonlinearity reads
(9)	DXi(z) • r,(z) / 0 (г G R2, i = 1,2).
Since we can also regard At as a function of w = (wi,W2), we can rewrite
(9) to read
(10)	° (w G R2, г Д
Owj
To see (9) is equivalent to (10), observe that if (10) fails, then
uWj	OZk OWj
J K=1	J
But since Y^k=i =	= 0 for г 7^ j, we see that (11) asserts that DXi
is parallel to Dw'. However, Dw1 is perpendicular to r,, and so we obtain
a contradiction to (9). Hence (9) implies (10), and the converse implication
is established in the same way.	□
11.3. SYSTEMS OF TWO CONSERVATION LAWS
595
Example (Barotropic compressible gas dynamics). We illustrate the fore-
going ideas by examining in detail Euler’s equations for barotropic compress-
ible gas dynamics, a special case of the general Euler equations (Example 2
in §11.1) when the internal energy e is constant. The relevant PDE are
(12)
Pt + (pv)x = 0
(pv)t + (pv2 + P)x = 0
(conservation of mass)
(conservation of momentum),
where we now assume
(13)
p = p(p)
for some smooth function p : R —> R. Formula (13) is called a barotropic
equation of state. We assume the strict hyperbolicity condition
(14)
p > 0.
Setting u = (u',u2) = (p,pv), we can rewrite (12), (13) to read
ut + F(u)x = 0,
for
F = (F1,F2) = (z2,(z2)2/zi+p(z1))
and z = (zi, z2), provided z\ > 0. Then
DE=(	°	4
\-(^)2+р'(^1)
Consequently
(15)	Ai = — -p'(zi)1/2, A2 = — + p'(z1)1/2.
Z1	Z1
In physical notation:
(16)	Ai = v — cr, A2 = v + cr,
for the sound speed
(17)	a :=p'(p)1/2.
Remembering (7), we consider next the ODE
(18)
±l(i) = v(xi(t),t) + cr(xi(t), t),
596
11. SYSTEMS OF CONSERVATION LAWS
(19)	i2(t) = v(x2(t),t) - cr(z2(t),t),
where a(x, i) := p'(p(x, £)) 2, t > 0. We know from (8) that the Riemann
invariant v1 = wx(u) is constant along the trajectories of (18) and v2 =
w2(u) is constant along trajectories of (19).
To compute w1 and w2 directly, let us carry out some computations on
the system (12). First we transform (12) in nondivergence form:
(20)	pt + pvx + pxv = 0,
(21)	ptv + pvt + pxv2 + 2pvvx + px = 0.
Multiplying (20) by <72 = p'(p) and recalling (13) gives us
(22)	pt + vpx + (j2 pvx = 0.
In addition (20), (21) combine to yield
(23)	pvt + pvvx + px = 0.
We now manipulate (22), (23) so that the directions Ai,A2 = v a
appear explicitly. To accomplish this, we multiply (23) by cr and then add
to and subtract from (22):
(24)	f Pt + (« + <x)px +pa(vt + (v + a)vz) = 0
I Pt + (v - cr)px - pa(vt + (v - cr)^) = 0.
We then deduce from (24) that
(25)
. ^[pW*),*)] -p(x2(t),t)cr(z2(t),t)^[i;(x2(t),t)] = 0.
As = a2^, we see
(26)	~~77 ± ~r = 0 along the trajectories of (18), (19),
p dt dt
provided p > 0.
Think now of the Riemann invariants as functions of p and v. Then
since v1 = w1(p, v) is constant along the curve determined by xi(-), we have
dw1 d . . . . dw1 d . , , . ..
up at	uv dt
11.3. SYSTEMS OF TWO CONSERVATION LAWS
597
This is consistent with (26) if
dw1 _ <r(p) dw1 __
dp p ' dv
We similarly deduce
dw2 _ a(p) dw2 _
dp p ’ dv
Integrating, we conclude that the Riemann invariants are, up to additive
constants,
i Г ^(s)	2 [p J
w1 = / ------- ds + v, w = / --------- ds — v.
Ji S	Ji s
We leave it as an exercise to check that w\w2, taken now as functions of
z = (zi, Z2), satisfy the definition of Riemann invariants.	□
11.3.2. Nonexistence of smooth solutions.
Illustrating now the usefulness of Riemann invariants, we establish the
following criterion for the nonexistence of a smooth solution:
THEOREM 2 (Riemann invariants and blow-up). Assume g is smooth,
with compact support. Suppose also the genuine nonlinearity condition
(27)	> 0 in R2 (г = 1,2, i / j)
holds. Then the initial-value problem (1) cannot have a smooth solution u
existing for all times t >0 if
(28)	either vx < 0 or v2 < 0 somewhere on R x {t = 0}.
Proof. 1. Assume for the time being that u is a smooth solution of (1).
Write
(29)	a .= vxx, b .= v2x,
where v = w(u), v = (v1, г/2), solves the system of PDE (6). We differentiate
the first equation of (6) with respect to x, to compute:
(30)	at + X-2ax + ~a2 +	= 0.
awi 0W2
598
11. SYSTEMS OF CONSERVATION LAWS
We employ then the second equation of (6), which we rewrite as
vt2 + A2v2 = (A2 - Ai)6.
Substituting this expression into (30) gives
(31)
, <ЭА2 2
Of + MO'X +	---a +
OWi
1 <ЭА2
A2 - Ai dw2
(vt + A2u2)
a = 0.
2.	To integrate (31), fix xq G R and set
(32)	£(t):=expf [	1	^-(ut2 + A2v2)(zi(s), s) dsY
\Jo A2 — Ai du>2	)
where
f ii(s) = A2(u(xi(s),s)) (s>0)
( }	U(0) = xo.
Next is the key observation from (8) that v1 is constant along the curve
(xi(s),s). Sowrite
v1(xi(s), s) = Vq =	0) (s > 0).
Thus we see that the expression	, considered as a function of
v = w(u), depends only on v2. Let us set
Then (32), (33) imply
£(i) = exp f f -^[7(v2(zi(s), s))] ds")
(34)	\Jo as	)
= exp(7(u2(zi(t),t)) — 7(u2(zq, 0))).
3.	We now transform (31) to read
d ,_ix —1 d , <ЭА2 j
dt((e“) ) ~ (£a)2dt(ea) ~ dwj
where a(t) := a(zi(t),t) and we assume a^0. Consequently
(e(i)o(i))-1 = (a(O))-1 +
Jo
11.4. ENTROPY CRITERIA
599
This equality in turn rearranges to become
(35)
a(t) = a(0)£ x(t)
l + a(0) [ \s)ds
Jo dw\
4.	Now in view of the system of PDE (6), v is bounded. Thus we
deduce from (34) that 0 < в < £(t) < © for all times t > 0, for appropriate
constants в, ©. Therefore it follows from (27) and (35) that a is bounded
for all t > 0 if and only if a(0) > 0, that is, if
*4(жо,О) > 0.
A similar calculation holds with v2 replacing u1. We conclude that if either
и* < 0 or else vx < 0 somewhere on R x {t = 0}, there cannot then exist a
smooth solution of (1), lasting for all times t > 0.	□
11.4. ENTROPY CRITERIA
In our study of Riemann’s problem in §11.2 we have taken Lax’s entropy
condition
(1)	Afc(tiy) < (r(ur, u/J
for some к E {1,..., m} as the selection criteria for admissible shock waves.
There is great ongoing interest in discovering other mathematically cor-
rect and physically appropriate entropy conditions of various sorts, with the
aim of applying these to more complicated integral solutions of our system
of conservation laws, so as to obtain uniqueness criteria, more information
concerning allowable discontinuities, etc.
One general principle, instances of which we have already seen for scalar
conservation laws in §4.5.1 and for Hamilton-Jacobi equations in §10.1, is
that physically and mathematically correct solutions should arise as the limit
of solutions to the regularized system
(2)	uf + Е(и£)ж - £uxx =0 in R x (0, oo)
as £ —» 0. The idea is to interpret the term “eu^” as providing a small
viscosity effect, which will presumably “smear out” sharp shocks. The hope
is to study various aspects of the problem (2) in the limit £ —» 0, and thereby
to discover more general entropy criteria, to augment Lax’s condition (1).
The next sections discuss aspects of this general program.
600
11. SYSTEMS OF CONSERVATION LAWS
11.4.1. Vanishing viscosity, traveling waves.
We begin our investigation of the parabolic system (2) by first seeking
a traveling wave solution, having the form
(t — nt \
--------j (гей, t>0),
where as usual the speed a and profile v must be found. Substituting (3)
into (2), we find v : R —* Rm, v = v(s), must solve the ODE
(4)	v =-ffv + f)F(v)v ( = T" ) •
\ CIS J
Assume now ui,ur G Rm are given, and furthermore
(5)
lim v = ui, lim v = ur, lim v = 0.
s—►—oo	s—++oo	s—>±oo
Then from (3) we deduce
(6)
lim ue(z, t) =
e—>0
Ul
ur
if x < at
if x > at.
Hence the limit as £ —> 0 of our solution to (2) gives us a shock wave
connecting the states ui,ur. The plan now is to study carefully the form of
a and v, and thereby glean more detailed information about the structure
of the shock determined by (6).
The first and primary question is whether there in fact exist a and v
solving (4), (5). Integrating (4), we deduce
(7)	v = F(v) - av + c
for some constant c 6 Rm. We conclude from (5) that
(8)	F(uz) — aui + c ~ F(ur) — aur + c.
Hence
(9)	F(tq) - F(ur) = a(ut - ur).
In view of (5), (8) and (9), our ODE (7) becomes
(10)	v = F(v) - F(uz) - <t(v - ui).
Now think of the left state щ as being given, and suppose we are trying
to build a traveling wave connecting и/ to a nearby state ur. From (9) we
see that necessarily ur e Sk(ui) for some к G {1,... , m}, and
(11)	a = a(ur,ui).
We refine this observation as follows:
11.4. ENTROPY CRITERIA
601
THEOREM 1 (Existence of traveling waves for genuinely nonlinear sys-
tems). Assume the pair (Ak,rk) is genuinely nonlinear for k =
Let ur be selected sufficiently close to щ. Then there exists a traveling wave
solution of (2) connecting щ to ur if and only if
(12)	ureS^(ui)
for some к 6 {1,..., m}.
Proof. 1. Assume first <r and v solve (4), (5). Then, as noted above,
necessarily ur E Sk(ui) for some к E {1,..., m}, a = a(ur, щ). Now set
(13)	G(z):=F(z)-F(uz)-<t(z-u;).
Our ODE (10) then reads
(14)	v = G(v),
and we have
(15)	G(uz) = G(ur) = 0,
according to (9). We compute
DG(uz) = DF(uz) - al:
and so the eigenvalues of DG at щ are {A^(uz) — a}^L1, with corresponding
right and left eigenvectors {rk, IfcjfcLv rk = rk(uif к = U(«z)-
2.	Now since ur 6 Sk(ui) and |ur — uz| is small, we know from Theo-
rem 2,(iii) in §11.2.3 that
Afe(ur) T Afc(uz) /I
CT = --------------+ o(|ur - uz|).
Thus
Afc(«Z) — n 4” o(lur «z|)-
In order that there be an orbit of the ODE (14) connecting щ at s = —oo
to ur E Sk(ur) at s = +00, it must be that Afc(uz) — a > 0; for otherwise the
trajectory would not converge to ui as s —> — oo. Thus if \ur — uz| is small
enough, Afc(ur) < Afe(uz), which is to say ur E S^(ui).
3.	We omit proof of the sufficiency of condition (12): see Majda-Pego
[М-Р].	□
602
11. SYSTEMS OF CONSERVATION LAWS
The preceding result employs the genuine nonlinearity assumption, but
the assertion holds in general, provided we introduce an appropriate variant
of Lax’s entropy condition (1). So let us suppose now ur G Sk(ui) for some
k G m} and furthermore
(16)
a(z, ui) > a(ur,ui) for each z lying
on the curve Sfc(iq) between ur and щ.
Condition (16) is Liu’s entropy criterion. (Observe this condition is au-
tomatic provided (Xk,^k) is genuinely nonlinear, ur G S^(ui), and ur is
sufficiently close to u/.)
We can motivate (16) by again seeking traveling wave solutions of system
(2). So assume щ is given. Then provided |ur — tq| is small enough, it turns
out that there exists a traveling wave solution ue(a?,t) =	v solving
(4), (5), if and only if the entropy condition (16) is satisfied. See Conlon
[CO] for a proof.
To make this all a bit clearer, we next present in detail a specific appli-
cation.
Example (Traveling waves for the p-system). Let us consider again the
p-system
u] — u? = 0 (compatibility condition)
и* — p(u1)I = 0 (Newton’s law),
under the usual strict hyperbolicity condition
(18)
p' > 0.
We investigate the existence of traveling wave solutions to the regularized
system
(19)
e,l e,2 n
Ut’ — Ux =0
£,2	/ g ix ___	s,2
ut' - p(ue'L)x = euxx.
Notice we have added the viscosity term only to the second equation. This
makes sense physically, as the first line of (17) is only a mathematical com-
patibility condition.
Assume now ue = v(£z^) is a traveling wave solution of (19), with
(20)
lim v = ui, lim v = ur, lim v = 0.
s—oo	s—>OC	s—>ioo
11.4. ENTROPY CRITERIA
603
Writing v = (u1,!?2), we compute from (19) that
(21)
-^1-v2 = 0 ('=^)
— (TV2 — ^(u1)' = v2.
