Lectures in
APPLIED MATHEMATICS Volume 3A
Partial Differential Equations
American Mathematical Society
LECTURES IN APPLIED MATHEMATICS Proceedings of the Summer Seminar, Boulder, Colorado, 1957
Volume I
Probability and Related Topics in Physical Sciences BY MARK KAC with G. E. UHLENBF.CK, A. R. HIBBS, and BALTH. VAN DER POL
Volume II
Fluid Mechanics By SYDNEY GOT DSTE.TN with J. M. BURGERS
Volume III
Partial Differential Equations By LIPMAN BERs, FRITL JOHN, and MARTIN SCHECHTER
with LARS GARDING and A. N. MILGRAM
Volume IV
Solid Mechanics By R. S. RIVI.IN with W. PRAGER
LECTURES IN APPLIED MATHEMATICS Proceedings of the Summer Seminar, Boulder, Colorado, 1957
AMERICAN MATHEMATICAL SOCIETY EDITORIAL COMMITTEE
ALSTON S. HOUSEHOLDER, Chairman Oak Ridge National Laboratory
MARK KAC Rockefeller University
H. J. GREENBERG International Business Machines Corporation
VOLUME III
PARTIAL DIFFERENTIAL EQUATIONS By
LIPMAN BERS
FRITZ JOHN MARTIN SCHECHTER Courant Institute of Mathematical Sciences, New York University
WITH SPECIAL LECTURES BY
Lars Girding University of Lund, Lund, Sweden
and the late A. N. Milgram University of Minnesota.
AMERICAN MATHEMATICAL SOCIETY
Providence, Rhode Island 02904
Copyright Q 1964 by John Wiley & Sons, Inc. All Rights Reserved This book or any part thereof may not be reproduced in any form without the written permission of the publisher. SECOND PRINTING, OCTOBER, 1966 BY JOHN WILEY & SONS, INC. THIRD PRINTING, 1971 BY THE AMERICAN MATHEMATICAL SOCIETY
International Standard Book Number 0-8218-0047-3 Library of Congress Catalog Card Number 63-19664
Printed in the United States of America
ARTHUR N ORTON MILGRAM
June 3, 1912-January 30, 1961
Arthur Norton Milgram died suddenly on January 30, 1961. He was a penetrating mathematician and an inspiring
teacher. The surviving authors, who were his friends, dedicate this volume to his memory.
Foreword
This is the third volume of the Proceedings of the Summer Seminar on Applied Mathematics, sponsored by the American Mathematical Society and held at the University of Colorado over the four weeks beginning June 23, 1957
The purpose of the Seminar was tutorial, to present to mature mathematicians the current status of the theory in several fields covered, and to pose for them some of the more pressing and interesting mathematical problems lying open in these fields. It could be described
as an endeavor to promote cooperation of mathematicians with theoretical physicists. The publication of these volumes is intended to extend the same information to a much wider public than was privi-
leged to actually attend the Seminar itself, while at the same time serving as a permanent reference for those who did attend. The program of the Seminar was organized by a committee of the American Mathematical Society with the following membership:
P. R. Garabedian A. S. Householder Mark Kac R. E. Langer
C. C. Lin Wm. Prager J. J. Stoker M. H. Martin, Chairman
Local arrangements, including the social and recreational program, were organized by a committee of the Department of Applied Mathematics, University of Colorado, as follows:
J. R. Britton R. Ben Kriegh L. W. Rutland
L. C. Snively
K. H. Stahl C. A. Hutchinson, Chairman
The indefatigable energy and enthusiasm of the chairmen, and the
cooperation of other members of the university staff, contributed
viii
FOREWORD
immeasurably to the successful execution of the plans for the seminar and to the enjoyment of the participants. The Seminar opened Sunday evening, June 23, with an address by Professor Richard P. Feynman, California Institute of Technology, on the subject "The Relation of Mathematics to Physics." The formal, technical sessions were held in the mornings, leaving the afternoons free for study and for holding informal discussion groups on related special topics. Several such groups were organized and met regularly throughout the period of the Seminar. A. S. HOUSEHOLDER
Preface
In June and July of 1957 we delivered lectures on the theory of partial differential equations at the Seminar on Applied Mathematics which the American Mathematical Society held at the University of
Colorado in Boulder. Mimeographed notes of these lectures were prepared at that time, and a printed version of the lectures was promised. We regret that pressure of other work delayed the publication for so long. The chapters on hyperbolic equations by F. John which appear here are an edited version of the original notes, in the preparation of which S. V. Parter gave valuable aid. The lecture notes on elliptic equations,
delivered by L. Bers, have been considerably expanded. Such an expansion would not have been accomplished had not M. Schechter agreed to assume co-authorship of this part of the book.
At the Seminar L. Girding and the late A. N. Milgram delivered two lectures, the text of which is included in this volume. No attempt has been made to achieve artificial unity of presentation
by avoiding all overlaps or by coordinating notation. The diversity of approaches, methods, and points of view is characteristic for the theory of partial differential equations. In the informal and stimulating
atmosphere of the Boulder Seminar this diversity became quite apparent; we hope that some of it is preserved in the printed version. The bibliography is far from complete; we list primarily the books and papers which we found helpful in preparing these lectures. This is neither a textbook for beginners, since most standard material is omitted, nor a reference book f o r specialists. I t is aimed, as the oral presentation was, at mathematicians not necessarily familiar with the field of partial differential equations who %%-ant to acquire an understanding of some of the problems and methods in this discipline. The technical prerequisites are very modest. A student familiar with the ix
x
PREFACE
fundamentals of real and complex function theory and with the elements of functional analysis should have no difficulty in following the presentation. LIPMAN BERS
FRITz JOHN
New York
March 1964
Acknowledgments
The authors are indebted to Anneli Lax for her help in rewriting the first part and to Alfred Schatz for the preparation of the index to that part. Concerning the second part, they would like to express their gratitude to Deborah Schechter for her help in preparing the manuscript, to Michael Herschorn and Joseph Ercolano for corrections, to Lesley M. Sibner for reading the proofs, and to Lynnel Marg for preparing the index. Appreciation is expressed to Helen Samoraj for expertly typing the entire manuscript. L. B.
F. J. M. S.
Contents Part I. Hyperbolic and ParaboUc Equations, by Farrz JoHN .
.
.
.
.
.
.
.
1. Equations of Hyperbolic and Parabolic Types
2. The Wave Operator . . . . . . 2.1. The one-dimensional wave equation
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
2.2. The initial value problem for the wave equation in three-space 2.3. Analysis of the solution . . . . . . . . . . . 2.4. The method of descent . . . . . . . . . . . 2.5. The inhomogencous wave equation . . . . . . . . 2.6. The Cauchy problem for general initial surfaces . . . . . . . . . . 2.7. Energy integrals and a priori estimates
1
4 4 10 13 16 17 18
24
2.8. The general linear equation with the wave operator as principal part . 2.9. Mixed problems
.
.
.
.
.
.
.
.
.
.
.
.
.
31
.
.
.
.
.
.
.
.
.
.
.
.
.
35
3. Cauchy's Problem, Characteristic Surfaces, and Propagation of Discontinuities
.
.
.
.
.
.
.
.
.
.
.
.
.
.
3.1. Notation . . . . . . . . . . . . . . . . 3.2. Relations between partial derivatives on a surface 3.3. Free surfaces. Characteristic matrix . . . . . . . 3.4. Cauchy's problem. The uniqueness theorem of Holmgren 3.5. Propagation of discontinuities . . . . . . . .
. .
. .
.
4. Linear Hyperbolic Differential Equations . . . . . . . 4.1. Solution of the homogeneous equation with constant coefficients by Fourier transform . . . . . . . . . . 4.2. Extension to hyperbolic systems of homogeneous equations with constant coefficients . . . . . . . . . . . 4.3. Method of decomposition into plane waves . . . . . 4.4. A priori estimates . . . . . . . . . . . . . 4.5. The general linear strictly hyperbolic equation with constant principal part . . . . . . . . . . . . . . 4.6. First-order systems with constant principal part . . . . 4.7. Symmetric hyperbolic systems with variable coefficients . . xi
38 38 40 43 45 53 62
64 69 71
74 78 82 87
CONTENTS
xii
5. A Parabolic Equation: The Equation of Heat Conduction 5.1. Parabolic equations in general . . . . . . . 5.2. The heat equation. Maximum principle . . . . 5.3. Solution of the initial value problem . . . . . 5.4. Smoothness of solutions . . . . . . . . 5.5. The boundary initial value problem for a rectangle
.
.
.
.
.
.
94 94 96 98
.
.
.
101
.
.
.
104
.
6. Approximation of Solutions of Partial Differential Equations by the Method of Finite Differences . . . . . . . . . . 6.1. Solution of parabolic equations . . . . . . . . . 6.2. Stability of difference schemes for other types of equations Bibliography
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
108 109 115 123
.
.
.
.
.
.
131
Part II. Elliptic Equations, by LIPMAN BERS and MARTIN SCHECHTER
1. Elliptic Equations and Their Solutions
I.I. Introduction
.
.
.
.
.
.
.
.
.
.
.
.
.
.
133
.
.
.
.
.
.
.
.
131
.
.
.
.
.
.
.
1.2. Linear elliptic equations . . . . . . . 1.3. Smoothness of solutions . . . 1.4. Unique continuation . . . . 1.5. Boundary conditions Appendix I. Elliptic versus Strongly Elliptic . Appendix II. "Weak Equals Strong" .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
134 135 139
.
.
.
.
.
.
.
.
141
.
.
.
.
.
.
.
.
.
.
.
.
143 144
.
.
.
.
.
.
.
.
.
.
.
2. The Maximum Principle . . . . . . . . . . 2.1. Second-order equations . . . . . . . . . 2.2. Statement and proof of the maximum principle . . 2.3. Applications to the Dirichlet problem . . . . . 2.4. Applications to the generalized Neumann problem . 2.5. Solution of the Dirichlet problem by finite differences 2.6. Solution of the difference equation by iterations . . . . . . . 2.7. A maximum principle for gradients 2.8. Carleman's unique continuation theorem . . . . .
. . 3. Hilbert Space Approach. Periodic Solutions 3.1. Periodic solutions . . . . . . . . . . . . . . 3.2. The Hilbert spaces Hi . . . . . . 3.3. Structure of the spaces H. . . . . . . . . . 3.4. Basic inequalities 3.5. Differentiability theorem . . . . . . . . . . 3.6. Solution of the equation Lit =f . Appendix I. The Projection 'T'heorem . . . . Appendix II. The Fredholm-Riesz-Schauder Theory .
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
150 150 150 152 154 155 158 160 162
164 164 165 167 170 174 175 177 183
CONTENTS
xiii
4. Hilbert Space Approach. Dirichlet Problem 4.1. Introduction . . . . . . 4.2. Interior regularity . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
190 190 190 192 193
4.3. The spaces H' andH.' H. 4.4. Some lemmas in Ho . . . . 4.5. The generalized Dirichlet problem 4.6. Existence of weak solutions . . 4.7. Regularity at the boundary . . . 4.8. Inequalities in a half-cube Appendix. Analyticity of Solutions . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. 5. Potential Theoretical Approach . . . . 5.1. Fundamental solutions. Parametrix . . . . . . . 5.2. Some function spaces . . . . . . 5.3. Fundamental inequalities . . . 5.4. Local existence theorem . . . . . . . 5.5. Interior Schauder type estimates. . . . . 5.6. Estimates up to the boundary . . . . . 5.7. Applications to the Dirichlet problem . . . 5.8. Smoothness of strong solutions . . . . . Appendix I. Proofs of the Fundamental Inequalities Appendix 11. Proofs of the Interpolation Lemmas .
.
.
.
.
.
.
.
.
.
.
.
.
196 198
200 202 207 211 211 216
220 228 231 235
237 240 242 250
6. Function Theoretical Approach . . . . . . . 6.1. Complex notation . . . . . . . . . . . . . 6.2. Beltrami equation . . . . . . . . . . . . . . 6.3. A representation theorem . . . . . . . . . 6.4. Consequences of the representation theorem . . . . . 6.5. Two boundary value problems . . . . . . . . . Appendix. Properties of the Beltrami Equation. Privaloff 's Theorem
254 255 257 259
7. Quasi-Linear Equations . . 7.1. Boundary value problems. 7.2. Methods of solution . . 7.3. Examples . . . . . Bibliography . . . .
.
.
261
263 267
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
282 282 284 286 291
Supplement I. Eigenvalue Expansions, by Lnxs GARDING
.
.
.
301
Supplement II. Parabolic Equations, by A. N. MILGRA.tit
.
.
.
327
.
.
.
341
.
Index
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
PART I
Hyperbolic and Parabolic Equations By Fritz John
CHAPTER 1
Equations of Hyperbolic and Parabolic Types
A general precise classification of partial differential equations will not be attempted here. Instead some of the intuitive notions underlying the conventional definitions of Ipperbolic and parabolic
equations will be indicated. Roughly speaking, one wants to include in this class the differential equations of mathematical physics that describe time-dependent processes, and then also any other equations in which one of the independent variables, t, plays a role similar to the time. The solutions u of such equations can be thought of as evoliing with t from a given initial state under the influence of certain boundary conditions.
One imagines that the differential equation (or system of equations) describes the laws governing the changes of some physical quantities u defined in some portion R of x1, ... , x,,-space
for all values of t subsequent to an initial time to. In addition to the differential equations we are given certain initial conditions in R for t = to which constitute a compressed history of the system,
as far as it affects the future. There are also prescribed conditions
on the boundary S of R for t > to, which correspond to the influences on R of processes going on outside R, not already contained in the differential equations. Back effects of R on the outside are neglected. The problems customarily treated consist in determining a solution u of the differential equations which satisfies additional conditions of this kind. The boundary conditions usually state that certain expressions depending on u and some of its derivatives take on prescribed values on S for t > to; the initial conditions make a similar requirement in R for t = t,,. These prescribed values are denoted by functions f. Any problem in which the solution u of a system of differential I
9
FRITZ JOHN
equations is to be determined from data f is said to be well posed if u exists for arbitrary f, is determined uniquely, and depends continuously on f.1 Here the terms "arbitrary" and "continuous" have to be interpreted properly to make this definition of "wellposed" meaningful. For most purposes it is adequate to think of "arbitrary functions" as all functions of class C3 with some fixed
s,2 provided their values are formally consistent on the intersections of different surfaces on which they may be prescribed. Moreover, continuous dependence of u on f shall stand for the requirement that uniform convergence off and its derivatives of order < s shall imply uniform convergence of u. Physically, the data f in a well-posed problem constitute a sufficient and independent set of causes, resulting in a stable effect u. Mathematically, we have a continuous mapping from the space of data f onto the space of solutions u. In the case of hyperbolic and parabolic equations there exist such well-posed problems in which u for points x in R and -times
tl > to depends only on initial conditions in R for t = to and boundary conditions on S for to < t < t1. Evolution of u in time means that once u and its derivatives are known for t = tl we can find u for t = t2 > ti directly by considering the values at tl as initial values and using the boundary conditions for ti < t < t2, without further reference to the history of the system for to <
t
Hyperbolic equations are distinguished from parabolic ones by the property that effects travel with finite speed. The value of a solution u of a hyperbolic equation at a point x at the time tl depends only on the initial values in a neighborhood of x for t = to, provided t1 - to is sufficiently small. At any interior point x of R the effects of the boundary conditions and of the initial conditions at far away points are not felt for some time subsequent to to. This makes it feasible to consider pure initial value problems for hyperbolic equations. 1 See the discussion in Hadamard [1], Chapter I and Courant-Hilbert [1], Vol. 2, pp. 226-231. 2 Functions of class C3 defined on a manifold are functions for which all derivatives of orders < s exist and are continuous. 3 For the connection with the notion of semigroup see Hille and Phillips [1], pp. 617-618.
HYPERBOLIC AND PARABOLIC EQUATIONS
3
It should be mentioned that the consideration of well-posed problems does not exhaust the subject of partial differential equations or their applications. One may think of the solution of a well-posed problem as predicting the outcome of an experiment for a given arrangement of apparatus. Instead one may ask what arrangements will produce a desired effect or what arrangement led to certain observed effects. Problems of this type are usually less accessible to known analytic methods.' In the following chapters we shall see how far the behavior of special hyperbolic and parabolic equations conforms to the intuitive picture contained in the preceding remarks. 4 See Pucci [ 1 ], John [ 1 ], Zirnmerman [ I ], Miranker [ 1 ].
CHAPTER 2
The Wave Operator
2.1. The One-Dimensional Wave Equation
We start with the simplest of all hyperbolic differential equations
lag - C2 a )U(X,t) = 0
(2.1)
where c is a given constant. (A physical interpretation is provided
by the transverse vibrations of a string under tension.) The variable x is commonly identified with "position" and t with "time." Solving equation (2.1) is trivial due to the fact that the wave operator in one space dimension can be factored: a2
ate - `
a2
a
a
a
aX2
at
` ax
at
a '
` ax
This permits us to write (2.1) as a system of two first-order equations (2.2)
(aT -+-
ca)u=v,
(a-`ax)
where the second equation involves only one of the two unknown functions u, it. Assume that u and v satisfy equations (2.2) in a convex region R
of the xt-plane. Then R can be covered by a set of parallel line segments s, given respectively by
x -{- ct = const. = y 4
THE WAVE OPERATOR
5
Along a segment s., dv = ax dx +
at
dt
ax
dx + c ax dt
=avd(x +ct) = 0 Hence v has a constant value along s,, which we denote by 0(y) : (2.3)
Then
v(x,t)
(a
=
0(y)
= ¢(x + ct)
a)
at
ax
Introducing a function a(y) for which an(y)
=
2c
(y)
the equation for u takes the form
(a + c
a atax
u=
ata a
+C
a
a(x + ct)
It follows by the same argument that u(x,t) - a(x + ct) is con-
stant along lines x - ct = const., and hence is of the form P(x - ct) with a suitable function ft. Consequently the general solution u of (2.1) in a convex region R is expressible in the form (2.4)'
u(x,t) = a(x + ct) + fi(x - ct)
where a and ft are functions of a single argument (d'Alembert's solution). For a point on the x-axis moving with velocity dx/dt = c to the
right the value of x - ct and hence also that of /3(x - ct) does not change. The graph of j4(x - ct) as a function of x is translated with velocity c along the x-axis. We say that f(x - ct) represents a "plane wave" traveling with velocity c. Formula (2.4) expresses that the general solution u of the wave equation (2.1) is obtained by superposition of two plane waves traveling respectively with velocities -c and c along the x-axis without changing shape.
FRITZ JOHN
6
The functions a, j9 appearing in (2.4) can be related immediately to the initial data of u. Let
for t = 0
at = g(x)
u = f(x),
(2.5)
Then clearly the functions a, fi in (2.4) are connected with the data f, g by the equations c(a'(x) - fi'(x)) = g(x)
a(x) + j9(x) = f(x),
Since
a(x + ct) + l3(c - ct) = J[a(x + ct) + 9(x + ct) + a(x - ct) 4-a+ f(x
- ct] +
x-
#Jz
d
c('(E)
we arrive at the formula (2.6)
u(x,t)
=f(x
ct)
2
f(x - ct) +
+ cl
f 2c f +
dE
Our derivation shows that any solution u of the "initial value
problem" (2.1, 2.5) which is of class C2 in the triangle with vertices (x + ct,0), (x - ct,0), (x,t) is represented by (2.6) and hence is determined uniquely by the initial values f, g. Conversely for f (x) of class C2, g(x) of class C1 the u represented by (2.6) is a solution of (2.1, 2.5) of class C2 as is easily verified. The initial
value problem for u satisfies all requirements for a well posed
problem. Not only is u determined uniquely but it depends continuously on the data in the sense that a change in f and g by at most E results in a change in u by at most (1 + t) e. Moreover the solution evolves in time. For f in C.2, g in C1 we obtain for any
t = t1 > 0 from (2.6) values (2.7)
u = A (x),
at
= g, (X)
where again f1 is of class C2 and g1 of class C1. Since the differential
equation does not change when we replace I by t - t1 the value of u(x,t) for any t > ti must be given by (2.8)
u(x,t)
=.fi(x + c(t - tl)) , f(x - c(t - t1)) 2
1+ 2c z-ctt-c,)
gl() d
Since u(x,t) is determined uniquely this formula for u goes over
THE WAVE OPERATOR
7
into equation (2.6) when we substitute for f, and g, their expressions in terms off and g. Formula (2.6) shows that the value of u(x,t) depends only on
the initial values f(y), g(y) with y satisfying ly - xl < ct. The interval x - ct < y < x + ct constitutes the domain of dependence on the initial data for u at the point (x,t). Its endpoints are the intersections with the x-axis of the characteristic lines of slope l 1c and -1 /c through the point (x,t). Conversely the values off and g at a pointy influence u at the time t only at points x with Ix - yI < a. The domain of influence of the data at a point (y,0) is then the region in the upper half of the xt-plane bounded by the characteristic lines through the point (y,0). Iff and g have their support in a set S of the x-axis then u(x,t) for fixed t will have its support
in a ct-neighborhood of S.5 In this sense initial disturbances propagate with velocity c. In applications one is rarely concerned with pure initial value problems of the type (2.1, 2.5) in which the space variable x ranges over all values from - co to + oo. More commonly the coordinate
x which locates points of the physical medium is restricted to a finite interval 0 < x < L. Prescribed initial conditions (2.5) are then only meaningful for 0 < x < L. Additional information in the form of "boundary" conditions at x = 0, L for t > 0 is then needed to determine the solution. We treat first the simpler case of such a "mixed" problem in the semi-infinite interval 0 < x < oo. Let u(x,t) be a solution of class C2 of (2.1) for x > 0, t >- 0. We try to satisfy the boundaryinitial conditions u = f (x), for x > 0, t = 0 ut = g(x) (2.9) for x = 0, t > 0 u = h(t) Clearly necessary for existence of u is that f is in C2, g in Cl for
x z 0 and h in C2 for t > 0. Moreover if u is to be in C2 for x = t = 0 our data must satisfy the "compatibility" conditions (2.10) h(0) =f(0), h'(0) = g(0), h"(0) = c2f"(0) 6 The support of a function f is the closure of the set of points in which f 0. A function f defined in Euclidean space will be of compact support if it vanishes everywhere outside a bounded set. The ct-neighborhood of a set S consists of all points having a distance < ct from some point of S.
FRITZ JOHN
8
We shall prove that these conditions are sufficient to guarantee existence and uniqueness of a solution u. If u exists it is necessarily representable in the form (2.4) for x > 0, t > 0. The boundary-initial conditions (2.9) require the functions a, fi to satisfy the relations (2.11 a) a (x) + fl (x) = f (x), cx'(x) - cfl'(x) = g(x) for x > 0 (2.11b)
fort > 0
a(a) ; (3(-ct) = h(t)
If we normalize x and fi so that a (0) = X3(0) = z f (0)
we have uniquely
a (x) = 2f(x) '
2c
jg(t) d
Jg(E) d
9(x) = 2f (x)
,
for x > 0. Then by (2.11 b)
fi(t) = -a(-t) + h(-t/c) =
h(...t/c)
- f(-t)
fu
2
d5
for t < 0. Substituting into (2.4) shows that any solution u of our problem must be given by (2 . 12 a )
u(x, t )
=
f (x + ct) -I- f (x - Ct)
r
2
1
f x+Ct
2c Jx-rt g ($)
d
for x > ct (2.12b)
u(x , t) = h ( t
.r
+ f (x + Ct) - f (ct -
cl
x)
2 1
+ 2c-
z -i- Ct
et -x
dd
for x < ct
One verifies easily that this u actually is a solution of our problem, if f, g, h satisfy the regularity and compatibility assumptions made
before. In particular the compatibility conditions (2.10) are needed to make u and its first and second derivatives continuous along the "characteristic" x - ct = 0 (see Fig. 1).