An integration using (20) gives
( av1 +v2 = avl 4- v? = avl. + v2
(221	<	1
\ v2 = a(v2 — v2) +p(u/) — Xu1) = a(v2 - u2) + ^(u1) — ^(u1),
for ui — (v1,^2), ur = (u1,!?2). In particular,
avj + v2 = av} + v2
av2 +p(u/) = (tv2 + p(X).
Solving these equations for <7, we obtain
(23)
2 _ P(Vr) ~P(tf)
& 1 1
X - v[
Suppose hereafter u1 > vj. In view of (18) we can take <7 > 0. In this
situation the Liu entropy criterion reads
,24)	P(zi) ~p(vi) > P(^r) ~P(^)
*1 -	v1 - u1
for all z on the curve Sk(ui) between ui and ur, z = (21,22).
We now claim the system of ODE (22), with asymptotic boundary con-
ditions (20), has a solution if and only if the entropy condition (24) holds.
To confirm this, combine the two equations in (22) to eliminate v2:
• 1 P(vly)-p(v}) x b x
v =-----------------(T[v — Vi) =: g{y ).
(T
Now ff(X) = 0 and <?(vr) — 0> according to (23). Thus in order that the ODE
(24) have a solution, with lims_+_oc. v1 = v1, linis-^ v1 = v1, we require
ff(zi) >0 torvi<z1<v^.
But this is precisely the entropy criterion (24). A similar calculation works
if u1 < v1.	□
604
11. SYSTEMS OF CONSERVATION LAWS
11.4.2. Entropy/entropy-flux pairs.
Both Lax’s and Liu’s entropy criteria provide restrictions on possible left
and right hand states joined by a shock wave (or a traveling wave for the
viscous approximation). It is however of considerable interest to widen still
further the entropy criteria, so as to apply to more general integral solutions
of our conservation laws.
One idea is to require an integral solution satisfy certain “entropy-type”
inequalities.
DEFINITION. Two smooth functions Ф, Ф : Rm —> R comprise an entro-
py/entropy-flux pair for the conservation law ut + Е(и)ж = 0 provided
(25)	Ф is convex
and
(26)	P$(z)PF(z) = РФ(г) (z E Rm).
To motivate condition (26), suppose for the moment u is a smooth so-
lution of the system of PDE u< + Е(и)ж = 0. We then compute
Ф(и)( + Ф(и)г = -ОФ(и) -и( + РФ(и) Ux
(	)	= (-ОФ(и)РР(и) + Т>Ф(и)) • их = 0
by (26). This computation says the quantity Ф(и) satisfies a scalar conser-
vation law, with flux Ф(и).
Now in general integral solutions of (1) will not be smooth enough, owing
to shocks and other irregularities, to justify the foregoing computation. The
new idea is instead to replace (27) with an inequality:
(28)	Ф(и)< + Ф(и)х <0 in R x (0, oo).
In applications Ф(и) will sometimes be the negative of physical entropy and
Ф(и) the entropy flux. The inequality (28) therefore asserts entropy evolves
according to its flux, but may also undergo sharp increases, for instance
along shocks.
Let us hereafter rigorously understand (28) to mean
(29)	f Г JZo ф(и>* + ф(иК dxdt > 0
[ for each v E Q?°(R x (0, oo)), v > 0.
We consider once more the initial-value problem
, ,	( ut + F(u)I = 0 in R x (0, oo)
1 J	\	u = g on R x {t = 0}.
11.4. ENTROPY CRITERIA
605
DEFINITION. We call u an entropy solution of (30) provided u is an in-
tegral solution and u satisfies the inequalities (29) for each entropy/entropy-
flux pair (Ф, Ф).
Let us now attempt to build for general initial data g an entropy solution.
As in §11.4.1 we expect such a “physically correct” solution u to be a limit
of solutions ue of approximating viscous problems
Г uf + F(ue)x ~£Uexx= 0 inRx(0,oo)
' '	[	u£= g on R x {t = 0}.
We assume u£ is a smooth solution of (31), converging to 0 as |ar| —► oo
sufficiently rapidly to justify the calculations below. Let us further suppose
{ue}o<e<i is uniformly bounded in L°° and furthermore
(32)	u£ —> u	a.e. as e —> 0
for some limit function u. (In practice it is extremely difficult to verify this
a.e. convergence.)
THEOREM 2 (Entropy and vanishing viscosity). The function u is an
entropy solution of the conservation law (30).
Proof. 1. Choose any smooth entropy/entropy-flux pair (Ф, Ф). Left mul-
tiplying (30) by 1?Ф(и£) and recalling (26), we compute
Ф(и£), + Ф(и£)х = еЕ>Ф(и£)и|ж
33	= ЕФ(и£)хх - £(Р2Ф(и£)и£) • U|.
As Ф is convex,
(34)	(Р2Ф(и£)и|) • u£ > 0.
2. Multiply (33) by v G C^°(R x (0,oo)), v > 0. We integrate by parts
and discover:
f [ $(us)vt + $(u£)vx dxdt
JO J—oo
= f [ е(Е>2Ф(и£)и£) • u£u — еФ(ие)и1:г dxdt
Jo J—oo
yoo yoo
>~	/ £$(u£fvxxdxdt,
JO J—oo
the last inequality holding in view of (34) and the nonnegativity of v.
606
11. SYSTEMS OF CONSERVATION LAWS
Now let e —» 0. Recalling (32) and the Dominated Convergence Theo-
rem, we obtain
I / $(u)vt 4- Ф(и)1Лг dxdt > 0.
J() J — OO
Thus u verifies the entropy/entropy-flux inequalities (29). If Ф and Ф are
not smooth, we obtain the same conclusion after an approximation.
3. Finally fix v 6 C“(R x [0, oo); Rm) and take the dot product of the
PDE in (31) with v. After integrating by parts we obtain
[	/" ue • vt + F(ue)vT + eu£ • nxx dxdt + [ g • V dx|t=o - 0.
Jo J —oo	J—oo
We send s —> 0, to deduce u is an integral solution of (30).	□
Example 1. In the case of a scalar conservation law (i.e. m = 1), for any
convex Ф we can find a corresponding flux function Ф, namely
Ф(г) = /" Q'lw'fF'^w^dw (z e R).
J Zq
See §11.4.3 following for an application.	□
Example 2. For the p-system we have m = 2. To verify (25), (26) we must
find Ф, Ф, with Ф convex and
_	, J 0	-1\	/Фг1\
(^’Ф^)	0 ) \Фг2)’
A solution is
z2 /"Z1
Ф(г) = -^ + p(w) dw, Ф(г) = -p(zi)z2 (z G R2).
4 Jo
Note Ф is convex, since p' > 0.	□
See Problems 4, 5 for other examples.
11.4.3. Uniqueness for a scalar conservation law.
As a further illustration of the ideas in §11.4.2, let us now consider again
the initial-value problem for a scalar conservation law
J ut + F(u)x = 0 in R x (0, oo)
' '	и = g on R x {t = 0}.
Hence the unknown и — u(x, t) is real-valued and F : R —» R is a given
smooth flux function.
In §3.4 we carefully studied the problem (35), making use of the primary
assumption that F be strictly convex, to derive the Lax-Oleinik formula (see
§3.4.2). Let us now drop the assumption that F be convex and devise an
appropriate notion of weak solution. As above, we introduce entropies:
11.4. ENTROPY CRITERIA
607
DEFINITION. Two smooth functions Ф, Ф : R —» R comprise an entro-
py/entropy-flux pair for the conservation law ut + F(u)x = 0 provided
(36)	Ф is convex
and
(37)	ф'(г)Е'(г) = Ф'(г) (г G R).
As noted in Example 1 above, for each convex Ф there exists a corresponding
flux Ф.
The entropy condition for и reads
Ф(и)г + Ф(п)1 <0 on R x (0, oo)
for each entropy/entropy-flux pair Ф, Ф. This means
Г /0°°	Ф(и)^ + Ф(u)vx dxdt > 0
( for each v G C“(R x (0, oo)), v > 0.
DEFINITION. We call и e C([0, oo), ^(R)) П L°°(R x (0,oo)) an en-
tropy solution of (35) provided и satisfies the inequalities (38) for each
entropy/entropy-flux pair (Ф,Ф), and u(-,t) —» g in L1 as t 0.
Remarks, (i) This definition supersedes our earlier definition of “entropy
solution” in §3.4.3.
(ii) Taking Ф(г) = ±z, Ф(г) = ±F(z) in (38), we deduce
roo
/	/ uvt + F(u)vx dxdt = 0
Jo J—oo
for all v > 0 and thus for all v e Cp(R x (0, oo)). It is an exercise to prove
then that
/•OO poo	poo
I I uvt + F(u)vx dxdt + I	gv dx\t=o = 0
J0 J—oo	J—oo
for all v G C'(lx [0, oo)), since и/, t) —» g in Ll. Thus an entropy solution
is an integral solution.	□
We discussed in §11.4.2 the construction of an entropy solution, and we
now prove uniqueness.
608
11. SYSTEMS OF CONSERVATION LAWS
THEOREM 3 (Uniqueness of entropy solutions for a single conservation
law). There exists—up to a set of measure zero—at most one entropy solu-
tion of (35).
As in the proof of Theorem 1 in §10.2, the basic idea will be to “double
the variables” in the problem.
Proof*. 1. Let и be an entropy solution of (35). Then
(39)	[ /" $(u)vt + Ф(г1)глг dxdt > 0
Jo J—co
for all v 6 C^°(R x (0, oo)), v > 0, where Ф is smooth, convex and
Ф(г) = f	dw
J ZQ
for any zq. Fix a E R and take
(40)	Фк(г') := 0k(z - а) (г E R),
where for each к = 1,...,	: R —> R is smooth, convex and
( (3k(z) —> \z\ uniformly
I Pk (г) —* 8ёп(г) boundedly, a.e.
Thus $k(z) —» \z — a| uniformly for z E R. A flux corresponding to (40) is
^k(z) = [ 0k(w - a)F'(w)dw.
J a
Consequently for each z
Фд.(г) —» f sgn(w — a)F'(w)dw = sgn(z — a)(F(z) — F(af).
J a
Putting into (39) and sending к —> oo, we deduce
(41)	f f \u — a\vt + sgn(u — a)(F(u) — F(a))vx dxdt > 0
Jo J—co
for each a E R and v as above.
2.	Next let й be another entropy solution. Then
(42)	f [ |й — d|vs + sgn(& - d)(F(u) — F(af)vy dyds > 0
Jo J—co
*0mit on first reading.
11.4. ENTROPY CRITERIA
609
where a 6 R and v G C^°(R x (0, oo)), v > 0.
Now let w 6 C“(R x R x (0, oo) x (0, oo)), w > 0, w = w(x,y,t,s).
Fixing (y, s) G Rx (0, oo), we take a = vjy, s'), v(x, t) = w(x, y, t, s') in (41).
Integrating with respect to y, s, we produce the inequality
(43)	Jo Jo J-oo J—oo
+ sgn(u(x, t) — u(y, sy)(F(u(x, t)) — F(u(y, s)))wx dxdydtds > 0.
Likewise, for each fixed (x,t) eKx (0,oo) we take a = u(x, t), v(y, s) =
w(x,y,t,s) in (42). Integrating with respect to x,t gives:
ws
(44)	Jo Jo
+ sgn(ii(y, s) — u(x, t)) (F(ii(y, s)) — F(u(x, t)))wy dxdydtds > 0.
Add (43), (44):
(45) 70 J°
+ sgn(u(x, t) - й(у, s)) (F(u(x, t)) - F(u(y, s)))
(wx 4- Wy) dxdydtds > 0.
3.	We design as follows a clever choice for w in (45). Select у to be
a standard mollifier as in §C.4 (with n = 1) and, as usual, write •qe{x') =
Ь(f) • Take
t + \ Х~У
w(x,y,t,s) := T]£[
\ “
(t — s\ (x + y t + s\
"'(“ГО’
where ф G C£°(R x (0, oo)), ф > 0. We insert this choice of w into (45) and
thereby obtain:
Jo Jo J-oo J-oo I	\ 2 ’ 2 /
(46) + sgn(u(x, t) - u(y, s)) (F(u(x, t)) - F(ii(y, s)))^ f	1
(x — y\	/1 — s\
r]£ I —-— I T]£ I —-— I dxdydtds > 0.
\ Zi J \ Zi J
Change variables by writing
- _ x+y 7 _ t+s
x, —	2 , t —	2
у =
610
11. SYSTEMS OF CONSERVATION LAWS
Then (46) implies
/OO roc
/ G(y, «МЫ») dyds > 0,
-oo J—oo
where
/•OO roo
G(y,s):= /	/ |u(x + y,t + s) - u(x - y,t - s)|</>t(x,t)
J 0 J—oo
(48)	+ sgn(u(x + y,t + s) — u(x — y,t — s))
(F(u(x + y, t + s)) — F(iz(x — y, t — 5)))</>ж(х, t) dxdt.
Now u(x + y,t + s) —> u(x, t), й(х — у, t — s) —> u(x,i) in L(oc as y,s —► 0.
Since the mappings (a,b) i—> |a — £>|, sgn(a — b)(F(a) — F(b)) are Lipschitz
continuous, we deduce upon letting e —» 0 in (47) that
/•oo roc
I / \u(x, t) — й(х, t)| </>t(x, t)
Jo J—oo
+ sgn(u(x, t) — й(х,Т)) (F(u(x,t)) — F(u(x,t)y) фх(х,1') dxdt > 0.