Of special interest is the case where the boundary function
THE WAVE OPERATOR
9
h(t) vanishes identically for t >_ 0. A solution u(x,t) will then exist for x > 0, 1 > 0, if f (x) is in C2, g(x) in C', and the conditions
f (0) = g'(0) =f"(0) = 0
are satisfied. If we continue f and g for negative x as odd functions of x :
g(-x) = -g(x) f ( - x) = -f (x), formula (2.12b) becomes formally identical with (2.12a). The solution u(x,t) will be defined for all x and t >_ 0 by the common
Fig. 1
formula (2.6) and will itself be odd in x. We can consider u as extended by "reflection" on the l-axis. A similar procedure can be used to find the solution of the mixed problem for the finite x-interval; this consists in finding a solution of (2.1) for
t >0
0 <x
which satisfies the initial conditions
tit = g(x) u =.f (x), and the boundary conditions
for 0 <x
u = hl(t)
for x = 0, t >_0;
u = h2(t)
for x=L,t>0
t=0
For the special case where hl and h2 vanish identically we can again obtain u as the solution of a pure initial value problem by
FRITZ JOHN
10
reflecting f and g with respect to the boundary points, i.e., by continuing f (x) and g(x) for all x in such a way that the equations
.f (x) Tf (-x) = 0,
(2.13)
f (x) + f (2L
- x) = 0,
g(x) + g(-x) = 0 g(x) -}- g(2L - x) = 0
hold identically. This requires that f, g, f" vanish at x = 0 and x = L, if discontinuities are to be avoided. The necessary continuation can be achieved most easily by representing f and g in the interval 0 < x < L by appropriate Fourier series , f (x) _n=1a sin "X L
(2.14)
g(x) _
n=1
ffnx
b.,, sin
L
The series will define functions f, g for all x which automatically satisfy (2.13). By (2.6) the solution u is then given by the expression 1
u
2
n=1a n( sin
*7rn(x + Ct)
L
L
+ sin
7rn(x
- ct)\
L
(cos 7rn (x + Ct) b n L
1
_ cos
7rn
(x - ct)
) This representation of u in terms of "progressive waves" can be 2irc n=1
L
transformed immediately into the expression 00
(2.15)
tj
t
u =n= I [an cos
L in terms of "standing waves."
bn sin -L
sin
7rnx
±
2.2 The Initial Value Problem for the Wave Equation in Three-Space
We consider next the three-dimensional analogue of equation (2.1), the equation of acoustic waves 2
(2.16)
L[u] = ate -
c2
u(X1iX21X310 = 0
where 0 is the three-dimensional Laplace operator. We shall use vector notation for the space variables and write (x1,x2,x3) = x
THE WAVE OPERATOR
11
so that u = u(x,t). The length of the vector x will be denoted by Ixi.
There exists an integral transformation which reduces (2.16)
to (2.1) based essentially on the spherical symmetry of the operator L.6 Given any function h(x) = h(x1,x2,x3), we can form the average of h on a sphere of radius r about the point x: (2.17)
ff h(x + ry) dSy
I(x,r) _
Ivl=1
1 is of class C8 in x and r for r > 0 if h is of class C8 in x. Moreover, for continuous h I(x,O) = h(x) (2.17a)
Decomposing a solid sphere into concentric spherical surfaces, we have
555
h(x -{- z) dz
f
R
o
Izl<_R
where dz = dz1 dz2 dz3. Hence for h in C2 A
f
r dr =J f f Oh(x + z) dz J
Izl <_ R
(2.17b)
=
ff
h2, (x + z) R dSz
IzI=R
= ff
h., (x + Ry)yR2 dSv = 4nrR2 aR I(x,R)
IuI=1
It follows by differentiation with respect to R that
fRrzI(xr) dr = R2 DI(x,R) =
aR R2
aR I(x,R)
or that I(x,r) is a solution of the partial differential equation? 2
(2.18)
irI(x,r) = are rI(x,r)
s This is the classical method of solution due to Poisson [1]. 7 Spherical averages in any number of dimensions satisfy analogous equations of the "Euler-Poisson-Darboux type." For three dimensions reduction to the wave equation is particularly simple. See Courant-Hilbert [1], Vol. 2, p. 700; Weinstein [I], John [2], p. 88.
FRITZ JOHN
12
We can consider the function rI(x,r) as resulting from the application of an operator S2, to the function h : r1(x,r) = S2,h(x)
(2.18a)
The operator transforms functions of x into functions of x for each value of the parameter r. Equation (2.18) expresses that a2
Os2,h = Q, Ah = are S2,h
(2.19)
Let now u(x,t) be a solution of (2.16) of class C2 satisfying the initial conditions
at = g(x)
u = f (x),
(2.20)
for t = 0
Then for any r > 0, t z 0 the function S2,u(x,t) satisfies the equation 0 = SZL[u] = L[S2,u]
_
`ate
- c2 ar2) S2,u
and the initial conditions at f2,u = S2,g
S2,u = SZ, f,
for r > 0, t = 0
and the boundary condition
forr=0,t>-0
S2,u=0 It follows from (2.12b) that QrU(x>t) = 2 [
,t+rf (x) -
(x)] +
2c
5
Qz g(x)
d
for 0 < r < ct. Since by (2.17a) u(x,t) = lim 1 nru(x,t) r-.O r
we see that a solution u of (2.16) of class C2 with given initial values f, g is necessarily represented by the formula u(x't) (2.21)
a
ar
r,
+ u
1
c
dug
' f (x + ccy) dsd = 4-7r at t JJ IWi=
fJg(x
t l
I
1
-17
y) dsy
THE WAVE OPERATOR
13
2.3. Analysis of the Solution
We see that the solution u of the initial value problem (2.16, 2.20) is determined uniquely. On the other hand, for any f of class CA+3 and g of class Ck+2 formula (2.21) actually defines a function u(x,t) of class Ck}2 which satisfies the differential equation
and initial conditions, as is easily verified with the help of relation (2.18). The solution u depends continuously on the data in the sense that small changes in f, g and the first derivatives off will result in only small changes in u. The initial value problem is well-posed.
One feature of basic importance in the theory of general
hyperbolic equations in more than one space dimension has to
be emphasized. In contrast to the one-dimensional case the solution u does not depend continuously on the data f, g in the crude sense that small changes in f, g necessarily result in small changes in u. The occurrence of the differentiation sign preceding
f in formula (2.21) makes this plausible. An explicit example will show that the solution can actually behave worse than the data, or that initial disturbances can he amplified. It is verified easily that an expression of the form (2.21a)
=.(IxI -;
u(x,t)
Ct) +,3(1x1 - Ct) IxI
for arbitrary functions x(r), (3(r) of class C2 is a solution of the wave equation." These special solutions with spherical symmetry can be thought of as composed of progressing waves traveling towards or away from the origin with speed c, analogous to the situation in the one-dimensional case given by formula (2.4). Taking x(r) _ ,3(r) = 2r¢(r) where 0(r) is even in r, we find the solution O(ct + IxI) + ¢(ct -- IxI) -; ,
u(x,t)
=
ct
- 0(c1 -- IxI )
2 IxI
2
c(ct) + ctO'(ct)
¢(ct + IxI)
for x = 0
8 In general u is singular for x = 0.
forx0
FRITZ JOHN
14
which corresponds to the initial values
u =f(x) = 0(IxI),
at
= g(x) = 0
for
t=0
It is clear that small changes in 0 will result in small changes in the solution u for t bounded and x bounded away from 0, while the resulting changes could be arbitrarily large at x = 0, t > 0. If for example fi(r) converges uniformly towards the continuous function Ire _-11, the limit of the corresponding u would become infinite for x = 0, 1 = 1 /c. Physically, we have the situation where a singularity can arise at the origin from regular initial data through, focusing of spherical waves into the origin. It is plausible that such peculiar roughness of the solution will be restricted to focal lines or surfaces. On the average u should not be worse than initially. We shall see later that indeed the integral of the sum of the squares of the first derivatives,
the total energy, stays finite, if finite initially, so that singular behavior of first derivatives stays confined to sets of measure 0. The preceding analysis shows that for hyperbolic equations in
more than one space dimension no bounds for u at individual points can be obtained in terms of corresponding bounds for the initial values. If we only consider solutions u with initial f in C3 and g in C2 we are sure that a solution u in C2 exists; taking initial
data u and au/at at a later time ti, we find that they are only of classes C2 and C', respectively, and that construction of u(x,t), for
t > ti from data at tl becomes doubtful. If we want "persistent"
initial conditions, such that data at later times depend continuously on data at earlier times, changes being measured in the same way at both times, we have to make use of global quantities, like L2-norms, rather than magnitudes at individual points.
Formula (2.21) shows that the value of u at a point x at the time t depends only on the values off, g, and of the first derivatives
off all taken on the surface of the sphere of radius ct about the point x. This surface constitutes the "domain of dependence" of u on the initial data. Conversely, an initial disturbance f, g confined to a small neighborhood of a pointy will only affect the value of u at the time t for x near the surface of the sphere of radius ct about the point y. This illustrates the fact that for the wave equation L[u] = 0 disturbances spread with velocity c.
THE WAVE OPERATOR
15
The wave equation in three space dimensions has the additional atypical feature that the effect at a point x of an initial disturbance confined to a pointy disappears completely for ct > l y - xl, i.e., after the spherical wave from y has passed the point x. This
property, known as "Huygens' principle in the strong form" corresponds to the existence of sharp signals. It is absent for most
other hyperbolic differential equations; while a disturbance generally takes a finite time to reach a point, it usually persists after the wave front has passed the point.9 Even for the wave equation in three-space the strong form of Huygens' principle ceases to be valid if "obstacles" are present. Consider for example solutions of the wave equation defined in
the exterior of the unit sphere lxi = 1 which have vanishing normal derivatives on the sphere for all t. (If u is interpreted as a velocity potential this condition expresses the absence of flow
through the boundary.) Special solutions u of this type with spherical symmetry are represented by formula (2.21a) if the functions a(s) and fl(s) are connected by the relation fj(s) = a(2
- s) + 2
dE 2-R
If here a(s) has its support in a finite interval 1 < s s S the initial values of u will vanish for lx) > S. If Huygens' principle in the strong form were valid the solution u(x,t) would have to vanish at any given x with lxi L, 1 for all sufficiently large 1. Actually however we find for the value of u in a point of the boundary lxi = 1 u
1 f d
= 2a(l + ct) - 2e-1-c'
e`a($) dd t
For ct > -1 + S this reduces to u=
s
d
-2e-1-cc
1
! For a discussion of Huygens' principle see Hadamard [1], Baker and Copson [1]. For the question of characterizing equations for which Huygens' Principle in the strong form is valid, see in addition to Hadamard the papers by Stellmacher [1, 2]. For a more generally valid formulation of Huygens' Principle in terms of singular behavior on wave fronts see R. Courant and
P. D. Lax[1]andP.D.Lax[I].
FRITZ JOHN
16
While u decreases exponentially in time it does not have to vanish for large 00 2.4. The Method of Descent
The derivation of the solution of the wave equation in threespace given here was based on the use of a linear operator SZT with the property (2.19). Such operators can be constructed also for other dimensions, and can be used to solve the corresponding wave equations (or other equations with spherical symmetry)." To obtain the solution of the two-dimensional wave equation (a2
(2.22)
at2 -
C2
a2, - Cl a2,) u (xt,x2it) - 0 axj
ax.;
(equation of vibrating membranes) it is simpler to use Hadamard's "method of descent."12 Let it be the solution of (2.22) with initial values (2.23)
au
u =f (x1,x2),
at
= g(x1,x2)
for t = 0
Then u can be looked at as the solution of the initial value problem (2.16, 2.20) for the special case in which u, f, g are independent of x3. The solution is given by (2.21), where now
0r.f
r 4-r
Jjf(x1
H- 91, x2 + rY2) dSp,
I1I=1
r 2ir
f (x1
--I
ryl, x2
y1s_yyg<1
92) dyl dye y'2
The domain of dependence of u(xl,x2,t) consists of the whole circular disc of radius ct about the point (x,,x2), though the boundary of the disc is weighted more heavily than the interior points. Huygens' principle in the strong form is not valid for equation (2.22) (nor for any hyperbolic equation in an even number of space dimensions). Signals originating at a point 10 Decay of solutions of the wave equation has been analyzed by C. S.