Rewriting x = x, t = t, we have therefore
(49)	[	[ a(x,t)$t(x,t) + b(x,t)</)x(x,t) dxdt > 0,
Jo J—oc
for
( a(x,t) := |u(x, t) — й(х,t)|
t b(x, t) := sgn(u(x, t) — u(x, ty)(F(u(x, t)) — F(u(x, t))).
4.	We now employ the inequality (49) to establish the L1-contraction
inequalities
(50)	/	- ^X’ ^dx-	“ й(Х’S>> । dx
\ for a.e. 0 < s < t.
To prove this assertion, we take 0 < s < t, r > 0, and let ф(х, t) = a(x)/3(t)
in (49), where
' a : R —» R is smooth,
< a(x) = 1 if |ar| < г, а(ж) = 0 if |x| > r + 1,
. |a/(;r)| < 2
and
' (3 : R —» R is Lipschitz,
/3(т) = 0 if 0 < r < s or т > t + 6,
(3(r) = 1 if s + б < т < t,
, /3 is linear on [s, s + 6] and [2, f + 6],
11.5. PROBLEMS
611
for 0 < 6 < t — s. We deduce
oo
I /	b(x,r)a'(x)b(r) dxdr.
S J	|x| <r+1}
s
t
Let r —* oo:
a(x, r) dxdr
Next let 6 —» 0 to deduce (50) for a.e. 0 < s < t.
5.	In light of (50) and the fact u(-, t), &(•, t) —> g in Ll as t —» 0, we at
last conclude и — й a.e.	□
11.5.	PROBLEMS
1.	Verify that the shallow water equations (Example 3 in §11.1) form a
strictly hyperbolic system, provided ф > 0.
2.	Define for z £ R, г 0, the matrix function
i /cos(|) sin(|) \
\sin(|) -cos(|)/’
and set B(0) = 0. Show that В is C°° and has real eigenvalues, but
we cannot find unit-length right eigenvectors {г1(г),Г2(г)} depending
continuously on z near 0. What happens to the eigenspaces as z —► 0?
3.	Confirm that the functions w^w2 computed for the barotropic gas
dynamics in §11.3.1 are indeed Riemann invariants.
4.	Suppose that Ф is an entropy for the shallow water equations. Prove
Э2Ф ,<Э2Ф
dv2	дф2
5.	Show that Ф = pv2/2 + P(p) is an entropy for the barotropic Euler
equations (from §11.3.1), provided P"(p) — p'^/p, p > 0. Confirm
that Ф is convex in the proper variables. What is the corresponding
entropy flux Ф?
6.	Assume that и is an entropy solution of the scalar conservation law
щ + F(u)x = 0, and that, as in §3.4.1, и is smooth on either side of a
curve {x = s(t)}.
612
11. SYSTEMS OF CONSERVATION LAWS
(i)	Prove that along this curve the left and right hand limits of и
satisfy the relations:
F(jz) > —---------—l—(z — ur) + F(ur) if ui < z < ur,
ur — Ul
and
F(z) < —----------—— (z — ur) + F(ur) if ur < z < ui.
ur — Ul
This is the condition E.
(ii)	What does condition E imply if F is uniformly convex?
11.6.	REFERENCES
T.-P. Liu wrote the initial draft of this entire chapter.
Section 11	.1
Section 11	.3
Section 11	.4
Section 11	.5
The shallow water equations are discussed in LeVeque [LV].
P. Colella helped me with the calculations in §11.1.2. The
proof of Theorem 2 follows suggestions of W. Han. See
Courant-Friedrichs [C-F], Xiao-Zhang [X-Z], Zheng [ZH]
for more.
I followed Logan [LO] for the example.
See Majda-Pego [M-P], Conlon [CO], Smoller [S] for more
about viscous traveling waves. The uniqueness theorem in
§11.4.3 is due to Kruzkov.
Problem 2 is taken from Kato [K, p. 111].
APPENDICES
A.	Notation
B.	Inequalities
C.	Calculus facts
D.	Linear functional analysis
E.	Measure theory
APPENDIX A: NOTATION
A.l. Notation for matrices.
(i)	We write A = ((aij)') to mean A is an m x n matrix with (i, j)th entry
ciij. (Sometimes, as in §8.1.4, it will be convenient to use superscripts
to denote rows.)
A diagonal matrix is denoted diag(di,..., dn).
(ii)	Mmx" = space of real m x n matrices.
§nxn = space of real symmetric n x n matrices.
(iii)	tr A = trace of the matrix A.
(iv)	det A = determinant of the matrix A.
(v)	cof A = cofactor matrix of A (See §8.1.4).
(vi)	AT = transpose of the matrix A.
(vii)	If A = ((a^)) and В = ((fy?)) are m X n matrices, then
m n
A . В —	bij,
i=i j=i
613
614
APPENDICES
and
(m n
22 2Z 4
t=i j=i
(viii)	If A G Snx" and, as below, x = (a?i,..., xn) G Rn, the corresponding
quadratic form is x • Ax = aijXiXj.
(ix)	If A g S"xn, we write
A >01
if x  Ax > 0\x\2 for all x G Rn.
(x)	We sometimes will write у A to mean ATy, for A G Mmx" and у G Rm.
A.2. Geometric notation.
(i)	R" = n-dimensional real Euclidean space, R = R1.
(ii)	= (0,..., 0,1,..., 0) = ith standard coordinate vector.
(iii)	A typical point in R" is x = (si,..., xn}.
We will also, depending upon the context, regard x as a row or column
vector.
(iv)	R” = {x = (a?i,..., Xn) G R" | xn > 0} = open upper half-space.
R+ = {x G R | x > 0}.
(v)	A typical point in R"+1 will often be denoted as (x, t) = (zi,..., xn, t),
and we usually interpret t —	= time.
A point x G R" will sometimes be written x = (x',xn) for x' =
(xi,...,x„_i) G R"-1.
(vi)	U, V, and W usually denote open subsets of R”. We write
V CC U
if V С V C U and V is compact, and say V is compactly contained
in U.
(vii)	dU = boundary of U, U = U U dU = closure of U.
(viii)	UT = U x (0,T].
(ix)	Гу = Ut — Ut = parabolic boundary of Ut-
(x)	B®(x, r) = {y G R" | |ar — y\ < r} = open ball in Rn with center x
and radius r > 0.
APPENDIX A: NOTATION
615
(xi)	B(x, r) = closed ball with center x, radius r > 0.
(xii)	C(x, t, r) = {у E Rn,s E R | |x — y| < r, t — r2 < s < t} = closed
cylinder with top center (x, t), radius r > 0, height r2.
(xiii)	a{n) = volume of unit ball 72(0,1) in R" =	
na(n) = surface area of unit sphere dB(0,1) in Rn.
(xiv) If a = (ai,..., an) and b = (bi,..., bn) belong to Rn,
i
n	/ n	\ 2
a • b — 57 aibi’ lal = I 57 ai I
2=1	\2=1 /
(xv) C” — n-dimensional complex space, C = complex plane.
If z E C, we write Re(z) for the real part of z, and Im(z) for the
imaginary part.
A.3. Notation for functions.
(i)	If и : U —> R, we write
u(a?) = u(a?i,... ,xn) (xEU).
We say и is smooth provided и is infinitely differentiable.
(ii)	If u, v are two functions, we write
и = v
to mean that и is identically equal to u; that is, the functions u, v
agree for all values of their arguments.
Let us also set
и := v
to define и as equaling v.
The support of a function и is denoted
spt u.
(iii)	u+ = max(u,0), u~ — — min(u,0), и = u+ — u~, |u| = u+ + u“.
The sign function is
{1	if	x > 0
0	if	x = 0
— 1	if	x < 0.
616
APPENDICES
(iv)	If u : U —► Rm, we write
u(x) = (u1^),..., um(x)) (x e U).
The function uk is the kth component of u, к = 1,..., m.
(v)	If E is a smooth (n — 1)-dimensional surface in Rn, we write
Lfis
for the integral of / over S, with respect to (n—l)-dimensional surface
measure. If C is a curve in Rn, we denote by
[ fdl
JC
the integral of f over C with respect to arclength.
(vi)	Averages:
/fdy = —- / fdy = average of f over the ball B(x,r)
B(x,r)	Ot(n)T
and
/ fdS=- tCn-ll fdS
j dB(x,r) not(n)r JdB(x,r)
= average of f over the sphere dB(x, r).
(1 if x 6 E
(vii) xe{x) = <
(0 if x f E.
Xe is the indicator function of E.
(viii) A function и : U —► R is called Lipschitz continuous if
|u(z) -u(y)\ < C\x~y\
for some constant C and all x,y EU. We write
Lipfu] sup
|u(a?) -u(y)|
k-y|
(ix) The convolution of the functions f, g is denoted
f *9-
APPENDIX A; NOTATION
617
Notation for derivatives. Assume и : U —> R, x E U.
(i)	|^(ж) = lim/l_>o u(x—$~u&, provided this limit exists.
(ii)	We usually write uXi for
(iii)	Similarly, = uXiX}., d~fx“dxk = uXiXjXk, etc.
(iv)	Multiindex Notation'.
(a)	A vector of the form a = (оц,..., an), where each component оц
is a nonnegative integer, is called a multiindex of order
|a| — ai + • • • + an
(b)	Given a multiindex a, define
Dau(x) :=	----£-5- = dx'    d£nu.
v ’ dx^1   dxnn 1 n
(c)	If к is a nonnegative integer,
Dku(x} := {Dau(x) | |a| = k},
the set of all partial derivatives of order k. Assigning some ordering
to the various partial derivatives, we can also regard Dku(x) as a
point in .
(d)	|Л| = IW) .
(e)	Special Cases: If к = 1, we regard the elements of Du as being
arranged in a vector:
Du = (uX1,..., uXn) = gradient vector.
If к = 2, we regard the elements of D2u as being arranged in a matrix:
Hessian matrix.
(v)	Au = ZT=1 uxiij — tr(Z)2u) = Laplacian of u.
(vi)	We sometimes employ a subscript attached to the symbols £>, D2,
etc. to denote the variables being differentiated. For example if
и = u(x,y) (x E Rn, у E Rm), then Dxu = (uX1,... ,uXn), Dyu =
(u^j,..., иУт).
618
APPENDICES
Function spaces.
(i)	Ck(U) = {u : U —» R | и continuous}
C(17) = {ti G | и uniformly continuous}
Ck(U} = {u : U —» R | и is fc-times continuously differentiable}
Cfc(C7) = {и E Ck(U) | Dau is uniformly continuous for all |a| < k}.
Thus if и E Ck(U\ then Dau continuously extends to U for each
multiindex a, |a| < k.
(ii)	C°°(C7) = {u : U —» R | и is infinitely differentiable} = Ck(U}
c°°(U)= nr=oc"W
(iii)	Cc(U), Ck(U), etc. denote these functions in C(U), Ck(U), etc. with
compact support.
(iv)	LP(C7) — {u : U —► R | и is Lebesgue measurable, ||it||£,p(V) < oo},
where
\\u\\lp(U) = ( [ l/l^xV (l<p<oo).
\Ju J
L°°(U) = {u : U —» R | и is Lebesgue measurable, ||u||l°°(c/) < °°}5
where
= ess SUPf/ |u|.
Zfoc(?7) = {u : U R | v E ЩУ) for each V EC U}.
(See also §D.l.)
(v)	11-Chi11£₽((/) — |||T»u|||lp(c/)
P2u||lp((7) = |||D2u|||Lp((7).
(vi)	Wk,p(U), Hk(U), etc. (fc = 0,1,2,..., 1 < p < oo) denote Sobolev
spaces: see Chapter 5.
(vii)	Ck'@(Wp Ck’@(U) (fc = 0,..., 0 < /3 < 1) denote Holder spaces: see
Chapter 5.
(viii)	Functions of x and t. It is occasionally useful to introduce spaces of
functions with differing smoothness in the x- and t-variables, although
there is no standard notation for such spaces. We will for this book
write
C12(C7r) = {u : UT R | u, Dxu, D$u, ut E C(UT)}.
In particular, if и E	then u, Dxu, etc. are continuous up to
the top U x {t = T}.
APPENDIX A: NOTATION
619
A.4. Vector-valued functions.
(i)	If now m > 1 and u : £7 —► Rm, u = (u1,..., um), we define
Dau = (Dau\ ..., Daum) for each multiindex a.
Then
Dku = {Dau | |o| = k}
and
/	\	1/2
i^ui = [ 22 pa«i2 j >
y|a|=fc	J
as before.
(ii)	In the special case к = 1, we write
/ du1
dxi
Du =
du1
= gradient matrix.
dum /
dxn ' mxn
(iii)	If m = n, we have
div u = tr(.Du) = 22 ux. = divergence of u.
i=i
(iv)	The spaces C(U; Rm), Z7(17;Rm), etc. consist of those functions u :
U -+ Rm, u =	with u* e C(U), etc. (i =
l,...,m).
Remark on sub- and superscripts. As illustrated above, we will adhere
to the convention of setting in boldface mappings which take values in Rm
for m > 1 (or else in Banach or Hilbert spaces). The component functions of
such mappings will be given superscripts. On the other hand, a typical point
x E Rn is not boldface and has components with sutecripts, x — (xi,..., xn).