Morawetz [1]. P. D. Lax and R. S. Phillips [1] for general star-shaped obstacles.
11 See Courant-Hilbert (1], Vol. 2, pp. 700-703.
12 See Hadamard [1]. p. 49. Other applications have been made by :Vsgeirsson [1]. [2].
THE WAVE OPERATOR
17
(yvy2) at the time t = 0 will reach a point (xl,x..,) a distance r away
at the time t - r/c and will be felt at (x,x2) thereafter. 2.5. The Inhomogeneous Wave Equation
We continue our study of questions associated with the three-
dimensional wave operator L. The notions and methods encountered in this study will serve as a model to be followed later for more general equations. The first problem to be considered is the solution of the initial value problem for the inhomogencous wave equation (2.24) L[u] = w(x,t) where the "impressed force" w is prescribed. Since the difference of any two solutions of (2.24) is a solution of the homogeneous wave equation for which we have uniqueness, it is clear that the
solution u of (2.24) is again uniquely determined by its initial data (2.20). It is sufficient to determine the solution of (2.24) with the special initial data (2.25)
a-
tt = 0,
0
for t = 0
The solution with the more general initial data (2.20) is then obtained by adding the right hand side of equation (2.21). The solution of the inhomogencous differential equation with
homogeneous initial data can be reduced to a set of initial value problems for the homogeneous differential equation by "Duhainel's integral" (method of impulses). Let t'(x,t,s) for any s > 0 denote the solution of L[U] = 0 with initial data (2.26)
I_r = 0,
at
= w(x,s)
for t _ 0
U will exist and be of class C2 for w in C2. It is verified immediately then that (2.27)
u(x,t) =
t
U(x,t - s,s) ds u
is the desired solution of (2.24, 2.25). Substituting for U its explicit expression from (2.21) corresponding to g = w(x,s), we find t
(2.28)
u(x,t)
=
1 C
K2,ct-Aw(x,s) ds 0
FRITZ JOHN
18
Using the expressions (2.18a, 2.17) for S2, we can write u in the form et
u(x,t) =
(2.29)
S2rwI x,t \
1
c2
fo
dr 4zrc2 Jo t
r
r dr c
if w \ z,t -
lz-zI=r 1
1 2 4irc
JJJ lz
iz -z)<et
- xl
/
w z
rI dSZ
-
z -c xl),t
dz
One sees that u(x,t) depends only on the values of w(z,s) in points (z,s) of four-dimensional space lying on the "backward characteristic cone"
lz -xl = c(t -s),
(2.30)
0 <s < t
with vertex (x,t). This cone describes the domain of dependence of
the point (x,t). Formulated differently an "impressed force" w at a point (z,s) only influences u(x,t) at points (x,t) on the "forward characteristic cone" with vertex (z,s), which is also given by (2.30).
Equation (2.29) connecting u and w will be written symbolically in the form (2.31)
u=
L-1 [w]
For w of class C2 the expression L-1 [w] is characterized by the properties L[L-1 [w] ] = w
L-1[w] = at L-1 [w] = 0
for t = 0
2.6. The Cauchy Problem for General Initial Surfaces
Let S denote a hyper-surface in xt-space. The general Cauchy problem for the equation L[u] = w consists in finding a solution u on one side of S for which u and an "out-going" first derivative are prescribed on S. For simplicity we take a surface S represented by an equation (2.31a)
t=
(x)
where ¢(x) is a positive function defined for all x of class C2 and
THE WAVE OPERATOR
where (2.32)
19
3
>0
We then look for a function u(x,t) of class C2 satisfying for t >_ 0(x) L[u] = w(x,t) for t = O(x) u, = g(x) u = f (x), (2.33b) where f, g, w are given (see Fig. 2). (2.33a)
t=o Fig. 2
The first step towards a solution of the problem consists in a reduction to the special case where f, g and the values of w and of its first and second derivatives on S vanish. For the reduction we only have to solve a simpler "truncated" Cauchy problem for a function u'(x,t), (2.34a) L[u] = w' for t > 0(x) (2.34b) for t = O(x) u' = f (x), u' = g(x) Here w' is not fully prescribed, but stands for any function of class C2 with the property that w" = w - w' vanishes with its first and second derivatives on S. Indeed, once a solution of (2.34a,b) is known, we have in u = u' + u" a solution of (2.33a,b), if u" denotes a solution of the reduced problem L[u"] = w" for t >- O(x) U" = ul = 0 for t = O(x) The truncated problem can be discussed more easily if we transform S into a coordinate plane through the introduction of
FRITZ JOHN
20
a new independent variable s = t - fi(x) instead of 1. Put
u= u' (x,S + O(x)) = v' (x,S) Problem (2.34a,b) then takes the form (2.35a) (2.35b)
w'(x,s -I- O(x)) for s > 0 v'(x,0) = f (x), v8(x,0) = g(x)
Here M is the second order differential operator defined by L[ri'] - .11[v']. It is easily verified that M [v'] is of the form M[v'] = L0[vse] - Ll [vs] + L2[v'] with
L = 1 - c2 L1 = 2c2
0."
fix
ax,
- c2(Ao)
-c2A
L2
L, is a differential operator of order k involving only x-differentiations. Put
(w(xs
(akv'(xs)) ask
asp
8=0
O(x)))'_0
The function w'(x,t) is to be such that (2.36)
u" .(x)
-
(aA LWW (x 1
tl
fork-0,1,2
at'
For a solution r' we have from (2.35b) vi = g(x) vo =f while differentiation of (2.35a) with respect to s and putting
(2.37)
s = 0 yields the recursion formula (2.38)
]
L i[va
L2[vx] = 't(x) at least for v' of class Ck-2 and w' of class CA. In particular we have l u[Z't -
for k = 0 L0[u'2] -- L, [g(x) ]
-` L2 [.f (x) ] = wo
Here the operator Lo stands for multiplication with the scalar
Q=1
-c2'10z,
THE WAVE OPERATOR
21
If Q vanishes at a point of S, we obtain the equation
L1[g(x)] + L2[f(x)] = w(x,0(x)) which represents a restriction on the data f, g, w. Points of S at which Q = 0 will be called characteristic. Geometrically they are points at which the normal to S forms with the t-axis an angle whose tangent is l 1c. If such points occur on S the truncated Cauchy problem, and hence also the original one, will not have a solution for general f, g, w. Surfaces S on which all points are characteristic are called characteristic surfaces. Examples are the characteristic cones
t=sy
Ix - zI c
Surfaces S for which Q # 0 at all points arc called noncharacteristic
or free. On a free surface formulae (2.37, 2.38) permit one to calculate successively all vx., at least formally, in terms off, g, w'. w sufficiently regular. Assume then that S is free, and f, We form the expression 4
v'(x,s) _
k-o
sk ra.(x)
-
k!
where vo, v' arc given by (2.37) and i'
, z, j,
v' have been deter-
mined recursively from (2.38), using the values (2.36) for the wi.. Then W, = M[v'] wu
}
sw' + 2 s2W
S3
(Ll[z'a]
- L2[US]) -
S4
24 La[va]
It is clear that w' and its first and second derivatives agree with W(x,s + O(x)) and its derivatives on s = 0. We have then in u'(x,t) = v'(x,t -- O(x)) a function with Cauchv data f; g on S
for which w" = w(x,t) - - L[u] = w (x,s -' 0(x)) - .11[v'] is of class C2 and vanishes with its first and second derivatives on S. Having solved the truncated Cauchy-problem for a free surface S we define more precisely w" by w"
w - L[u'] 0
for0
for t > fi(x) 0(x)
The extended function w" is in C2 for t > 0 and all x. We form
FRITZ JOHN
22
the solution u" = L-'1 [w"]
of the equation L[u"] = w" with vanishing initial data on t = 0. Putting u = u' + u" we have
L[u] = L[u] + L[u"] = L[u'l + w" In particular for t O(x) the function u will be a solution of L[u] = w. On S
u=f+u",
u==g+ui
Hence u will have the prescribed Cauchy data on S, if it can be shown that u" and ul vanish on S.
Thus our Cauchy problem with S as initial surfaces can be solved if u" = u, = 0 on S for every u" with for 0 < t < O(x) L[u"] = 0
u=uj =0
fort=0
This is certainly the case, if for any point (x,t) of S the conical domain (2.39)
Iz - xI < c(t - s),
0<s
in zs-space can be shown to lie in the region (2.40)
0 < s < O(z)
where L[u"] = 0. For then the backward characteristic cone with
vertex (z,s), the domain of dependence of (z,s), will also lie "below" S; hence u"(z,s) = 0 for all (z,s) satisfying (2.39) apd hence also u"(x,t) = u'(x,t) = 0. It can be proved that (2.39) implies (2.40) if we assume that Q>0 for all x (2.41) We have then indeed for t = O(x) and for (z,s) satisfying (2.39)
cMz) - s) = c(t - s) + cMz) - 0(x))
z-xJ+cod,ocd O(x+a(z-x))da _ Iz -xI +cl I (z, -x;)0, dot >_0 Jo i
THE WAVE OPERATOR
23
since by (2.41) i
(z, - xi) x, l C 1
(z - .17 1
<-Iz -x1 c Surfaces t = O(x) satisfying (2.41) will be called "spacelike."
Their normals form an angle less than arc tan (11c) with the t-axis. The Cauchy problem with initial conditions prescribed on
a spacelike surface has a solution, if the data are sufficiently regular. The geometric property of spacelike surfaces actually used in the preceding construction is that a characteristic cone with vertex on such a surface does not intersect the surface. It can be proved that the Cauchy problem for a "timelike" initial surface (for which Q < 0) in general has no solution. This is plausible since on a timelike surface some points will lie in the domain of influence of others, so that the Cauchy data on some parts of the surface cannot be prescribed completely independently from those in other parts. We can interpret the relation O(x) < t as describing for each t a region Rt in x-space. The set Rt increases monotonically with the time t. The Cauchy-problem then consists in determining a function u satisfying for each t the equation L[u] = w in Rt and having prescribed values u =f, u, = g on the boundary O(x) = t of R,. The normal velocity of the moving surface O(x) = t in x-space can be defined as the speed with which the surface advances in
the direction of its normal. It is easily seen that that speed is given by 02
Hence for a characteristic S the level surface 0 = const. = I in x-space propagate with normal velocity c, for a spacelike S with normal velocity > c, and for a timelike S with normal velocity < c. In the latter case influences from u in the interior of Rt, which travel with speed c, overtake the moving surface 0 = t and affect the data on that surface.
FRI'Z'Z JOHN
24
2.7. Energy Integrals and a Priori Estimates
Very often precise information on the solution of a simple equation permits to establish existence and uniqueness of solutions of more complicated equations that in some sense are near to the simple one. For the wave operator L such information is furnished by integral identities and inequalities that can be interpreted as special instances of the law of conservation of energy.'3 With the help of energy inequalities one can then pass to more general equations having the wave operator as principal part. We start with the algebraic identity (2.42)
utL[u] =
2a
[c2
uT, -f-
u1]
- c2
a- t
ulux
We can think of u as a displacement, u1L[u] as the work done per unit time and unit volume by the impressed force L[u] = w, and
E=
(2.43)
2 [c2
uj'
+
u2]
as energy per unit volume. Let R denote a region in x-space with boundary B. Integrating (2.42) over a cylindrical region in xt-space with cross section R and plane faces t = 0, T, we obtain (2.44)
jjJ'fuL[u] dx dt = f f E dx - f Edx _f I'dt fJ c2u, do dS XER
0
J
t=T
JJ t-u
B
Here d/dn denotes differentiation in the direction of the exterior normal of B, and dS the surface element of B; then c2ut(du/dn) dS gives the flux of energy through the clement dS, in unit time. Identity (2.43) immediately yields a uniqueness theorem for a mixed boundary-initial value problem (see Fig. 3). Let u be a
solution of the differential equation L[u] = w for x in R and t 0. Let the initial values u =f(x), ut = g(x) be given for x in R and t = 0. In addition either u or du/dn have prescribed
-
values h on the boundary B of R for t > 0. Then u is determined 13 See Zaremba [ 11, Friedrichs and Lewy [ 1 ], Leray [ 1], [2]. Garding [2]. [3].
THE WAVE OPERATOR
25
uniquely for all x in R and t > 0. Indeed let it denote the difference
of two solutions. Then u = ut = 0 for t = 0 and hence also E = 0 for t = 0. Moreover ut(du/dn) = 0 on B, and L[u] = 0. It follows that the integral of the non-negative expression E over R
vanishes for t = T. Hence E = 0 for any t = T 0. Then it must be constant, and thus u = 0 since u vanishes for t = 0. t
B
t=0
B
Fig. 3
More generally we can integrate identity (2.42) over any region in xt-space described by 0'(x) < t < i2(x)
x e R,
(see Fig. 4). Here R is a region in x-space with boundary B. For any hyper-surface t = 4(x) in xt-space the quantities - Tzl
dx,
- Ox2 dx,
- 0?2 dx,
dx
represent the products of the direction cosines of the normal with the hyper-surface element. It follows then from (2.42) that (2.45)
5555 u1L[u] dx dt zER
41
t=42
x-555 E' dx t=41
- f f f c2u du dS dt t
0
FRITZ JOHN
26
Here for j = 1, 2
+ui)
E'
2`1
-
C2
This identity gives a more precise uniqueness theorem. Let the Cauchy data u and ut vanish on the surface t = #1(x) for x in R. t
R
B
t=0
Fig. 4
Then El = 0. Moreover let u vanish on B for7'#' S t < #2. If also L[u] = 0 for ,/,1 < t < 02. Then
fffE2dX = 0 t=Az)
If here
1 -C2L(#z) 2 z0 i.e., if the hyper-surface t = #2(x) is spacelike or characteristic, it follows that au(x,#2(x)) ax{
=0
for i = 1, 2, 3
Consequently u is constant on t = 02(x) and hence = 0, since u vanishes on t = #2(x) for x e B. We can choose for the surface
THE WAVE OPERATOR
27
t = 2(x) the backward characteristic cone
Ix - x°I = c(t° - t) for a vertex (x°,t°) with x° E R, 0'(x°) < t°. We find then that a solution u ofL[u] = w with initial data u =f, ut = g on t = 01(x)
t=41 R
t=0
Fig. 5
and boundary values u = h(x,t) for x c- B, is determined uniquely at (x°,t°) by the values of w, f, g, h taken at points (x,t) lying in the region
Ix - x°I < c(t° - t) which represents the domain of dependence of the point (x°,t°) (see Fig. 5). If the hyper-surface t = 0'(x) is itself characteristic the values of ut do not have to be prescribed on that surface. Indeed u = 0 on t = 01(x) then already implies E1 = 0. Of special interest is the case where R is the whole x-space. To avoid convergence difficulties we assume that u has finite
support in x. More precisely for every T > 0 there shall be a bounded region in x-space outside of which u vanishes for 0 < t <
T. If u =f, ut = g for t = 0 and L[u] = w, identity (2.44)
reduces to (2.46)
fJf t=T
[C2
uz, + ui ] dx
JJJ 2
[c2 i J.fx,2
+ g2] dx
+ Jfff O
di
FRITZ JOHN
28
For w = 0 we find that the expression
ff
u?' + u2 j dx
Lc
is independent of 1. In other words, in the square mean the first partial derivatives of u(x,t) at the time t behave no worse than those at the time 0. This is in contrast to the fact found earlier that u at the time t can behave worse point-wise than at the time 0. For f = g = 0 and E defined by (2.43) we have from (2.46) (2.47)
fffE t= T
dx =
f f f f up
dx dt
OJ
ff55 0
u dx dt
w2 dx dt ffff ` 0
jffJ2E dx dt ffffw2 dx dt O
O
Then
//ff5f2Edxdt
w2 dx dt
0
0
and, since E = 0 for t = 0,
Ifs" 2E dx dt S U
w2dxdi
Tds fo
0
It follows from (2.47) that (2.48)
fffEdx I=T
w2(x,t) dx di O
The magnitude of a function f (x) can be measured by the quantity
H[f(x)] =fff
f2(x) dx
THE WAVE OPERATOR
29
(the square of its "Hilbert norm"). Then (2.48) can be written in the form rOT
(2.49) nc2 Y H[ux,(x,T) + ;H[ui(x,T)] < TJ H[w(x,t)] dt This represents an estimate for the first derivatives of u = L-1[w] in terms of w itself. We can then estimate u itself by observing that for u(x,O) = 0
H[u(x,T)] =ffJ u2(x,T) dx =JfJ < f f fdx T
JOT
u2 (x,t) dt
s I ,H[w(x,t)] 240,
J o T Jo f oT
`J T ut(x)t)
=T
dt)2 dx
T H[ut(x,t)]
dt
J
dt ds
2sJ TH[w(x,t) di ds = T3JTH[w(x,t)] dt
We combine the estimates for u and its first derivatives by introducing the quantity K[u(x,t)] = Maximum (H[u],H[uxl],H[uxq],H[u.3],H[u,])
Then (2.50)
K[L-1 [w(x,t)]] < ¢(T)
r7H[w(x,t)] dt o
where (2.51)
q(T) = max (T3,2c-2T,2T)
Analogous estimates hold for all space derivatives of u. Let azi "s-}-"3
D"
axi i &"I ax;3
denote any partial derivative operator. Following L. Schwartz
we denote the order al + x2 ; x3 of the operator by (xl. We define higher order norms by Hk[w(x,t)] = max H[D"w(x,t)] I"I
Kk[u(x,t)] = max K[D"u(x,t)] IA:5k
FRITZ JOHN
30
For w sufficiently regular D'xL-lw = L-1D"w
Hence for ka) < k
K[D"L-1[w(x,t)]] < O(TJ H[D°`w(x,t)] dt 0
< O(TJ
Hk[w(x,t) ] 0
Then also Kk[L-1[w(x,t)]] < 4(T) J0 THk[w(x,t)]
(2.52)
dt
We have here estimates for the derivatives of a solution u of L[u] = w with vanishing initial values in terms of derivatives of w. These estimates concern the square integrals of the quantities
involved. For applications it is important that the value of a function at any point can be estimated in terms of integrals of squares of the function and of a certain number of its derivatives. More precisely we have for a function h(x) of three independent variables an inequality of the form (2.53)
h2(x) < y2H2[h]
where y is a universal constant. To prove (2.53), which is a special case of a lemma of Sobolev,14 we start with identity (2.17b) which yields dr
f h(x +
dw
r-2
rr JJ
Oh(x + y) dy
lyl
I:I=1
I1f
r-2
a,rr3 / N
f
(Oh) 2 dy
Ivl
3,
Jar
H2[h]
Integrating with respect to r gives then 4irh (x) - f h (x ± J
dw
I
S V 48TrrH2[h]
l=1-1
14 For other proofs and generalizations of this lemma see Sobolev [1], Nirenberg [1], and pp. 77 of the present volume.
THE WAVE OPERATOR
31
Multiplying by r2 and integrating once more with respect to r from 0 to 1 leads to
8irh(x) -J f fh(x + y) dy s
irH2[h]
I1'II<1
Then I
s ffJ/z(x + y) dy +
fah(x) I
ivl<1
H2[h] + 8$ H-2[h] Squaring leads to an inequality of the type (2.53). Combining (2.52, 2.53) we find a point-wise estimate for L-1[w] in terms of w and its first derivatives: y2K1[L-1[w(x,T)]] y2H2[L-1[w(x,t)]] < (2.54) IL-1[w(x,T)](2 <
S
S 720(T)foTHi[w(x,t)] dt 2 .8
The General Linear Equation with the Wave Operator as Principal Part
We are now able to attack the solution of the initial value problem for a general differential equation-of the type
L[u] = N[u] + w
(2.55)
where N is any first-order linear operator : (2.56)
N[u] _
3
ai(x,t)ux, + a4(x,t)ut + a5(x,t)u i=1
The initial values of u are prescribed:
for t = 0 We assume that f, g, w and the a, are of class C. In order to give the coefficients in the first order operator compact support we replace N by (x)N, where fi(x) is some (2.57)
u = f (x),
ut = g(x)
function of class C°° with (2.57a)
1
(x) =
0
for lxl < r for Ixl > 2r
FRITZ JOHN
32
(r is a fixed positive number). Instead of (2.55) we first solve the equation
L[u] = (x)N[u] + w
(2.58)
with initial conditions (2.57). Any derivative of order loci < 3 is a linear com-
bination of terms Dau and D''u= with 1#1 < 4, lyi < 3, with coefficients formed from the derivatives of the a; of orders <3. Since the tat vanish for lxi > 2r, their derivatives are bounded for fixed t and all x. It follows that there exists a bound A(t) such that for all u and loci < 3 H[D"C(x) N[u(x,t)]] < A(t) K3[u(x,t)]
Then also (2.59)
H3[C(x)N[u(x,t)]] S A(t)K3[u(x,t)]
We now solve (2.57, 2.58) by iteration. Define a sequence of functions Uk(X,t) inductively by the conditions
u° =f (x) + tg(x)
(2.60a)
(2.60b)
L[u1Ct1] = (x)N[uk] + w
uk+1 =f
ui ' 1 = g for t = 0 These relations uniquely determine a sequence of functions Uk(X,t) of class C. Subtracting the recursion formulae for two successive uk we find that uk+1 _ uk = L-1[z(x)N[uk fork > 1 uk-1]] (2.60c)
-
It follows from (2.52, 2.59) that T
(2.60d) K3[uk+l(x,T) - uk(x,T)] < 0(T) I
< ¢(T)
T
uk-1]] dt 0
A(t)K3[uk(x,t) - uk-l(x,t)] dt ) .o From this formula one obtains by induction the estimate K31uk+I(x,7
- uk(x,T)] < I T"(C(T))(B(fl)k
where
B(T) = (max
C(T) = max
- u°(x,t)]
THE WAVE OPERATOR
33
More generally then, since VK3 satisfies the triangle inequality, K3[uk+'
- uk] <
k CTkBke " TB i
This shows that the uk form a Cauchy sequence in the K3 norm.
The transition to point-wise convergence is then achieved by means of the relation (2.53), which yields, e.g., for any derivative D°` of order Iai < 2 - Dauk(x,T)12 < y2H2[D2(ukii - uk)] y2K3[uk+, - uk] C Y2H4[uk+' - uk] < ID%k+i(x,T)
In this way one shows that the uk and their first and second x-derivatives and the ui and their first x-dcrivatives converge uniformly for all x and 0 < t < T. Since utl_i = e'2
i (x)N[uk]
-w
it follows that also the u i converge uniformly. There exists then a limit function u(x,t) of class C2 which satisfies (2.57, 2.58). In fact by the same type of argument, using norms of higher order, one finds that u is in C. The value of the solution uk+l of (2.60b, c) at a point (x°,t°) only depends on the values off, g, w, , u', a, at points (x,t) of the backward characteristic cone with vertex (x°,t°) : (2.61)
Ix - x°I < c(t° - t),
0 < t < t°
Since the backward characteristic cone from any point (x,t) of (2.61) lies itself in the set (2.61) it follows by induction that the values of all uk(x°,t°) are determined by f, g, w, , a, for (x,t) in (2.61). The same holds then for the value of u(x°,t°). If r > I x°I + ct°
Z(x) will have the value 1 in the cone (2.61), and hence u(x°,t°)
will then no more depend on r or the particular choice of the function t;(x). For r oo and corresponding functions fi(x) we obtain a function u defined for all (.r°,t°) with to > 0, which satisfies (2.55, 2.57). In this way a solution u(x,t) of (2.55, 2.57) of class C°° for all x and t > 0 is obtained for f, g, w, a, in Cr. The
domain of dependence of this solution for any point is the backward characteristic cone with that point as vertex.