Matrix-valued mappings will also be set in boldface, and their compo-
nent functions written with either superscripts or a mixture of sub- and
superscripts, depending on the context.	□
A.5. Notation for estimates.
Constants. We employ the letter C to denote any constant that can be
explicitly computed in terms of known quantities. The exact value denoted
by C may therefore change from line to line in a given computation. The
big advantage is that our calculations will be simpler looking, since we con-
tinually absorb “extraneous” factors into the term C.
620
APPENDICES
DEFINITIONS, (i) (Big-oh notation.) We write
f - 0(g) as x -> x0,
provided there exists a constant C such that
\f(x)\<C\g(x)\
for all x sufficiently close to xq.
(ii) (Little-oh notation.) We write
f - o(g) as x -> x0,
provided
limLM
—0 |5(x)|
= 0.
Remark. The expression “O(<?)” (or “0(5)”) is not by itself defined. There
must always be an accompanying limit, for example “as x —> xq” above,
although this limit is often implicit.	□
A.6. Some comments about notation.
The foregoing notation is largely standard within the PDE literature,
with a few significant exceptions:
(i) We employ the symbol “£>u”, and not “Vu”, to denote the gradient
of the function u. The reason is that “D2u” then naturally denotes the
Hessian matrix of u, whereas “V2u” would be confused with the Laplacian.
The multiindex notation also looks better with the letter D.
(ii) Most books and papers on partial differential equations denote by
“Q” the open subset of Rn within which a given PDE holds.
As indicated above, we will instead mostly use the symbol “C7” for such
a region. The advantages are several. First of all, since a typical solution is
denoted u, it makes sense to denote its domain by U, and not to switch to
a Greek letter. Furthermore, once we call a given open set U, the letters V
and W are then available for subregions.
Lastly, it is important to save Q as the standard symbol for a probability
space. Many important partial differential equations have probabilistic rep-
resentation formulas (cf. Freidlin [FD]), and although such are beyond the
scope of this book, it seems wise to avoid the possibility of future notational
confusion.
APPENDIX В: INEQUALITIES
621
APPENDIX В: INEQUALITIES
B.l. Convex functions.
Definition. A function f : Rn —> R is called convex provided
(1)	f(jx + (1 - rfy) < rf(x) + (1 - r)f(y)
for all x,y € Rn and each 0 < т < 1.
THEOREM 1 (Supporting hyperplanes). Suppose f : Rn —> R is convex.
Then for each x e Rn there exists r e Rn such that the inequality
(2)	f(y) >f(x)+r-(y-x)
holds for all у G R”.
Remarks, (i) The mapping у н-> f(x) + r • (y—x) determines the supporting
hyperplane to f at x. Inequality (2) says the graph of f lies above each
supporting hyperplane. If f is differentiable at x, r — Df(x).
(ii) If f is C2, then f is convex if and only if D2f > 0. The C2 function
f is uniformly convex if D2f > 01 for some constant 0 > 0: this means
n
(x,eeRn).
ij=l
□
THEOREM 2 (Jensen’s inequality). Assume f : R —> R is convex, and
U C Rn is open, bounded. Let и : U —> R be summable. Then
(3)	f udx^ < j f(u)dx.
Remember from §A.3 the notation Jf 7udx = udx = average of и
over U.
Proof. Since f is convex, for each p € R there exists r € R such that
/(?) > /(p) + r(<7 - P) for all q G R.
Let p = f-yUdx, q = u(x):
f(u(x)) > f udx^ + r ^u(x) — j- udx^ .
Integrate with respect to x over U.
□
Convex functions are discussed more fully in §3.3.2 and §9.6.1.
622
APPENDICES
В.2. Elementary inequalities.
Following is a collection of elementary, but fundamental, inequalities.
These estimates are continually employed throughout the text and should
be memorized.
a. Cauchy’s inequality.
a2 b2
(4)	ab<~ + — (a,btR).
Proof. 0 < (a — b)2 = a2 — 2ab + b2.	□
b. Cauchy’s inequality with e.
b2
(5)	ab < ea2 + — (a, b > 0, e > 0).
Proof. Write
aft = ((2f)V20) ((^5)
and apply Cauchy’s inequality.	□
c. Young’s inequality. Let 1 < p, q < <x>, 1 + 4 = 1. Then
ap bq
(6)	ab <-----1---(a, b > 0).
P Q
Proof. The mapping x h-> ex is convex, and consequently
ab — elosa+losb = g£logoP+jl°g&« < IgiogaP + lglogfc« _	+
“ P q P q'
□
d. Young’s inequality with e.
(7)	ab < eap + C(e)b9 (a, b > 0, e > 0)
for C(e) = (ep)-9//pQ-1.
Proof. Write ab = ((ep)1/pa) (^7?) and apply Young’s inequality. □
e. Holder’s inequality. Assume 1 < p, q < oo, | + 1 = 1. Then if
и G LP(U), v G Lq(U}, we have
(8)	/ \uv\dx < ||u||LP({/)||v||L9({/).
Ju
APPENDIX В: INEQUALITIES
623
Proof. By homogeneity, we may assume ||u||£P = ||v||l« = 1. Then Young’s
inequality implies for 1 < p, q < oo that
[ |uv|<fc < - [ |u|pdx + - [ |v|9dx = 1 — ||n||LP||v||Lg.
Ju Q Ju Q Ju
□
f. Minkowski’s inequality. Assume 1 < p < oo and u, v 6 Then
(9)	IIм + vIIlp(V) <	•
Proof.
IIм +	+ v\Pdx - /^u + WIP-1(M + dx
< \u + v|pdx^ ((/ |n|pdx^ + |v|pdx^
= ||u + г>||рр(^ (||u||LP(t7) + ||^||zzP(£7))-
□
Remark. Similar proofs establish the discrete versions of Holder’s and
Minkowski’s inequalities:
< (SLi ЫРЙП=11М%
(H=i + WTP < (ZLi Ыр)" + (П=1 \ьк\^,
for a = (ai,..., an), b = (iq,..., bn) € R” and l<p<oo, | + | = 1.	□
g. General Holder inequality. Let 1 < pi,... ,pm < oo, with A- + T 4
•  • + ~ = 1, and assume uk E Uk (U) for k = 1,..., m. Then
P	m
(11)	/ hl Umjdx < PI
JU	fe=l
Proof. Induction, using Holder’s inequality.	□
(10)
h. Interpolation inequality for F-norms. Assume 1 < s
and
1 _ g ! (1-0)
r s t ’
Suppose also и € LS(U) П Т4(17). Then и € Lr(U), and
(12)	Wlw < 11«|1Ь(с/)11«)11ч^).
624
APPENDICES
Proof. We compute
[ |u|refc= [ |u|<>r|u|(1-<')’’da;
Ju Ju
\ — / \ (1~9)r
< fy |u|6r3? dxJ	fy	(!-e)r dam
We have invoked Holder’s inequality, which applies since у +	= 1.
□
i.	Cauchy—Schwarz inequality.
(13)	|x-y| < |x||y| (x,yeRn).
Proof. Let e > 0 and note
0 < \x ± cyl2 = \x\2 ± 2ex  у + e2|т/|2.
Consequently
±x-y< ^k|2 + ||y|2.
Minimize the right hand side by setting c = provided у / 0.	□
Remark. Likewise, if A is a symmetric, nonnegative n x n matrix,
(14)
n
aijxiyj
i,j=l
(z,yeRn).
j.	Gronwall’s inequality (differential form).
(i)	Let ??(•) be a nonnegative, absolutely continuous function on [0,T],
which satisfies for a.e. t the differential inequality
(15)
r/(t) < <?!>(t)77(t) + V>(*),
where 0(f) and ip(t) are nonnegative, summable functions on [0, Т]. Then
(16)
77(f) < eJo^s)ds
7/(0) + [ ^(s) ds
Jo
for all 0 < t < T.
(ii)	In particular, if
77' < фт] on [0, T] and 77(0) = 0,
then
77 = 0 on [0,T],
APPENDIX В: INEQUALITIES
625
Proof. From (15) we see
4 fr?(s)e-;o^Wc/r) =e-AWd4r/(s) -4>(sh(s)) < e";o^)^(s)
as \	/
for a.e. 0 < s < T. Consequently for each 0 < t < T, we have
t	f^s	ft
?7(^)e_^o	< 7^(0) + / e”l<> ^r')drip(s') ds < 77(0) + / i[)(s)ds.
Jo	Jo
This implies inequality (16).	□
k.	Gronwall’s inequality (integral form).
(i)	Let £(£) be a nonnegative, summable function on [0, T] which satisfies
for a.e. t the integral inequality
(17)
f ew^+c2
Jo
for constants Ci, C2 > 0. Then
(18)
£(<)< C2(l + Ci^Clt)
for a.e. 0 < t < T.
(ii)	In particular, if
C(0 <Ci f £(s)ds
Jo
for a.e. 0 < t < T, then
£(t) = 0 a.e.
Proof. Let T](t) := f^(s)ds; then t) < C^rj -f- C2 a.e. in [0, Т]. According
to the differential form of Gronwall’s inequality above:
77(f) < eClt(r?(0) + C2t) = C2teClt.
Then (17) implies
e(i) <	+ C2< C2(l 4- CiteClf).
626
APPENDICES
APPENDIX C: CALCULUS FACTS
C.l. Boundaries.
Let U C Rn be open and bounded, к G {1,2,... }.
DEFINITION. We say dU is Ck if for each point x° G dU there exist
r > 0 and a Ck function 7 : Rn-1 —> R such that—upon relabeling and
reorienting the coordinates axes if necessary—we have
U П B(x°,r) = {x G B(x°,r) I xn > 7(xi,... ,xn-i)}.
Likewise, dU is C°° if dU is Ck for к = 1,2,, and dU is analytic if the
mapping 7 is analytic.
The boundary of U
DEFINITIONS, (i) If dU is C1, then along dU is defined the outward
pointing unit normal vector field
v=
The unit normal at any point x° G dU is i/(x°) = v = (17,..., pn).
(ii) Let u€C\U). We call
APPENDIX C: CALCULUS FACTS
627
Straightening out the boundary
the (outward) normal derivative of u.
We will frequently need to change coordinates near a point of dU so as
to “flatten out” the boundary. To be more specific, fix x° G dU, and choose
r, 7, etc. as above. Define then
( yi = Xi =: Фг(х)	(г = 1,... ,n - 1)
l Уп = xn -7(xi,...,zn_i) =: Фп(х),
and write
у = Ф(х).
Similarly, we set
( Xi = yi =: Фг(?/)	(г = 1,... ,n — 1)
I xn = yn +	 ,Уп>) =: фп(у),
and write
x = Ф(г/).
Then Ф = Ф-1, and the mapping x i-+ Ф(х) = у “straightens out dU" near
a?0. Observe also that det Ф = det Ф = 1.
C.2. Gauss-Green Theorem.
In this section we assume U is a bounded, open subset of Rn, and dU is
C1.
THEOREM 1 (Gauss-Green Theorem). Suppose и G C1 (U): Then
(1)	/ uXi dx = / uvl dS (г = 1,..., n).
Ju	JdU
628
APPENDICES
THEOREM 2 (Integration-by-parts formula). Let u,v € C^U). Then
(2)	/ uXivdx = — I uvXidx + I uvv1 dS (i = 1,..., ri).
Ju	Ju Jau
Proof. Apply Theorem 1 to uv.	□
THEOREM 3 (Green’s formulas). Let u,v E C2(U). Then
(0 fu dx — $du dS,
(ii) Du  Du dx = — fи и Av dx + fdu ^u dS,
(iii) uAv - vAu dx = fdu u^ -v^dS.
Proof. Using (2), with uXi in place of и and v = 1, we see
/ uXiXidx= / uXivldS.
Ju	JdU
Sum i = 1,..., n to establish (i).
To derive (ii), we employ (2) with v = uXi. Write (ii) with и and v
interchanged and then subtract, to obtain (iii).	□
C.3. Polar coordinates, coarea formula.
Next we convert n-dimensional integrals into integrals over spheres.
THEOREM 4 (Polar coordinates).
(i) Let f : Rn —> R be continuous and summable. Then
for each point xq E Rn.
(ii) In particular
for each r > 0.
Theorem 4 is a special case of
APPENDIX C: CALCULUS FACTS
629
THEOREM 5 (Coarea formula). Let и : Rn —> R be Lipschitz continuous
and assume that for a.e. r € R the level set
{x G Rn | u(x) — r}
is a smooth, (n — 1)-dimensional hypersurface in Rn. Suppose also f : Rn —>
R is continuous and summable. Then
[ f\Du\dx = [ ( [ fdS] dr.
JTSU1	J-oo yJ{u=r}	J
Theorem 4 follows from Theorem 5 by taking u(x) = |x — xq|- See
[E-G, Chapter 3] for more on the coarea formula. The word “coarea” is
pronounced, and sometimes spelled “со-area”.
C.4. Convolution and smoothing.
We next introduce tools that will allow us to build smooth approxima-
tions to given functions.
Notation. If U C Rn is open, e > 0, write U( := {x G U | dist(x, dU) > e}.
□
DEFINITIONS, (i) Define r? G C°°(Rn) by
( Cexpf^M if |x| < 1
rfix) := <	'•l 1	7
I 0	if kl > 1,
the constant C > 0 selected so that rjdx = 1.
(ii) For each e > 0, set
r^x) :=	(I) .