FRITZ JOHN
34
We prove that for a, in C°° a solution u of class C°° of (2.56, 2.57) is also determined uniquely by f, g, w. It is sufficient to prove that a solution u of
L[u] = N[u]
(2.62a)
u=u,=0
(2.62b)
for t = 0
vanishes identically. We first prove this under the further assump-
tion that u has finite support in x, that is, that there exists for every s an r such that u(x,t) = 0 for
IxI>r,
0
Let indeed Z(x) be a function with the properties (2.57a). Then u is also a solution of
L [u] = (x)N[u] It follows as in (2.60d) that K3[u(x,T)] S
(T)
for all x and 0 < t < s
rT A(t)K3[u(x,t)] dt
for 0 < t _< s
From this we obtain by induction the estimates K3[u(x,T)] S k+ Tk(C(T)) (B(T))k
for 0 S T< s
where B(T) is defined as before and C(T) = Max. K3[u(x,t)] O
It follows for k
oo that K3[u(x, T)] = 0 and hence also u(x, T) =
0 for 0 < T < s. Since s was arbitrary, u vanishes identically. Let now u be a solution of (2.62a, b) of class C°° which does not necessarily have finite support in x. Take any r > 0 and a corresponding fi(x) with the properties (2.57a). Let
N[4u] = w(x,t)
Then w is a function of class C' which vanishes for lxi < r and jxj > 2r. By our preceding existence proof there exists a solution v(x,t) of
L[v] - N[v] = w(x,t)
THE WAVE OPERATOR
35
with vanishing initial data. Then iu - v is a solution of
v] = N[lu - v] with vanishing initial data. Moreover the constructed solution v has the proper domain of dependence and hence vanishes for
fix) >ct+2r Thus u - v has finite support in x, and hence vanishes identically.
Then u=vforlxi
Iteration can also be used to establish existence of solutions in the small for certain types of nonlinear differential equations with the wave operator as principal part, for example, for equations of the form15
L[u] = F(x,t,u) 2.9. Mixed Problems
We conclude this chapter on the wave operator yvith some remarks about more complicated problems for the wave equation that arise in applications.154 We mentioned before the boundary-
initial value problem and proved uniqueness of its solution. There a solution u of L[u] = 0 is to be found for x in a domain D and t >_ 0 from prescribed values of u and u, in D for t = 0 and of u or du/dn on the boundary B of D for t > 0. Somewhat similar in character are problems involving several media as 15 See K. Jorgens RI.
11 For mixed problems for more general equations see Campbell and Robinson [I], Duff (1], Wilcox [l], Ladyzenskaya [1].
FRITZ JOHN
36
they occur, e.g., in electrodynamics, where the fields inside and outside a wire have to be determined and different differential equations hold in the two media. In the simplest case one has to determine a function u(x,t) for t > 0 which satisfies one wave equation in a region D of x-space and another wave equation with a different c outside of D. The values u and ut are prescribed for all x and t = 0. In addition, one has on the boundary B of D two transition conditions derived from conservation laws which state that au and d (du/dn) are continuous across B, where a and
i have different constant values in D and its complementary region.16
Explicit solution of such problems by quadratures are available only in a very few cases, e.g., where D is a half-space. For domains
with a more complicated geometry one usually falls back on the
method of separation of variables. One considers the special solutions u(x,t) which are simply harmonic in t with frequency co, i.e., the solutions of the form (2.63) u(x,t) = a(x) cos cot + b(x) sin cot = Re
[O(x)e-«']
Here 4(x) = a(x) + ib(x) is a solution, in general complexvalued, of the reduced wave equation (2.64)
0i + 220 = 0,
2 = co/c
Problems involving boundary conditions which do not depend on t are then formally reduced to problems for the elliptic equation (2.64). If, for example, u is a solution of L[u] = 0 for x in a domain D of the form (2.63) which is required to vanish on the boundary B of D, then the parameter 22 must be an eigenvalue A for D which can be taken and 0 a corresponding eigenfunction as real. The solution u for arbitrary initial values f, g is then represented formally by a series of the form x
u = 2; (a cos (c),t) ± b sin (cA t) c (x) n=1
(in analogy to (2.15)) where the constant coefficients a,,, b are determined from the initial conditions. Convergence of the series 16 See C. Muller [1].
THE WAVE OPERATOR
37
in the square mean follows from the completeness of the eigenfunctions." Other types of mixed problem arise in the study of scattering of waves by an obstacle. In the simplest situation one looks for a solution u of L[u] = 0 defined for x outside a bounded region D, which has vanishing normal derivative on the boundary D of B and whose initial conditions are those corresponding to a plane wave coming in from infinity in the direction of the x. ,-axis. Decomposing the incoming wave into components of the form e'l(zl-``) the problem is reduced to that of determining a complexvalued solution O(x) of (2.64) outside D with vanishing normal derivative on B, for which 0' = O(x) -- e1 ' satisfies the Sommerfeld radiation condition
lim
r(
r
- i A)
0
uniformly for
17 See Courant-Hilbert, Vol. I, Chapter VI.
1
CHAPTER 3
Cauchy's Problem, Characteristic Surfaces, and Propagation of Discontinuities
3.1. Notation
In this chapter we consider functions u(x) in an (n + 1)dimensional Euclidean space with points x = (x0, .
(3.1)
. . ,
For differentiation we use the operational symbol
Di=, a axi
(3.2)
i = 0,1,...,n
The D; can be combined into a gradient vector
D = (Do,
(3.3)
... ,
The general derivative of the function u(x) is of the form DooDil
(3.4)
with non-negative integers aa,
... D, ^u . .
... ,
. , a which can be combined
In order to keep formulae from becoming too cumbersome we make use of the notation introduced by L. Schwartz [1]. For two vectors into a vector a = (CEO,
S=
(S0, S1, . .
.
a = (ao, .
, Sn),
. .
, an)
where the a, are non-negative integers, we denote by
`
the
monomial (3.5)
Expression (3.4) is
boo
... fin"
then written simply as D°u. The order 38
CAUCHY'S PROBLEM
39
ao + al + ... + a of the derivative will be denoted by jai. In addition we define a ! by oc! al ! ...
In this simplified notation many of the formulae involving functions of several variables assume the same form as for functions
of a single variable. For example the power series expansion of an analytic function f (x) = f (xo, x1, ... , becomes
.f (x +y) = I
(3.6a)
i
yc'D°f(x)
For the special function f (x) = (xo + xl +
... + xn)k with a
positive integer k, formula (3.6a) yields for x = 0 k!
(3.6b)
Ixi=k 0C!
Leibnitz' rule for the derivative of a product of two functions takes the form (3.6c)
Dx(f (x)g(x))
_ Fr=
a!
Y!,
(DIf (x)) (D"g(x) )
The general linear differential operator of order m can be written in the form (3.7)
L=
A.(x)Dx IxI <m
If there are several unknown functions u1i u2, ... , UN, they can be combined into a column vector
u =
The general linear system of differential equations of order m then takes the form (3.8)
L[u] _
A.(x)Dxu = B(x) jaj5m
where now the A. are matrices with N columns and as many rows as there are equations, and B is a column vector. We shall only consider the case of N differential equations of order m
FRITZ JOHN
40
In that case the A,, are square matrices of order N. Similarly the general m-th order system of quasi-linear equations is given by for N functions u1,
.
. .
, u..,..
1 A,,D"u = B
(3.9)
I'll =nj
where the Ax are square matrices and B a column vector, all depending on x, u and derivatives of order < m. More general nonlinear systems can be reduced to quasi-linear ones by differentiation. 3.2. Relations between Partial Derivatives on a Surface
Let S be a hyper-surface in xo ... x,, -space given by an equation 0(x) = 0(xo, x1, ... , x.) = 0
We put
ao
i=0,...,n
ax=
We assume that on S the vector rJ = Do = (Doo, ... , Dn0) does not vanish, so that there is a unique normal with direction 77
defined in each point of S. Let u be an N-vector defined in a neighborhood of S. For simplicity we assume that both 0 and u are of class C. We say that u vanishes of order k on S, if (3.10)
D"u=0
for jai
onS
Equivalent to (3.10) is the requirement that u is of the form (3.11)
u = 0k(x)Y(x)
where y is of class C, x in a neighborhood of S. That (3.11) for
regular y implies (3.10) is obvious. In order to prove (3.11) from (3.10) we apply a nonsingular coordinate transformation i = i(x) of class C°° locally, in which io = q(x). The existence of such a transformation is guaranteed by the assumption 77 0 on S.18 Condition (3.10) is invariant under regular coordinate 18 If, e.g., '70 r 0 at a point P of S we use in a neighborhood of P the trans-
formation
z, =x1 go =O(x), which has a nonvanishing Jacobian.
for i = I,...,n
CAUCHY'S PROBLEM
41
transformations. Let a'1[
w,(x)
=
axc;
Then w, = 0 for i < k and £o = 0. Expanding u with respect to xo we find from Taylor's theorem Xk
u=
(k
1
-o l )
1
fo (1 - s) k-1wx (szo, x1, ... , x,,)
ds
It is clear from this formula that
y = 0-ku =
(X0)-ku
is of class C°` with respect to the variables a and hence also with respect to X.
Let u vanish to the order k on S. Let x be a point of S. Then the expression lI
u(x +y) N"I
y"D"u(x) = v(y)
k oc.
considered as a function ofy has the property that all its derivatives
of orders
0
dt°
fort =Oandi =0,1,...,k
also. Then 0 = Jim t-kv(ty)
and hence for anyy Jim t-ku(x t-.0
y" Dxu (x)
+ y)
On the other hand, by (3.11, 3.6b) Jim t-ku(x + ty) 1-0
= urn two
u(x + ty)
(x + y) Ok
= y(x) lim
tk
n
i=U
(x + y) = y(x)(lim t t-.0
t-_0
= y(x)
(x + ty)
k
kl
-y l,y,) = y(x) 121=k OCI
ty )
k
FRITZ JOHN
42
Comparing coefficients ofy" we find for u vanishing of order k on
S that (3.12)
for x on S and lal = k
D"u(x) = k! i1"y(x)
It follows in particular, using (3.6b) that for x on S (3.13) II
a rj"Dxu = y I
i
r7 2"
a! = (?J + ... + ?12)AY = I rll2ky Ix
x
where Till denotes the length of the vector 71. We define the k-th normal derivative of a function u(x) for x on S by dkU
k!
raD"u (x)
= 1 111-k 1:
dv
Ixl=k M!
(Symbolically dk
k
n
dv =
(d
-
where the D, are meant to commute with the rl;.) For u vanishing to the order k on S relation (3.13) yields nI Y(x) ==
l
dA'u
k! dVk
Then (3.12) furnishes an expression for an arbitrary derivative of order k in terms of the k-th normal derivative: a
(3.14)
D"u(x) =
(1I11
Il du
for xon Sand lal =k
For functions u of class C°° the values of the normal derivatives
of orders <s on S uniquely determine all derivatives of orders <s on S. Indeed let v and w be two functions for which all j-th normal derivatives agree on S for j = 0, 1, . . . , s. Then for the difference u = v - w d'u dvj
=O
on Sforj =0, 1,...,s
It follows by induction that
D"u=0
on S for
Ss
CAUCHY'S PROBLEM
43
For if this relation has been proved for loci < k, where k < s then (3.14) applies and yields
Dau=0
on Sfor loci =k
3.3 Free Surfaces. Characteristic Matrix Let
L = I Ax (x) Do' jaj<_m
be an m-th order linear operator with coefficients Aa(x) which are square matrices of order N whose elements are functions in C'.
We associate with L and an arbitrary vector the characteristic matrix
m! I Aa(x)"
A=
I al -m
Let P(x,E) be the determinant of A(x,e). Then P is a form of degree mN in , the characteristic form of L. The surface S is called free or noncharacteristic with respect to
the operator L, if
P(x,rt(x)) ; 0 for x on S Here q is again the gradient of 0 which has the direction of the
normal of S. The surface S is called characteristic at a point x if P(x,77(x)) = 0. If this holds for all x on S then S is a characteristic surface and the operator L is tangential to S. Notice that these notions depend on the order m assigned to L. If, say, all Aa with lal = m vanish, so that L can also be considered as an operator or order <m then every S is characteristic for L taken as an m-th order operator. For S free the rank of the matrix A(x,r?(x)) is precisely N on S, whereas it is less than N on a characteristic surface. If the rank is precisely N - 1 along S we call S a simple characteristic surface. For a single equation the Aa(x) are scalars and the characteristic condition reduces to Aarla = 0 kI=m
For the wave operator
L =Do -c2(Di +D? +D3)
FRITZ JOHN
44
the characteristic condition for
(a 1
1)2
- c-
'
becomes
R G,-'), + )L
a-- )2I 3
=0
in accordance with p. 21. The Cauchy data (or initial data) of a vector-valued function u on S with respect to the m-tli order operator L consist of the values of the set of "?n functions u,
au
a,,,-iu
along S. Prescribing the Cauchy data of u on S is equivalent to prescribing in a consistent fashion all derivatives of u of orders <m on S. If S is free with respect to L then the Cauchy data of u on S together with the values of L[u] on S uniquely determine all D-"u w vith HxH m on S. For if zv and u have the same Cauchy data and L[v] = L[u] on S then w = v - u vanishes of order m on S. It follows that w = 01"y with y regular near S. Then by (3.12)
0 = L[v] - L[u] = L[w] = m! 114a(x)n"y loci =n?
A(x,n(x))y
on S. Since by assumption the matrix A(x,q) is nonsingular we must have 0 and then also Daze = m! ify = 0 for jxl = m On a free surface S the Cauchy data and the values of L[u] on S are independent of each other. Indeed if u has certain Cauchy
data on S then any v of the form v = u + 0-y with regular y has the same Cauchy data. We can give L[v] any values on S by determining the vector y on S from the system of equations L[v] = L[u] + A(x, I(x)) y and continuing y as a function in C" into a neighborhood of S. Conversely if S is not free the Cauchv data and the values of L[u] on .S are not independent. Indeed for vanishing Cauchy data L[u] must be of the form L[u] = A(x,27) y
C \UCHY'S PROBLEM
45
If S is characteristic at a point x the matrix 11(x,,)) is singular and the range of values of L[u] is restricted. If the surface S is free with respect to S then the Cauchy data of u on S together with the values of L[u] in a neighborhood of S uniquely determine all derivatives of u on S. (It follows then that an analytic solution u of a system of equations L[u] = B is
determined uniquely by its Cauchy data on a free surface S.) Indeed, it has been shown already that all Dxu with 1a1 r m are determined uniquely on S. At any point x of S one of the components of the vector q does not vanish, say r/k f 0. For any i the surface S is then also free with respect to the operator DLL near x. Indeed the characteristic matrix of DLL is given by (m m! + i) ! (qk(x))iA(x,n(x))
and hence is again nonsingular. Moreover the values of D.L[u] = DLB
are known on S. Then all D''u on S with 1a.1 m i can be found if the values of all D''3u with 1#1 , m + i are known on S. Induction over i proves that all D'"u are determined on S. Similar considerations apply to quasi-linear systems of equations of type (3.9). On the hyper-surface S the values of the coefficients
A,7 and B are determined by the Cauchy data on S. For given Cauchy data we can form the expression P(x,ir(x)) as before. We call S free "with respect to L and the given Cauchy data," if P(x,r1(x)) ' 0 along S. Again the Cauchy data on S will determine all derivatives on S of a solution u of (3.9) provided the Cauchy data are such that S is free. We see that for nonlinear equations a surface S may be characteristic or not depending on the particular solution u considered, more precisely depending on the Cauchy data of u on S. 3.4 Cauchy's Problem. The Uniqueness Theorem of Holmgren
For a system of m-th order differential equations (3.9) the Cauchy problem (or initial value problem) consists in determining a
solution for which the Cauchy data are prescribed on a hypersurface S. This problem cannot be expected to have a solution
46
FRITZ JOHN
without further restrictions on S or the Cauchy data. We have seen that at a point where S is characteristic the Cauchy data and the values of L[u] are in general inconsistent. Thus existence
of a solution can in general be expected only if S is free with respect to the Cauchy data. Even for S free the Cauchy problem will in general not have a solution, unless S in some sense is also spacelike. (See the discussion of the initial value problem for the wave equation in Chapter 2.)
The situation is simpler if one restricts oneself to analytic equations with analytic Cauchy data prescribed on an analytic surface S. We know already that in that case a solution u of the Cauchy problem which is analytic in a neighborhood of S is determined uniquely, provided S is free. There also exist always locally analytic solutions of such an analytic Cauchy problem. This is asserted by the theorem of Cauchy and Kowalewski. More
precisely the theorem states: Let the A,, and B be analytic functions of their arguments (i.e., of x and the various Yu with #I < m) in a certain domain. Let S be an analytic hyper-surface. The prescribed Cauchy data on S determine at each point x all JYu with Ii3I < m. Assume that the x on .S and corresponding DOu are in the domain of analyticity of the Aa and B. Let x° be a point of S at which S is free (with respect to the Cauchy data). Then there exists an analytic solution u of the differential equations in a sufficiently small neighborhood of x° taking the prescribed Cauchy data on S in that neighborhood. This theorem is very general inasmuch as it applies to analytic equations regardless of type (elliptic, hyperbolic, or otherwise). From the point of view of physical applications however the restriction to analytic equations, analytic solutions, and analytic data is much too rigid. Since analytic functions have the property of unique continuation there is no clear cut domain of dependence: data prescribed in an arbitrarily small neighborhood on S of a point x° already determine u everywhere. There is no room
to prescribe boundary-data on some surfaces in addition to Cauchy data on others. Hence we cannot deal with an isolated physical object. As far as behavior of solutions in the large is concerned the theorem of Cauchy and Kowalewski only fits a universe which can be constructed completely by analytic continuations from its smallest portion. In spite of this seeming
CAUCHY'S PROBLEM
47
inadequacy the theorem is very useful as a tool due to the fact that general real continuous functions can be approximated by analytic ones. We shall not go into the proof of this classical theorem, which
can be found in many books on partial differential equations.19 We only recall the main features of the usual proofs : S is transformed analytically into the plane x0 = 0. In a neighborhood of
this free plane, the solution is expanded into a formal power series. Then a "majorant" initial value problem is formed which
is so simple that it can be solved explicitly and thus can be shown to have an analytic solution. Finally, the convergence of the formal power series solution for the actual problem is established by comparison with the majorant problem. The theorem of Cauchy and Kowalewski gives existence of
analytic solutions of the Cauchy problem in the small, if the differential equation and all the data are analytic, and if the initial surface is free. It is clear moreover that no other analytic solutions can exist than that given by the formal power series whose coefficients are determined uniquely. However, this still leaves open the possibility that other, nonanalytic, solutions of the Cauchy problem exist. Actually for linear analytic equations there exist no other solutions. We have indeed the famous UNIQUENESS THEOREM OF HOLMGREN. The Cauchy problem for a linear system of differential equations (3.8) with analytic coefficients A,,(x) and with Cauchy data prescribed on an analytic free initial surface
S has at most one solution in a neighborhood of 5.20 (Neither the data
nor the solution are assumed to be analytic.)
The proof is based on a simple argument reducing uniqueness of one Cauchy problem to existence of solutions of another one. Consider a region R in x-space with boundary B. Let the differential operator L be defined by (3.15)
L = I A,, (x) Da I'll :5 rn
19 See, e.g., Courant-Hilbert [ I ] Vol. 2, Goursat [ 1 ], Petrowskii [ I ]. For extensions see P. C. Rosenbloom [2], A. Friedman [1], Hellwig [1]. 20 See Holmgren [1], John [4]. It is sufficient to assume that a is defined and of class Cm on one side of S, of class C-1 on S. The assumption that S is analytic is not essential. For extensions of the theorem to cases where S is not free see Hormander [2].
FRITZ JOHN
48
where the A. are square matrices. Let u and v be respectively a column and a row vector such that at every point of B either all DNu or all DOv with 1#1 < m - 1 vanish. Integration by parts then givesr us Green's identity J RV
II
dx = f
.1 (-1) I x D"vA,u dx
R (1,215 m
IaI
Let now the boundary B of R consist of two parts S' and S", where all Cauchy data of u vanish on S', all Cauchy data of v vanish on S. Let L[u] = 0 in R and define
M[v] _
(3.16)
(-1) I"I D"vAa = w(x) IaI<m
Then, by Green's identity,
0 = fR wu dx
(3.17)
If for each w we can find a solution v of (3.16) with vanishing Cauchy data on S", we could conclude from (3.17) that u = 0 in R. For then we could take for w the transposed vector u* of u, and the result would follow. The same conclusion can be made if we can at least find such v for every w of a dense set, i.e., if every continuous w in R can be approximated uniformly by a sequence of w for which there exist solutions v of (3.16) in R with vanishing Cauchy data on S". Again for w = u*
in R. Now the problem of finding a solution v of (3.16) with vanishing Cauchy data is itself a Cauchy problem for the transposed vector v* of v. We can write (3.16) in the form 0
M*[v*] _ IaI<m
(-1)I-ID"A*(x)v* = w*
We see that the Cauchy problem for L with data on S' has at most one solution in R, if the Cauchy problem for M* with vanishing Cauchy data on S" can be solved for a dense set of functions w. Here the operator M* has the principal part
(-1)," I Az(x)D" 1=I=m
CAUCHY'S PROBLEM
49
and the same characteristic equation as L:
0 = determinant (-1) "' 1 A*ij" I«I =m
_
(-1)'"N determinant (
Ax1"1
S" is free with respect to M* if it is free with respect to L. It suggests itself to use w from the class of analytic functions, which a
S
Fig. 6
is dense. The theorem of Cauchy-Kowalewski guarantees existence locally of a solution v of M[v] = w with vanishing Cauchy data on S", if S" is analytic. However these solutions v cannot be expected to exist throughout the full region R. Thus we have to modify this idea, breaking up R into sulliciently small pieces. Let now S be the free analytic initial surface and S a bounded
portion of it. We deform S analytically keeping its boundary fixed. More precisely, we consider a family of surfaces S, with common boundary given by an equation 0(x,2) = 0 for 0 < A < a, where 0 is analytic in x and A. For any A1, A2 with 0 <, Al < 12 < a the Sx with Al < ) < 22 shall cover a region R),12 bounded by S,, and Sx2 (see Fig. 6). Choose the function qi(x,).) so that all S, with 0 < A S a are free with respect to the linear operator L. We shall prove that then a solution u of L[u] = B defined in R .a is deter-
mined by its Cauchy data on So. For the proof we need the following lemma.
FRITZ JOHN
50 LEMMA.
There exists a positive a such that for any A,
0
,u
with
µ
the Cauchy problem
M[v] = w
in R;,,,,
DRv
=0
on S,, for I#1 < m
possesses solutions v for a set of w which is dense in the space of functions continuous in RA,,,.
The uniqueness theorem follows easily once the lemma is proved. For, let L[u] = 0 in Ro,, and let the Cauchy data of u vanish on S0. (It is, of course, sufficient to consider the case of vanishing Cauchy data.) We can find a sequence of values ,L, A,,
and
. . . ,
A. such that
0 = AO < Al < A2 < ... < Ak = a Ai+1-2 <e
forj=0,...,k-1
For a dense set of w there exist solutions of M[v] = w in
Ro,A1
with vanishing Cauchy data on Sat. It follows that u = 0 in R0,21.