We call 77 the standard mollifier. The functions r)e are C°° and satisfy
/ rydx = 1, spt(?7e) C B(0, e).
JR"
DEFINITION. If f : U -+ R is locally integrable, define its mollification
fe =T]t* f in Ue.
That is,
/€(x) = / 77e(x-y)/(y)dy = /	7?6(y)/(x-y)dy
JU	.7.8(0,e)
for x G Ue.
630
APPENDICES
THEOREM 6 (Properties of mollifiers).
(i)	г e C°°(uef
(ii)	fe f a.e. as e —> 0.
(iii)	If f E C(U), then fe —> f uniformly on compact subsets ofU.
(iv)	If 1 < p < oo and f E Lfoc(17), then f( -> f in tfoc(U).
Proof. 1. Fix x E Uc, i E {1,... ,n}, and h so small that x + het E Uf .
Then
/e(x + hej -/£(х)	1 f 1 Г (x + het-yX
-------h--------= F Jr h I” (-----e----) “ ’'(“)] M ’
1 fir (x + hei-y\ (x-y\\
= F Jv h [4 (----i“ 4 (“Г)]	dy
for some open set V CC U. As
1
h
f x + het — y\ fx — у
q ( -------------- - r? -----------
\ e /	\ e
1 &q
e dxi
x-y
e
uniformly on V, ^(x) exists and equals
IJfxx~ywy'ldy-
A similar argument shows that Dafe(x) exists, and
Daf(x) = [ Da^x - y)f(y) dy (x E U€f
Ju
for each multiindex a. This proves (i).
2. According to Lebesgue’s Differentiation Theorem (§E.4),
(3)	lim/ |/(y) -f(x)\dy = 0
r~+0 J B(x,r)
for a.e. x E U. Fix such a point x. Then

rf(x - y)[/(y) - f(x)\dy
<4 [	I №)-/(*) I
<C'/ l/(sO - f(x)\dy -> 0 as e —> 0,
J B(x,e}
APPENDIX C: CALCULUS FACTS
631
by (3). Assertion (ii) follows.
3.	Assume now f E C(U). Given V CC U we choose V CC W CC U
and note that f is uniformly continuous on W. Thus the limit (3) holds
uniformly for x E V. Consequently the calculation above implies /e —» f
uniformly on V.
4.	Next, assume 1 < p < oo and f E Lpoc(U). Choose an open set
V CC U and, as above, an open set W so that V CC W CC U. We claim
that for sufficiently small 6 > 0
(4)	II/||lp(v) < II/||lp(W)
To see this, we note that if 1 < p < oo and x E V,
1 T]e(x-y)f{y)dy
B(x,e)
I l/p(x-y')T]^p(x-y)\f(y)\dy
B(x,e)
i-i/p
i/p
Since r]f(x — y) dy = 1, this inequality implies
I ~y)|/(y)|pdy dx
B(x,e)	j

provided e > 0 is sufficiently small. This is inequality (4).
5.	Now fix V CC W CC U, 6 > 0, and choose g E C'(W) so that
II/ - ffllLP(W) < $
Then
II/ ~ /IIlp(V) < II/ - /IIlp(V) + II/ - SIIlp(V) + II# “ /IIlp(V)
<	2|| f ~ 9||lp(w) + II/ _ 9IIlp(v) by (4)
<	26 + ||ye — р||ьр(у).
Since g£ —> g uniformly on V, we have limsupe^0 Ц/ — /||lp(v) < 26.	□
632
APPENDICES
С.5. Inverse Function Theorem.
Let U C Rn be an open set and suppose f : U —> Rn is C1, f =
(J1,.. .,fn). Assume x0 G U, z0 = f(xo)-
Notation. Remember from §A.4 that we write
/ f1 f1 \
DI =	' •. I = gradient matrix of f.
\ fn	fn	)
' h\ ’  • Jxn ' nxn
□
DEFINITION.
Jf = Jacobian off = |detDf| =
Э(Л ,/”)
d(xi,...,zn)
THEOREM 7 (Inverse Function Theorem). Assume f € C1((7;Rn) and
Jf(xo) ± 0.
Then there exist an open set V G U, with xq G V, and an open set W C R”,
with zq G W, such that
(i) the mapping
f : V -> W
is one-to-one and onto, and
(ii) the inverse function
f1 : W V
APPENDIX C: CALCULUS FACTS
633
is C1.
C.6. Implicit Function Theorem.
Let n, m be positive integers.
Notation. We write a typical point in Rn+m as
(x,y) =
for x e Rn, у € Rm.	□
Let U C Rn+m be an open set and suppose f : U —> Rm is C1, f =
(f1,-  Assume (хо,Уо) eU, z0 = f(x0,y0)-
Notation.

rm fm
Xn Jy\
Dt =

rm
Ут / mx(n-bm)
= (Dxf, Dyf) = gradient matrix of f.
□
DEFINITION.
Jyf = | det Dyf | =
д(У1,...,Ут')
634
APPENDICES
THEOREM 8 (Implicit Function Theorem). Assume f G C*1(6r;Rm) and
J^f (rro, уо) ф 0.
Then there exists an open set V C U, with (a?o, yo) € V, an open set W C
Rn, with xq E W, and a Cl mapping g : W —> Rm such that
(i)	g(a:o) = yo,
(ii)	f(z,g(z)) = z0 (x G IV),
and
(iii)	if (ж, y) G V and f(x, y) = zq, then у = g(x).
(iv)	If f G Ck, then g E Ck (k = 2,...).
The function g is implicitly defined near xo by the equation f(x, y) = zq.
C.7. Uniform convergence.
We record here the Arzela-Ascoli compactness criterion for uniform con-
vergence:
APPENDIX D: LINEAR FUNCTIONAL ANALYSIS
635
Suppose that {fk}kLi is a sequence of real-valued functions defined on
Rn, such that
\fk(x)\<M	= zer)
for some constant M, and the {fkYjf=i are uniformly equicontinuous. Then
there exists a subsequence {f^YfLi Q {A}£i and a continuous function f,
such that
fkj —> f uniformly on compact subsets of Rn.
To say the {fk}kLi are uniformly equicontinuous means that for each £ > 0,
there exists 6 > 0, such that |z — y| < 6 implies |Д(х) — Д(у)| < £, for
z,yeRn, fc = l,....
APPENDIX D: LINEAR FUNCTIONAL ANALYSIS
D.l. Banach spaces.
Let X denote a real linear space.
DEFINITION. A mapping || || : X —> [0, oo) is called a norm if
(i)	||u + v|| < ||u|| + ||u|| for all u,v e X.
(ii)	||Au|| = |A|||u|| for all и € X, A € R.
(iii)	||u|| = 0 if and only ifu = 0.
Inequality (i) is the triangle inequality.
Hereafter we assume X is a normed linear space.
DEFINITION. We say a sequence	С X converges to и € X,
written
Uk —> u,
if
lim \\uk — u|| = 0.
fc—ЮО
DEFINITIONS, (i) A sequence	С X is called a Cauchy sequence
provided for each e > 0 there exists N > 0 such that
— u/|| < e for all к, I > N.
(ii)	X is complete if each Cauchy sequence in X converges; that is, when-
ever	is a Cauchy sequence, there exists и E X such that
converges to u.
(iii)	A Banach space X is a complete, normed linear space.
636
APPENDICES
DEFINITION. We say X is separable if X contains a countable dense
subset.
Examples, (i) Lp spaces. Assume U is an open subset of Rn, and 1 < p <
oo. If f : U —> R is measurable, we define
fesssup^l/l ifp = oo.
We define LP(U) to be the linear space of all measurable functions f :
U —> R for which ||/||lp((7) < oo. Then 1/(11} is a Banach space, provided
we identify two functions which agree a.e.
(ii) Holder spaces. See §5.1.
(iii) Sobolev spaces. See §5.2.	□
D.2. Hilbert spaces.
Let H be a real linear space.
DEFINITION. A mapping ( , ) : H x H —> R is called an inner product
if
(i)	(u, v} = (y, u} for all u,v G H,
(ii)	the mapping и (u, v} is linear for each vEH,
(iii)	(и, и} > 0 for all и G H,
(iv)	(u, u) = 0 if and only ifu = 0.
Notation. If ( , ) is an inner product, the associated norm is
(1)	IM| := (u,u)1/2 (ибЯ).
□
The Cauchy-Schwarz inequality states
(2)	|(u,u)|<MM (u,v€H).
This inequality is proved as in §B.2. Using (2), we easily verify (1) defines
a norm on H.
DEFINITION. A Hilbert space H is a Banach space endowed with an
inner product which generates the norm.
Examples, a. The space L2(U} is a Hilbert space, with
(f,g} = [ fgdx.
Ju
APPENDIX D: LINEAR FUNCTIONAL ANALYSIS
637
b. The Sobolev space H1(17) is a Hilbert space, with
(/,5) = [ fg + Df  Dgdx.
JU
DEFINITIONS, (i) Two elements u,v G H are orthogonal if (u, v) = 0.
(ii) A countable basis {wfc}^=1 С H is called orthonormal if
r(wfe,w/) = 0 (k,l = 1,...; к ф I)
t ||Wfc|| = 1 (fc = l,...).
If и G H and	С H is an orthonormal basis, we can write
ОС
u = 52(w,wfe)wfc,
fc=l
the series converging in H. In addition
OO
imi2 = 52(w,wfc)2.
fc=i
DEFINITION. If S is a subspace of H, S1 = {u G H | (u, v) = 0 for all
v G S} is the subspace orthogonal to S.
D.3. Bounded linear operators.
a. Linear operators on Banach spaces.
Let X and Y be real Banach spaces.
DEFINITIONS, (i) A mapping A : X —>Y is a linear operator provided
A [Au + pv] = XAu + pAv
for all u,v G X, X, p G R.
(ii) The range of A is R(A) := {u G Y | v = Au for some и G X} and
the null space of A is N(A) := {u G X | Au = 0}.	□
DEFINITION. A linear operator A : X —> У is bounded if
||A|| := sup{||Au||y I ||u||x < 1} < 00.
It is easy to check that a bounded linear operator A : X —> Y is contin-
uous.
638
APPENDICES
DEFINITION. A linear operator A : X —> У is called closed if whenever
Uk —> и in X and Auk —> v in Y, then
Au — v.
THEOREM 1 (Closed Graph Theorem). Let A : X —> Y be a closed,
linear operator. Then A is bounded.
DEFINITIONS. Let A : X —> X be a bounded linear operator.
(i) The resolvent set of A is
p(A) = {77 G R I (A — rjl) is one-to-one and onto}.
(ii) The spectrum of A is
<r(A) = R — p(A).
If 77 G p(A), the Closed Graph Theorem then implies that the inverse
(A — 77/) ' : X —> X is a bounded linear operator.
DEFINITIONS, (i) We say 77 G <7(A) is an eigenvalue of A provided
N(A - gl) + {0}.
We write <xp(A) to denote the collection of eigenvalues of A; <rp(A) is the
point spectrum.
(ii) If 77 is an eigenvalue and w 0 satisfies
Aw = gw,
we say w is an associated eigenvector.
DEFINITIONS, (i) A bounded linear operator и* : X —> R is called a
bounded linear functional on X.
(ii) We write X* to denote the collection of all bounded linear functionals
on X; X* is the dual space of X.
DEFINITIONS, (i) If и G X, u* G X* we write
(u*,u)
to denote the real number u*(u). The symbol ( , ) denotes the pairing of X*
and X.
(ii) We define
||u*|| := sup{(u*, u) I ||u|| < 1}.
(iii) A Banach space is reflexive if (X*)* = X. More precisely, this
means that for each и** E (X*)*, there exists и E X such that
(u**,u*) = {u*,u) for all и* E X*.
APPENDIX D: LINEAR FUNCTIONAL ANALYSIS
639
b. Linear operators on Hilbert spaces.
Now let H be a real Hilbert space, with inner product ( , ).
THEOREM 2 (Riesz Representation Theorem). H* can be canonically
identified with H; more precisely, for each u* G H* there exists a unique
element и E H such that
(u*,v) = (u,v) for all v E H.
The mapping u* *—> и is a linear isomorphism of H* onto H.
DEFINITIONS, (i) If A : H —> H is a bounded, linear operator, its
adjoint A* : H —> H satisfies
(Au, v) = (u, A*v)
for all u,v E H.
(ii) A is symmetric if A* = A.
D.4. Weak convergence.
Let X denote a real Banach space.
DEFINITION. We say a sequence	С X converges weakly to
и E X, written
Uk u,
(u*,uk) -> (u*,u)
for each bounded linear functional и* E X*.
It is easy to check that if u^ —> u, then Uk —u. It is also true that any
weakly convergent sequence is bounded. In addition, if u^ u, then
||u|| < liminf ||ufc||.
k~+OO
THEOREM 3 (Weak compactness). Let X be a reflexive Banach space
and suppose the sequence	С X is bounded. Then there exists a
subsequence	C	and uE X such that
U)qj 1 u.
In other words, bounded sequences in a reflexive Banach space are
weakly precompact. In particular, a bounded sequence in a Hilbert space
contains a weakly convergent subsequence.
Mazur’s Theorem asserts that a convex, closed subset of X is weakly
closed.
640
APPENDICES
IMPORTANT EXAMPLE. We will most often employ weak conver-
gence ideas in the following context. Take U C IR" to be open, and assume
1 < p < oo. Then
the dual space of X = IX(U) is X* = Lq(U},
where | = 1, 1 < q < oc. More precisely, each bounded linear functional
on IX(17) can be represented as f >—> gf dx for some g 6 Lq(U). Therefore
fk~^ f weakly in LP(U)
means
(3)	/ gff, dx—> / gf dx as к —> oo, for all g 6 Lq(U).
Ju Ju
Now the identification of Lq(U) as the dual of IX(U) shows that
IX(U) is reflexive if 1 < p < oo.
In particular Theorem 3 then assures us that from a bounded sequence in
£p(17) (1 < p < oo) we can extract a weakly convergent subsequence, that
is, a sequence satisfying (3). This is an important compactness assertion,
but note very carefully: the convergence (3) does not imply that fi- —> f
pointwise or almost everywhere. It may very well be, for example, that the
functions {fkfkLi oscillate more and more rapidly as к —> oo. See Problem
1 in Chapter 8.	□
D.5. Compact operators, Fredholm theory.
Let X and Y be real Banach spaces.
DEFINITION. A bounded linear operator
K.X^Y
is called compact provided for each bounded sequence {и/с}^5=1 С X, the
sequence {Kuk}^=1 is precompact in Y; that is, there exists a subsequence
such that {KukjYj^ converges in Y.
Now let H denote a real Hilbert space, with inner product ( , ). It is
easy to see that if a linear operator К : H —> H is compact and u,
then Kuk —> Ku.
THEOREM 4 (Compactness of adjoints). If К : H —> H is compact, so
is К* -H H.
APPENDIX D: LINEAR FUNCTIONAL ANALYSIS
641
Proof. Let	be a bounded sequence in H and extract a weakly
convergent subsequence	и in H. We will prove K*uk] —► K*u.
\\K*ukj - K*u\\2 = (K*ukj - K*u,K*[ukj - u])
= (KK*ukj - KK\ukj - u).
Now since K* is linear, K*ukj —> K*u, and so KK*ukj —> KK*x. Thus
K*ukj K*u.	3	3	□
THEOREM 5 (Fredholm alternative). Let К : H —> H be a compact
linear operator. Then
(i)	ДГ(.Г — K) is finite dimensional,
(ii)	R(I — K) is closed,
(iii)	Rfl — K) = N(I — К*)\
(iv)	N(I — K) = {0} if and only if R(J — К) = H,
and
(v)	dimN(Z-A) =dim7V(Z-A*).
Proof. 1. If dim N(I — K) = +oo, we can find an infinite orthonormal set
C N(I - K). Then
Kuk = uk (fc = l,...).
Now ||ufc-wd|2 = ||ufc||2-2(ufc, u/) + ||u(||2 = 2 if к I, and so ||Z<ufc-Kty || =
y/2 for к I. This however contradicts the compactness of K, as {Kuk}f=1
would then contain no convergent subsequence. Assertion (i) is proved.
2.	We next claim there exists a constant 7 > 0 such that
(4)	||u — Ku\\ > 7||u|| for all и E N(I — ZC)1.
Indeed, if not, there would exist for к = 1,... elements uk E N(J — K)1
with ||ufc|| = 1 and — ATtk|| < Consequently
(5)	uk - Kuk -> 0.
But since	is bounded, there exists a weakly convergent subsequence
ukj u. By compactness Kukj —> Ku, and then (5) implies ukj —> u. We
therefore have и E N(I — K) and so
(ukj,u) = 0 (> = !,...).
Let kj —> 00 to derive a contradiction to (4).
642
APPENDICES
3.	Next let	C R(I — K), vk —> v. We can find uk E N(I — A')1
solving Uk — Kuk — Vk- Using (4) we deduce
||vfc - vi\\ > 7IH - uj|.
Thus Uk —> и and и — Ku = v. This proves (ii).
4.	Assertion (iii) is now a consequence of (ii) and the general fact
7?(A) = A^A*)1 for each bounded linear operator A : H —> H.
5.	To verify (iv), let us suppose to start with that N(I — K) = {0}, but
= (/ — K)(H) С H. According to (ii) Hi is a closed subspace of H.
Furthermore H2 = (/ — K)(Hi) C Hi, since I — К is one-to-one. Similarly if
we write Hk = (I — K)k(H) (k = 1,...), we see that Hk is a closed subspace
ofH, Hk+i CHk (k = l,...).
Choose uk E Hk with |fyfc|| — 1, uk E H^r+1. Then Kuk — Kui =
—(uk — Kuk) + (ui — Kui) + (uk — ui). Now if к > I, Hk+i C Hk Q Hi+1 C Hi.
Thus uk — Kuk, ui — Kui, uk E Hi+i. Since ui E Hj^, ||u/|| = 1, we deduce
— Kuzll > 1 (k, I = 1,...). But this is impossible since К is compact.
6.	Now conversely assume R(I — К) = H. Then owing to (iii) we see
that N(I — K*) = {0}. Since K* is compact, we may utilize step 5 to
conclude R(I - К*) = H. But then AT(7 - К) = 7?(/ - AT*)1 = {0}. This
conclusion and step 5 complete the proof of assertion (iv).
7.	Next we assert
dimNfy-A) > dim7?(7 — AT)1.
To prove this, suppose instead dim N(I — K) < dim R(I — K)^. Then there
exists a bounded linear mapping A : N(I — K) —> R(I — A')1 which is one-
to-one, but not onto. Extend A to a linear mapping A : H —> R(I — A')1 by
setting Au = 0 for и E N(I — A')1. Now A has a finite dimensional range
and so A, and thus К + A, are compact. Furthermore N(I — (K + A)) =
{0}. Indeed, if Ku + Au = u, then и — Ku = Au E R(I — A')1; whence
и — Ku = Au = 0. Thus и E N(I — K) and so in fact и = 0, since A is
one-to-one on N(I — K). Now apply assertion (iv) to AT = К + A. We
conclude R(I — (К + A)) = H. But this is impossible: if v E R(I — A')1,
but v 7?(A), the equation
и — (Ku + Au) = v
has no solution.
APPENDIX D: LINEAR FUNCTIONAL ANALYSIS
643
8.	Since R(I — К*')‘ = N(I — K), we deduce from step 7
dimN(7 - K*) > dim R(I -K*)1
= dim N(I - K).
The opposite inequality comes from interchanging the roles of К and K*.
This establishes (v).	□
Remark. Theorem 5 asserts in particular either
f for each f E H, the equation и — Ku = f
(a)	{ ,	.	, .
[ has a unique solution
or else
( the homogeneous equation и — Ku = 0
[ has solutions и / 0.
This dichotomy is the Fredholm alternative. In addition, should (/3) obtain,
the space of solutions of the homogeneous problem is finite dimensional, and
the nonhomogeneous equation
(7)	u- Ku = f
has a solution if and only if f E N(J — K*)1.	□
Now we investigate the spectrum of a compact linear operator.
THEOREM 6 (Spectrum of a compact operator). Assume dimH = 00
and К : H —> H is compact. Then
(i) 0 E a(K),
(ii) cr(K) - {0} = crp(K) - {0},
and
( <t(K) — {0} is finite, or else
( crfiK) — {0} is a sequence tending to 0.
Proof. 1. Assume 0 cr(A). Then К : H —> H is bijective and so
I = К о К-1, being the composition of a compact and a bounded linear
operator, is compact. This is impossible, since dim H = 00.
2. Assume rj E <t(K~), rj 0. Then if N(K — rjl) = {0}, the Fredholm
alternative would imply R(K — gl) = H. But then 77 £ p(K), a contradic-
tion.
3. Suppose now {%}^=1 is a sequence of distinct elements of <r(A) — {0},
and 7]k —► T We will show 77 = 0.
644
APPENDICES
Indeed, since Vk € &p(K) there exists Wk 0 such that Кищ = г]к^к- Let
Hk denote the subspace of H spanned by {wi,..., w^}. Then Hk Q Hk+i
for each к = 1, 2,..., since the	are linearly independent.
Observe also (К — т]кГ)Нк Q Нк - \ (к = 2,...). Choose now for к =
1,... an element Uk E Hk, with Uk E H^_r and ||ufc|| = 1- Now if к > I,
H^ CHtC Hk-i С Hk. Thus
Кик _ Кщ
Vk Vi
{Kuk - T)kUk) {Кщ - тцщ)
------------------------------\-Uk~Ul
Vk	VI
since Kuk — VkUk, Кщ — Viui, ul € If Vk —* V 0, we obtain a
contradiction to the compactness of K.	□
D.6. Symmetric operators.
Now let S : H —> H be bounded, symmetric, and write
m := inf {Su, u), M := sup {Su, u).
llull=1	||u||=l
LEMMA (Bounds on spectrum). We have
(i) сг(5) C [m, M], and
(ii) m, M e o’(S').
Proof. 1. Let v > M. Then
(щи — Su, и) > (77 — M)||u||2 (u £ H).
Hence the Lax-Milgram Theorem (§6.2.1) asserts 77I — S is one-to-one and
onto, and thus 77 6 p{S). Similarly 77 £ p(S) if p < m. This proves (i).
2. We will prove M £ сг(5). Since the pairing [u,v] := {Mu — Su,v)
is symmetric, with [u, u] > 0 for all и £ H, the Cauchy-Schwarz inequality
implies
\{Mu — Su,v)\ < (Mu-Su,u)1/2(Mv- Sv,v)1/2
for all u,v £ H. In particular
(6)	\\Mu - Su\\ < C{Mu - Su, u)1/2 (и £ H)
for some constant C.
Now let {ua }^-] С H satisfy ||ufc|| = 1 (& = !,...) and {Suk, uk) —> M.
Then (6) implies ||A4ufc — Su/JI —> 0. Now if M £ p(S), then
uk = {MI - S)~1{Muk - Suk) 0,
a contradiction. Thus M £ cr(S), and likewise m £ сг(5).	□
APPENDIX E: MEASURE THEORY
645
THEOREM 7 (Eigenvectors of a compact, symmetric operator). Let H
be a separable Hilbert space, and suppose S : H —> H is a compact and
symmetric operator. Then there exists a countable orthonormal basis of H
consisting of eigenvectors of S.
Proof. 1. Let {gk} comprise the sequence of distinct eigenvalues of S,
excepting 0. Set go = 0. Write Hq = N(S), Hk = N(S — gkl) (fc = 1,...).
Then 0 < dim До < oo, and 0 < dim Hk < oo, according to the Fredholm
alternative.
2.	Let и E Hk, v E Hi for к I. Then Su = gkU, Sv = ryu and so
gk(u, v) = (Su,v) = (u, Sv) = gi(u, v).
As gk / gi, we deduce (u,v) = 0. Consequently we see the subspaces Hk
and Hi are orthogonal.
3.	Now let H be the smallest subspace of H containing Ho,H\,... .
Thus H = {)>2^=o akuk | m E {0,... }, rtfc G Hk, ak E IR}. We next demon-
strate H is dense in H. Clearly S(H) С H. Furthermore S(HV) С H!:
indeed if и E H^~ and v E H, then (Su, v) = (u, Sv) = 0.
Now the operator S = is compact and symmetric. In addition
cr(S) = {0}, since any nonzero eigenvalue of S would be an eigenvalue of S
as well. According to the lemma then, (Su,u) = 0 for all и E HL. But if
u, v E H\
2(Su, v) = (S(u + v),u + v) — (Su, u) — (Sv, v) = 0.
Hence S = 0. Consequently H1 C N(S) С H, and so H1 = {0}. Thus H
is dense in H.
4.	Choose an orthonormal basis for each subspace Hk (k = 0,...),
noting that since H is separable, Hq has a countable orthonormal basis. We
obtain thereby an orthonormal basis of eigenvectors.	□
Most of these proofs are from Brezis [BRI]. See also Gilbarg-Trudinger
[G-T, Chapter 5] and Yosida [Y],
APPENDIX E: MEASURE THEORY
This appendix provides a quick outline of some fundamentals of measure
theory.
E.l. Lebesgue measure.
Lebesgue measure provides a way of describing the “size” or “volume”
of certain subsets of IR".
646
APPENDICES
DEFINITION. A collection Л4 of subsets ofW1 is called a tr-algebra if
(i)	0, Rn e M,
(ii)	A E Nt implies Rn — A G Л4,
and
(iii)	if{Ak}^=1 с Nt, then иГ=1 Ak, ПГ=1 Ak G Nt.
THEOREM 1 (Existence of Lebesgue measure and Lebesgue measurable
sets). There exist a a-algebra Nt of subsets ofW1 and a mapping
| | : Л4 —> [0, +00]
with the following properties:
(i)	Every open subset ofW1, and thus every closed subset of№n, belong
to Nt.
(ii)	If В is any ball in R", then |B| equals the n-dimensional volume of
(iii)	If {Ak}^! С Л4 and the sets {Afc}*^ are pairwise disjoint, then
(1)
Ак = УУ |Afc| (“countable additivity”).
fc=i	fc=i
(iv)	If A С B, where В E Nt and |B| = 0, then A E Nt and | A\ = 0.
Notation. The sets in Nt are called Lebesgue measurable sets and | • | is
n-dimensional Lebesgue measure.
Remarks, (i) From (ii) and (iii), we see that |A| equals the volume of any
set A with piecewise smooth boundary.
(ii) We deduce from (1) that
(2)
|0| = 0,
and
(3)
Ад- < УУ IAfc|	(“countable subadditivity”)
fc=l k=l
for any countable collection of measurable sets {A^}^
Notation. If some property holds everywhere on Rn, except for a mea-
surable set with Lebesgue measure zero, we say the property holds almost
everywhere, abbreviated “a.e.”.	□
APPENDIX E: MEASURE THEORY
647
E.2. Measurable functions and integration.
DEFINITION. Let f : Rn —> R. We say f is a measurable function if
Г\и)ЕМ
for each open subset U C R.
Note in particular that if f is continuous, then f is measurable. The
sum and product of two measurable functions are measurable. In addition
if {fk}^=Q are measurable functions, then so are limsup/k and liminf Д.
THEOREM 2 (Egoroff’s Theorem). Let {fk}kLi,f be measurable func-
tions, and
fk~> f a.e. on A,
where A C Rn is measurable, |A| < oo. Then for each e > 0 there exists a
measurable subset E C A such that
(i) \A — E\<e
and
(ii) fk~>f uniformly on E.
Now if / is a nonnegative, measurable function, it is possible, by an
approximation of / with simple functions, to define the Lebesgue integral
[ f dx.
JR"
Cf. §E.5 below. This agrees with the usual integral if f is continuous or
Riemann integrable. If f is measurable, but is not necessarily nonnegative,
we define
[ f dx = f f+ dx — [ f~ dx,
JRn JRn	JRn
provided at least one of the terms on the right hand side is finite. In this
case we say f is integrable.
DEFINITION. A measurable function f is summable if
[ \f\dx < oo.
JR"
Remark. Note carefully our terminology: a measurable function is inte-
grable if it has an integral (which may equal +oo or —oo) and is summable
if this integral is finite.	□
648
APPENDICES
Notation. If the real-valued function f is measurable, we define the essen-
tial supremum of / to be
ess sup f := inf{/r £ IR | \{f > /r}| = 0}.
□
E.3. Convergence theorems for integrals.
The Lebesgue theory of integration is especially useful since it provides
the following powerful convergence theorems.
THEOREM 3	(Fatou’s Lemma). Assume the functions {fk}kLi>f are
nonnegative and summable. Then
[ lim inf fk
dx < lim inf / fkdx.
k—°° JR"
THEOREM 4	(Monotone Convergence Theorem). Assume the functions
{fk}k^i are measurable, with
fi < fi < • • • < fk < fk+i < • • • 
Then
/ lim fk dx = lim / fk
JR" к—юо	k—>oc jRn
dx.
THEOREM 5	(Dominated Convergence Theorem). Assume the functions
are integrable and
fk—' f a.e.
Suppose also
\fk\ < 9 a.e.,
for some summable function g. Then
[ fkdx -> [ f dx.
JR"	JR"
E.4. Differentiation.
An important fact is that a summable function is “approximately con-
tinuous” at almost every point.
APPENDIX E: MEASURE THEORY
649
THEOREM 6	(Lebesgue’s Differentiation Theorem). Let f : IR" —> IR be
locally summable.
(i) Then for a.e. point xq E IR",
У fdx^>f(x0) as г—O).
J B(x0,r)
(ii) In fact, for a.e. point xq E IR",
(4)	-Z |/(x) - /(x0)| dx -> 0 as r -> 0.
J B(x0,r)
A point xq at which (4) holds is called a Lebesgue point of f.
Remark. More generally, if f E L^JW1) for some 1 < p < oo, then for a.e.
point xo E IRn we have
У |/(x) - /(x0)|pdx -> 0 as r —> 0.
J B(x0,r)
□
E.5. Banach space-valued functions.
We extend the notions of measurability, integrability, etc. to mappings
f : [0, T] -+ X
where T > 0 and X is a real Banach space, with norm || ||.
DEFINITIONS, (i) A function s : [0, T] —> X is called simple if it has
the form
m
(5)	s(t) = XEi^i (0 < t < T),
2 = 1
where each Ei is a Lebesgue measurable subset of [0, T] and Ui E X (i =
1,..., m).
(ii	) A function f : [0, T] —+ X is strongly measurable if there exist simple
functions Sk : [0, T] —> X such that
sfc(f) —► f(t)	for a.e. 0 < t < T.
(ii	i) A function f : [0, T] X is weakly measurable if for each u* EX*,
the mapping t ь-> (u*,f(t)) is Lebesgue measurable.
650
APPENDICES
DEFINITION. We say f : [0, T] —> X is almost separably valued if there
exists a subset N С [0, T], with |7V| = 0, such that the set {f(t) | t G
[0, T] — N} is separable.
THEOREM 7 (Pettis). The mapping f : [0, T] —+ X is strongly measurable
if and only if f is weakly measurable and is almost separably valued.
DEFINITIONS, (i) 7/s(t) = £Х1 ХеА^щ is simple, we define
(6)
m
dt := \Ei\ui.
i=l
(ii) We say f : [0, T] —> X is summable if there exists a sequence {sA.})Y1
of simple functions such that
(7)
||sfc(t) — f(t)|| dt —» 0 as к —> oo.
(iii) Iff is summable, we define
(8)
fT
dt = lim / Sfc(t)dt.
Jo
THEOREM 8 (Bochner). A strongly measurable function f : [0, T] —> X
is summable if and only if t ||f (t)|| is summable. In this case
< Г \m\\dt,
Jo
and
u*, f f(t)dt\= I (u*,f(t))dt
Jo / Jo
for each и* E X*.
A good book for measure theory is Folland [F2], See Yosida [Y, Chap-
ter V, Sections 4-5] for proofs of Theorems 7, 8.
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INDEX
a priori estimates, 504
action functional, 116
adjoint, 639
bilinear form, 302
of elliptic operator, 302
admissible states, 589
Airy’s equation, 4, 173
almost everywhere (a.e.), 646
almost separably valued function, 650
amplitude, 207
analyticity, 226-233
of harmonic functions, 31
of solutions to heat equation, 62
of solutions to PDE, 229
Arzela-Ascoli compactness criterion, 274, 455,
540, 634
asymptotics, 157, 199-221
averages, 616
backwards uniqueness
heat equation, 63
Banach space, 241, 249, 635
Barenblatt’s solution, 182
barrier, 346
basis
of eigenfunctions, 169, 335, 353, 360, 363,
364, 388, 390, 391, 493
orthonormal, 637
beam equation, 4
Bernoulli’s law, 196
Bessel potential, 187
biharmonic equation, 345
bilinear form, 296
estimates for, 299-301
time-dependent, 351, 378, 403
bistable equation, 175
blow-up
conservation law, 597
nonlinear heat equation, 513
porous medium equation, 171
Bochner’s Theorem, 650
boundary, 626
Ck, 626
analytic, 626
normal to, 626
straightening, 103, 224, 228, 320, 325, 627
boundary data
admissible, 106
compatibility conditions on, 104
noncharacteristic, 104, 106
boundary-value problem
eigenvalue problem, 334
homogenization of, 218
Laplace’s equation, 41
nonlinear boundary conditions, 489
nonlinear eigenvalue problem, 464
nonlinear Poisson equation, 507, 518
Poisson’s equation, 27, 42
quasilinear elliptic equation, 491, 505
second-order elliptic equation, 293, 296
semilinear elliptic equation, 482, 514
Browder-Minty method, 497, 540
Burgers’ equation, 5, 140, 141, 143
vanishing viscosity, 204
viscous, 195
calculus of variations, 11, 43, 117, 123, 431-
490
Cauchy data, 222
Cauchy problem
for boundary data, 222
for heat equation, 47
Cauchy sequence, 241, 249, 635
655
656
INDEX
Cauchy-Kovalevskaya Theorem, 10, 228-233
chain rule, 291
characteristic equation, 398
characteristics, 10, 19, 97-115, 152, 200, 217,
399
and boundary conditions, 103—106
and local solution to PDE, 106-110
applications of, 110-115
crossing, 112
derivation, 97
examples of, 99-102
projected, 98
summary of characteristic ODE, 98
Clairaut’s equation, 93
Closed Graph Theorem, 417, 638
coarea formula, 396, 628
coercivity, 443, 453
cofactor matrix, 411, 440
compact
linear operator, 640
mapping, 503
compact embeddings, 271
complete integral, 92
complete linear space, 635
condition E, 612
conservation law, 5, 8, 10, 136-162
characteristics for, 112, 143
decay in sup-norm, 157
decay to N-vtave, 158
derivation, 567
implicit solution, 114
initial-value problem, 136
integral solution, 137, 148
Lax-Oleinik formula, 146
Rankine—Hugoniot condition, 140
systems, 6, 12, 567-606
viscous, 234
constitutive relation, 569, 578, 595
constraints
incompressibility, 472
integral, 463
one-sided, 467
point wise, 470
contraction, 499, 501, 529
semigroup, 414
control theory, 551
controls
admissible, 551
optimal, 11, 560
converting nonlinear into linear equations,
194-199
convex function, 523, 621
domain, 523
subdifferential of, 163, 523
supporting hyperplane, 621
convexity, 447
uniform, 131, 143, 144, 621
corrector problem
for Green’s function, 34
for homogenization, 221
cost, 552
running, 552
terminal, 552
Courant minimax principle, 347
Crandall, М.-Lions, P.-L., 564
critical
point, 118, 432, 476-481
value, 477
critical exponent, 486, 514
curvature, 201
mean, 435, 489
principal curvatures, 216
cutoff function, 310
d’Alembert’s formula, 68, 69, 78, 82, 88
Deformation Theorem, 477, 481
DeGiorgi—Nash estimates, 462
degree, 487
Derrick-Pohozaev identity, 517
difference quotients, 277—279
and weak derivatives, 277
definition of, 277
differential equations
in Banach spaces, 413
in Hilbert spaces, 479, 528
diffusion, 200
and transport, 199-204
equation, 4, 44
DiPerna, R., 165
Dirac measure, 25, 35, 48, 200, 201
Dirichlet’s boundary condition, 294
Dirichlet’s principle, 42, 434
generalized, 434
dispersion, 173
dissipation, 173
divergence form PDE, 294, 350
domain of dependence
generalized wave equation, 395
wave equation, 84
Dominated Convergence Theorem, 134, 154,
211, 412, 452, 510, 606, 648
doubling variables, 547, 608
Douglis, A., 165
dual space, 638
of Hl, 283
duality of Hamiltonian and Lagrangian, 122
Duhamel’s principle, 49, 81
dynamic programming, 11, 551, 552
Egoroff’s Theorem, 446, 647
eigenfunction, 169, 305
basis, 335
INDEX
657
eigenvalue, 11, 169, 305, 334, 574, 575, 638
nonsymmetric elliptic operator, 340-344
principal, 11, 336-344, 512, 537
symmetric elliptic operator, 334-340
eigenvector, 334, 575, 638
left, 574
right, 574
eikonal equation, 5, 93, 96, 564
generalized, 395
elasticity
linear, 6
nonlinear, 457, 475
elliptic regularization, 487
ellipticity, 294
energy estimates
quasilinear elliptic equation, 494
second-order elliptic equation, 299
second-order hyperbolic equation, 381
second-order parabolic equation, 354
symmetric hyperbolic system, 405
energy methods
heat equation, 62-65
Laplace’s equation, 41-43
wave equation, 83-85
entropy condition, 12, 143, 149, 589, 599
Lax’s, 589
Liu’s, 602
entropy solution
single conservation law, 150
system, 604, 607
entropy/entropy-flux pair, 604, 607
envelopes, 94
construction of solutions using, 94
equipartition of energy, 88
essential supremum (ess sup), 648
Euler equations, 569, 578
barotropic, 595, 611
incompressible, 6, 196
Euler’s formula for Jacobians, 476
Euler-Lagrange equation, 434, 450
harmonic maps, 470
one-dimensional, 117
system, 454
systems, 438
Euler—Poisson-Darboux equation, 70, 75
exponential solutions, 172
extensions, 254-257
factoring PDE, 67, 398
Fatou’s Lemma, 532, 648
feedback control, 553
first variation, 433
first-order equation, 9, 91-165
complete integral, 92
notation for, 91
first-order hyperbolic system, 11, 400-412
fixed point theorem
Banach’s, 498, 500, 502, 537
Brouwer’s, 441, 493
Schaefer’s, 342, 503, 507
Schauder’s, 502
fixed-point methods, 11, 498-507
Fokker—Planck equation, 4
Fourier transform, 10, 182-190, 282-283, 292,
408
and Sobolev spaces, 282
applications, 186-190
definition of, 183, 184
properties, 183, 184
Fredholm alternative, 220, 641, 643
free boundary problem, 470
fully nonlinear PDE, 2
function spaces
BMO, 276
Ck, 618
Ck’\ 241
C2, 618
Hk, 245
Ha, 283
H'1, 283
246
Lp, 618, 636
Wk'P, 244
Wk'p, 245
Banach space-valued, 285-289
fundamental solution
heat equation, 46, 180, 188
Laplace’s equation, 22, 25
Schrodinger’s equation, 188
Galerkin’s method, 169, 353, 380, 493
Gauss-Bonnet Theorem, 488
Gauss-Green Theorem, 20, 627
general integral, 96
generator of semigroup, 414-418
genuine nonlinearity, 581, 588, 594
geometric notation, 614—615
geometric optics, 207-217, 395
Gidas, B.-Ni, W.-Nirenberg, L., 538
gradient flow, 529
Green’s formulas, 33, 34, 628
Green’s function, 33-41
derivation of, 33-35
for ball, 40
for half-space, 37
symmetry of, 35
Holder spaces, 240-241
Hadamard’s example, 234
Hamilton’s differential equations, 115, 116,
120
658
INDEX
Hamilton-Jacobi equation, 5, 10, 11, 94, 96,
115-136, 171, 395, 539
characteristics for, 114, 116, 123
Hopf-Lax formula, 124
semiconcavity condition, 130
solution a.e., 128
uniqueness, 132
weak solution, 132
Hamilton—Jacobi—Bellman equation, 557
and optimal controls, 560
Hamiltonian, 119, 557
duality with Lagrangian, 122
harmonic function, 20
harmonic mapping, 470
heat ball, 52
heat equation, 4, 9, 44—65, 172
strong maximum principle, 54
backwards uniqueness, 63
derivation, 44
energy methods, 62—65
fundamental solution, 46
infinite propagation speed, 49
initial-value problem, 47
maximum principle, 57
mean-value formula, 52
nonhomogeneous problem, 44, 49
nonlinear, 511
regularity, 59, 61
uniqueness, 56, 58
Helmholtz’s equation, 3, 305
Hilbert space, 636
Hille-Yosida Theorem, 418, 421, 423
hodograph transform, 197
homogeneous PDE, 2
homogenization, 218-221, 235
Hopf’s Lemma, 330, 333, 341, 343, 344, 519
Hopf-Cole transformation, 195
Hopf-Lax formula, 10, 121-136, 539, 560-
563
Huygens’ principle, 80
hyperbolic system
constant coefficients, 408-412
nonlinear, 573
symmetric, 402-408
hyperbolicity, 398, 401
for symmetric system, 401
strict, 401, 573, 578, 595
uniform, 403
hyperelastic materials, 457
Implicit Function Theorem, 105, 212, 217,
465, 518, 576, 577, 585, 633
incompressibility, 472-476
inequality
Cauchy’s, 622
with e, 622
Cauchy-Schwarz, 624, 636
Gagliardo-Nirenberg-Sobolev, 10, 262-266
general Sobolev-type, 269
Gronwall’s
differential form, 624
integral form, 625
Holder’s, 622
Harnack’s, 32, 86, 333, 370
interpolation, 290
Jensen’s, 621
Minkowski’s, 249, 623
Morrey’s, 10, 266-269
Poincare’s, 265, 275-276, 448
Young’s, 622
with e, 622
initial-value problem
Burgers’ equation, 140
conservation law, 136, 144, 150, 568, 604,
606
Hamilton-Jacobi equation, 115, 124, 129,
135, 539, 542, 546, 550, 560
heat equation, 47, 188, 191
quasilinear parabolic equation, 194, 540
Schrodinger’s equation, 188
telegraph equation, 190
transport equation, 18, 85
viscous Burgers’ equation, 195
wave equation, 67, 189, 192, 208
initial/boundary-value problem
beam equation, 427
heat equation, 57, 88, 168
parabolic equation, 511
quasilinear parabolic equation, 535
reaction-diffusion system, 499
second-order hyperbolic equation, 378
second-order parabolic equation, 350
wave equation, 69, 83
inner product, 636
integral solution, 137, 148, 570, 607
integration by parts formula, 627
interior ball condition, 330, 519
Inverse Function Theorem, 106, 197, 198,
591, 592, 632
inversion through sphere, 39
irreversibility, 189, 538
irrotational fluid equations, 197
fc-contact discontinuity, 588
fc-rarefaction wave, 582
fc-shock wave, 589
fc-simple wave, 580
Kirchhoff’s formula, 73
Kolmogorov’s equation, 4
Korteweg-deVries (KdV) equation, 5, 174
Kovalevskaya’s example, 235
Kruzkov, S., 612
INDEX
659
Lagrange multiplier
as eigenvalue, 464
as function, 471
integral constraints, 464
pressure as, 472
Lagrangian, 116, 432, 437
duality with Hamiltonian, 122
null, 439, 441, 487, 488
Laplace transform, 191-194, 417
applications, 191-194
definition of, 191
Laplace’s equation, 3, 9, 20-43, 295
analyticity, 31
derivation, 20
Green’s function, 33-41
Harnack’s inequality, 32
Liouville’s Theorem, 30
mean-value formulas, 25
regularity, 28, 29
Laplace’s method, 204
Lax-Milgram Theorem, 297, 299, 301, 345,
483, 644
Lax-Oleinik formula, 10, 144-162, 206
Lebesgue measure, 645
Lebesgue point, 186, 649
Lebesgue's Differentiation Theorem, 281, 630,
648
Legendre transform, 121, 122, 561
classical, 198
Leibniz’ formula, 12, 247
linear degeneracy, 581, 587
linear operator, 637
bounded, 637
closed, 638
symmetric, 335, 639
linear PDE, 2
Liouville’s
equation, 3
Theorem, 30
Lipschitz continuity, 616
and differentiability a.e., 279
and weak partial derivatives, 279
lower semicontinuity, 523
weak, 445, 525
majorants, 227, 231
maximum principle, 10
Cauchy problem for heat equation, 57
strong, 339, 341
heat equation, 54
Laplace’s equation, 27
second-order elliptic equation, 330-333
second-order parabolic equation, 375-
377
weak
second-order elliptic equation, 327-329
second-order parabolic equation, 368-
370
Maxwell's equations, 6, 88
Mazur’s Theorem, 449, 525, 639
mean-value formulas
heat equation, 52
Laplace’s equation, 25, 26
measurable function, 647
method of descent, 74, 79
minimal surface equation, 5, 198, 435
minimax methods, 347, 480
minimizer, 433, 438, 443
mollifier, 250, 629
momentum, generalized, 119
Monge-Ampere equation, 5
Monotone Convergence Theorem, 447, 648
monotone vector field, 492
monotonicity, 11, 524
strict, 497
Morse Lemma, 212
Mountain Pass Theorem, 480, 485
moving plane method, 521-522
multiindices, 12, 617
Multinomial Theorem, 12, 227
A'-wave, 159
Navier-Stokes equations, 6
Neumann boundary conditions, 345, 425
Newton’s law, 66, 119, 569
noncharacteristic
boundary data, 104, 106
surface, 223, 224
noncharacteristic surface, 226
nondivergSnce form PDE, 294, 350
nonexistence of solutions, 511-517
nonhomogeneous problem
heat equation, 49, 51
transport equation, 19
wave equation, 81
nonlinear eigenvalue problem, 464
norm, 241, 635, 636
normal derivative, 627
normal to surface, 626
notation for estimates, 619
notation for functions, 615-619
notation for matrices, 613—614
null Lagrangian, 439, 441, 487, 488
О, о notation, 619
obstacle problem, 467
Oleinik, O., 165
optimality conditions, 553
orthogonal elements, 637
660
INDEX
p-Laplacian, 5
p-system, 568, 606
traveling waves for, 602
Palais-Smale condition, 477, 478, 480, 484
parabolic
boundary, 52
cylinder, 51
parabolic approximation, 204
parabolicity, 350
partial differential equation
definition of, 1
partition of unity, 251, 253, 256, 260, 290
penalty method, 537
Perron-Frobenius Theorem, 334
phase, 207
phase plane, 176
Plancherel’s Theorem, 183
plane wave, 172, 401
Poisson’s equation, 20, 23, 295
nonlinear, 5, 435
Poisson’s formula
ball, 41, 86
half-space, 37, 87
wave equation, 74, 80
Poisson’s kernel
ball, 41
half-space, 37
polar coordinates, 628
polyconvexity, 456
porous medium equation, 5, 170, 180
Barenblatt’s solution of, 182
potentials, 196, 198
projected characteristics, 98
propagation speed, 182
finite, 78, 84, 163, 394-397
infinite, 49, 375
proper function, 523
quasilinear PDE, 2
Rademacher’s Theorem, 128, 281
radial solution, 22
Rankine-Hugoniot condition, 140, 141, 144,
572
rarefaction curve, 580
rarefaction wave, 141, 582
Rayleigh's formula, 336
reaction-diffusion equation, 511
bistable, 175
scalar, 5
system, 6, 499
reflection, 69, 255, 521
reflexive space, 638
regularity, 8
heat equation, 59, 61
Laplace's equation, 28, 29
minimizers, 458-462
second-order elliptic equation, 308-326
second-order hyperbolic equation, 387-393
second-order parabolic equation, 358-367
Rellich-Kondrachov Compactness Theorem,
272
resolvent, 191
identity, 417
nonlinear, 526
resolvent set, 417, 638
response of system to controls, 551
retarded potential, 82
Riemann invariants, 593-599
blow-up, 597
Riemann’s problem, 12, 154-157, 579-592
local solution of, 590
Riesz Representation Theorem, 284, 298, 299
scalar conservation law, 570
Schauder estimates, 462
Schrodinger’s equation, 4, 173, 234
second variation, 436, 488
second-order elliptic equation, 10, 293-347
bilinear form for, 296
boundedness of inverse, 306
eigenvalues and eigenfunctions, 334-344
existence theorems, 301-306
Harnack’s inequality, 333
maximum principles, 326-333
regularity, 308-326
boundary, 316-326
interior, 309-316
second-order hyperbolic equation, 11, 377-
400, 402
as semigroup, 423
finite propagation speed, 394-397
in two variables, 397-400
canonical forms, 399
regularity, 387-393
weak solution of, 380-387
second-order parabolic equation, 11, 349-
377
as semigroup, 421
Harnack’s inequality, 370-374
maximum principles, 367-377
regularity, 358-367
weak solution of, 351-358
semiconcavity, 130, 131
semigroup, 191
linear, 11, 412-424
nonlinear, 11, 523, 528-536
semilinear PDE, 2
separable space, 636
separation of variables, 167-171
shallow water equations, 570, 611
shock set, 583
INDEX
661
shock wave, 8, 140, 143, 573, 583, 589, 599,
600, 604
nonphysical, 141, 589
signature of matrix, 210
similarity solutions, 45, 172-182
simple function, 649
simple waves, 579
singular integral, 95
singular perturbation, 199
Sobolev inequalities, 261-271
Sobolev space, 10, 241-292
approximation by smooth functions, 250-
254
compactness, 271-274
completeness of, 249
convergence in, 245
definition of, 244
differentiability a.e., 280
extensions, 254-257
fractional, 283
norm, 245
traces, 257-261
soliton, 175
solution
classical, 7
weak, 8, 508, 542
sound speed, 579, 595
spectrum, 305, 638
compact operator, 643
compact, symmetric operator, 644
complex, 340
point, 638
spherical means, 67, 70
star-shaped
level sets, 517
set, 514, 515
state of system, 551
state space, 568
stationary phase, 208-217
Stokes’ problem, 472
strain, stress, 578
strict hyperbolicity, 573
strongly measurable function, 649
subdiflferential, 523
subscripts, superscripts, 438, 619
subsolution, 88, 327, 346, 508, 537
summable function, 647
supersolution, 327, 508, 537
system
analytic PDE, 228
conservation laws, 567-606
constant coefficient, hyperbolic, 408-412
Euler’s equations, 196
Euler-Lagrange, 437-441, 453-457
irrotational flow, 197
ODE, 176
reaction-diffusion, 499
semilinear hyperbolic, 573
system of partial differential equations, 2
Taylor’s formula, 13, 31, 32, 226
telegraph equation, 4, 190, 426
terminal-value problem
Hamilton-Jacobi equation, 557
heat equation, 63
transport equation, 152
test function, 242
traces, 257-261, 291
transport equation, 3, 9, 18-19, 67
traveling wave, 172, 174, 176, 234
conservation law, 573, 600-602
uniqueness
backwards in time, 63
Cauchy problem for heat equation, 58
conservation law, 151, 607
Hamilton-Jacobi equation, 132-135
heat equation, 62
in calculus of variations, 449
Poisson’s equation, 27
quasilinear elliptic equation, 497
viscosity solution, 547
value function, 552
vanishing viscosity method, 204, 205, 540,
543, 605
variational formulation, 296
variational inequality, 467-470, 489, 537
version of a function, 269
viscosity solution, 539-565
definition of, 542
existence, 550
uniqueness, 546, 547
wave
N-, 159
plane, 172, 401
rarefaction, 141, 582
shock, 143, 573, 583, 589, 599, 600, 604
simple, 579
traveling, 172, 174, 176, 234, 573, 600-602
wave equation, 4, 9, 65-85, 173
d’Alembert’s formula, 68
derivation, 66
dimension
even, 78-80
odd, 74-78
one, 19, 67-69, 75
three, 71-73
two, 73-74
equipartition of energy, 88
generalized, 4
662
INDEX
wave equation (continued)
Kirchhoff’s formula, 73
method of descent, 74, 79
nonhomogeneous, 65
nonlinear, 5, 569, 578
Poisson’s formula, 74
wave speeds, 172, 401
weak continuity of determinants, 454
weak convergence, 444, 495, 540, 639
weak partial derivative, 242-254
definition of, 242
examples of, 243-244, 246-247
properties of, 247-248
uniqueness of, 243
weak solution
conservation law, 150-154, 570, 607
Euler-Lagrange equation, 450
first-order hyperbolic system, 401, 403
Hamilton-Jacobi equation, 129-136
second-order elliptic equation, 295-307
second-order hyperbolic equation, 378-387
second-order parabolic equation, 351-358
transport equation, 19
weakly measurable function, 649
well-posed problem, 7, 234
Yosida approximation, 526, 530