Then also the Cauchy data of u on SA1 vanish. Repeating the argument we find that u vanishes also in RA1.AS, etc. We finally arrive at the conclusion that u vanishes throughout Ro,a.
It remains to prove the lemma. We know from the CauchyKowalewski theorem that if w is analytic and x any point of a
free analytic surface there exists in a neighborhood of x an analytic solution v of M[v] = w with vanishing Cauchy data on that surface. Let now a be a compact free portion of an analytic surface, and let w be analytic in a neighborhood of a. For any x on a we have a sphere of radius r(x) about x and in that sphere an analytic solution v of M[v] = w with vanishing Cauchy data on a. A finite number of spheres of radius +r(xk) about points xk of S covers a neighborhood of a. If the spheres with superscripts
k and j overlap and, say, r(xk) > r(x'), the sphere of radius r(xk) about xk will contain x'. Then the two analytic solutions v corresponding to xk and x' will have the same derivatives at x' and hence be identical. Thus the construction defines without ambiguity a solution v of M[v] = w in a full neighborhood of a with vanishing Cauchy data on a.
Let now w be an entire function of x and of parameters
CAUCHY'S PROBLEM
51
o, $1, . . . , n which form a vector $. Let the SA form the family of free surfaces described before. For every E with IEl < 1 and ,u
with 0 < ,u < a we wish to determine a solution v = v(x,y,E) of M[v] = w(x,$) with vanishing Cauchy data on S,,. We look at the equation M[v] = w as a system of partial differential equations f o r a function v of the 2n + 3 independent variables x0, x,n, ,u,
... ,
o, ... , $,,, for which no derivatives with respect to the
last n + 2 variables happen to enter the equations. We ask for a solution for which all derivatives of v of order <m with respect to x0, ... , n vanish on the analytic hyper-surface 0
in x0,
... , -space. The points (x, u,E) with I$I <1 xonS, 0<4u
form a free analytic compact hyper-surface in x, 1u, $-space. (The
characteristic form is not changed by the additional variables.) It follows that there exists a solution v(x, in a full neighborhood of this hyper-surface. Thus there exists a 6 > 0 such that for any u, with 0 c µ < a,
IEI < 1
the function v(x, u,E) is defined for x in a 6-neighborhood of S,,. Consequently there exists an e > 0 such that v(x, u,E) is defined in Rx.,, for µ - A < E and all with ICI < 1. It remains to choose the family of functions w depending on the parameters so that their linear combinations for 1 are dense. Take w(x,4) = et ;
Then every monomial xa
= xo ... x,aan 20
can be approximated uniformly in every bounded set of x-space by linear combinations of the w(x,fl) with 1#1 < 1. We obtain indeed such approximations from the formula t -) xa =E--o1im 11 k=0 'k
Since the linear combinations of the monomials are known to be dense, the same holds for the linear combinations of the w(x,).
52
FRITZ JOHN
It follows that the equation M[v] = w has a solution v in RA.,, with vanishing Cauchy data on S,, for a dense set of w, provided It - A < e. This proves the lemma and completes the proof of the uniqueness theorem. We see that the solution u of the Cauchy problem is determined uniquely at all points that can be reached by deforming a portion
Fig. 7
of the initial surface analytically through free surfaces with the same boundary. How far into space uniqueness can be established by this method depends on the geometry of characteristic surfaces.
For elliptic equations the characteristic form is definite and every surface is free. In that case a solution u defined in a connected domain will be determined uniquely by its Cauchy data on any analytic initial manifold So. (This follows also from the analyticity of all solutions.) If L[u] = 0 is the wave equation and R a region bounded by a characteristic cone and a spacelike surface S, it is plausible that
the whole interior of R can be swept out by spacelike surfaces with common boundary, and hence that u is determined uniquely in the whole interior of the cone by its Cauchy data on S (see Fig. 7). Uniqueness of the Cauchy problem for linear equations with nonanalytic coefficients is much harder to establish. Of course, the argument given here for uniqueness of solutions of analytic equations will work, whenever the corresponding existence
CAUCHY'S PROBLEM
53
theorem for the "adjoint" equation M[v] = w has been established. The difficulty is that the Cauchy problem will in general have no solution, except in the hyperbolic case. References to uniqueness results for nonanalytic equations will be found in the second part of this book. 3.5. Propagation of Discontinuities
If S is a free surface we can compute all higher derivatives on S of a solution of a differential equation from the Cauchy data on
S. If the equation and the data are analytic a power series for the solution of the Cauchy problem can be obtained in this way. Even in the nonanalytic case two solutions of the same equation agreeing in their Cauchy data on a free surface S must agree in all higher derivatives, as far as they exist and are continuous. Consider the situation where S divides some portion of space into two regions R' and R" and solutions u', u" of class C'° are defined respectively in R', R". In general u' and u" will not "fit together"
along S. We can look at u' and u" as defining a solution a in R = R' -I- R", which is discontinuous on S. Suppose that on S, u' with its derivatives of order <m agrees
with u" and its derivatives of order <m. Then, if S is free, u' and u" also have the same higher derivatives on S. We see that continuity on the free surface S of all derivatives of u of orders <m implies continuity of any higher derivatives. Thus, if a is to be a solution regular on each side of S, of class C111-1 on S, but
discontinuous in some derivative of order m in each point of S, then S has to be a characteristic surface. It is often easier to describe how discontinuities of solutions behave than to "solve" the differential equation. Such a description can yield important qualitative information about a solution. For example, wave fronts and shocks can be interpreted as discontinuity surfaces for some of the dependent variables. Geometrical optics can be considered as the study of the propagation of discontinuities of solutions of Maxwell's equations.
We shall restrict ourselves here to the case of a linear system of m-th order equations (3.18)
L[u] _
A,,(x) D"u = w(x) kkI=m
FRITZ JOHN
54
for u = (u1, ... , u,,). Our aim will be to discuss propagation of jumps in the m-th order derivatives of a solution for which all lower order derivatives are continuous.21 The prospective discontinuity surface S shall be given by an equation
(x) = 0 where the gradient vector
rl = (DOO, ... , Dn0) shall not vanish along S. To simplify computations it is conveni-
ent to introduce a calculus of formal "derivatives" of L with respect to o.22 We associate with the mth order differential operator L (whose coefficients may be matrices) and with a scalar O(x) the commutator Lcl> = LqS - c¢L
this means that Let is the differential operator defined by D1>[u] = L[ou] - OL[u]
It is easily seen that Lcl> is of order at most m - 1. If L is a "zero-order" operator, i.e., L[u] = A(x)u where A is a matrix, then
-
DI) = A0 ¢A = 0 If L is a first-order operator of the form L = A(x)D,, then L(l) is the zero-order operator AD=O - OAD, = A(D=o) = An;
More generally we define a sequence of derived operators DO) LM, Lc2>, (3.19)
... by
Lco> = L,
Lcx-ri>
=
(L(' )) cu = L(k) 0 - OLck>
21 The discussion is based on the treatment of the case of systems of first order equations given by Courant and P. D. Lax [1]. These authors also
study the case where u itself becomes discontinuous. For shock discontinuities propagating along noncharacteristic manifolds see Courant-Friedrichs [1]. 22 Sec John [5]. These formal derivatives of L also lead to useful forms of Green's identity.
CAUCHY'S PROBLEM
55
By induction Lck) is at most of order m - k. It follows that L(k) is identically zero for k > m and that L(") is a zero-order operator. By induction (3.20)
r!
r =O
r)
kk!
=I(-1)r
L(k)
)
(
and conversely k
Lok =
(3.21)
k
r!(k-r)! !
r
x rLcr)
One easily verifies that for two operators L1, L2 (L1L2)") = L(1)L2 + L1L(21)
More generally "Leibnitz' rule" holds: (L1L2)(k) =
(3.22)
k
!
ro r! (k - r) !
Lcr)k-r)
Let L1, L2 be respectively of orders ml, m2. It follows from (3.22)
for k = m1 + m2 that (LIL2)(m`+m2)
= mi + m2
t
' Llml)L2mq)
m1! mil!
Similarly for k operators L1, L2, .... Lk of orders m1, m2, (L1L2
...
Lk)cml+... +mk) _
(m1 + ... + mk) ! m1 ! m2 !
...mk
Lzml) L2 s)
!
..,mk
... Lk k)
If the factors Li are all of order 1 we find (L1L2
... Lk) (k) = k ! L(l) L.,l) ... L(')
Applied to a kth order derivative
D" =Do0...Dn^,
jai =ao +... +an =k
the formula yields
= k! (D(O'))"o ... (D;,'))" = k! rj'o ... 27n° = k! 2 This result permits us to find an explicit expression for the mth derived operator of the mth order operator (Da) (k)
77
L= IA1D" 1"I <_ m
FRITZ JOHN
56
We have L(m)
=
I Ax(D2)(m) 1a1:5m
= m! I
Ax71a
IaI =n,
= A(x,'1(x))
is the characteristic matrix of L. We are now ready to discuss the propagation of singularities along a hyper-surface S given by 0 = 0. We consider a solution u of L[u] = 0 with jump discontinuities in its mth derivatives on S. More precisely let u' and u" be two functions of class C°° in a neighborhood of S satisfying where
L[u] = L[u"] = 0 Let u' and u" agree on S with their derivatives of orders <m: Dxu' = Dxu"
on S for ( a l < m
Let there exist for each x on S a j with (f l = m such that D'3u'
DIzu"
at x
If we define u = u' on "one side" of S and u = u" on the other side, we have in u a solution with jump discontinuities in its mth derivatives of the kind that will be considered here. The D°`u with jai < m will be continuous on S while for each x on S a certain Dxu with 1#1 = m will jump by the amount D'lu'
- Dxu"
in passing from one side of S to the other one. Since u' - u" vanishes of order m on S we have
u' - u" _ 0'"y where y is regular in a neighborhood of S. Then on S by (3.12)
D'u'-Ddu"= m!gyt- 0 Hence y
0 on S. Since however on S
0 = L[u] - L[u"] = A(x,77(x))y it follows again that A(x,i7) must be a singular matrix on S. Thus
the surface S of discontinuity must be a characteristic surface. Assume that S is a simple characteristic surface, i.e., that the
CAUCHY'S PROBLEM
57
rank of A(x,ii) is precisely N - 1 in the points of S. There exists then for each x on S a solution C :t- 0 of A(x,i1(x))C = 0
(3.23)
determined uniquely within a factor. We can pick a particular solution C of (3.23) on S and continue it in such a way into a neighborhood of S that the extended vector C will be of class C°°, though possibly multiple-valued in the large (by normalizing for example C to be of length 1 and constant along the normals of S). From the assumption that A(x,?)) has rank N - 1 it follows
that along S Y = AC.
where A = A(x) is a nonvanishing scalar. We can also think of A as continued into a neighborhood of S as a function of class C'. Then AC, vanishes along S and, hence, is of the form
y-AC=4E where the vector E is again of class C° near S. Thus u' - U" = 0»,AC 0"1+lE If A is known on S then y = AC is determined and so is the jump in any mth order derivative of u across S: D''u'
- D'3u" = m ! iy = m ! r/AC
We see that A measures the magnitude of the jumps in the mth derivatives of u (for a fixed choice of C). The object is now to
derive an ordinary differential equation for . connecting the magnitude of the jumps at different points. We observe that on S for any i = 0, ... , n
0 = D,L[u'] -- D,L[u"] = D,L[u' - u"] = D;L[c6"'AC] +
Applying (3.21) with L replaced by D,L and (3.22) we find that on S (3.24)
0 = (D,L)"m)[AC] D,L(")[AC] -4- m(D,)(>)D,,,_i)[i.C]
+ (m + 1) (D,)ci DiL(m)[AC] +
mm,D",_1)[AC]
+ (m + 1)77,L'111)[E]
FRITZ JOHN
58
Now for S characteristic there exists in addition to the column vector C satisfying (3.23) also a row vector I' for which
rA(x,i1(x)) = 0 and hence
on S rL("') = 0 Multiplying (3.24) from the left with q,r and summing over i (3.25)
we obtain (3.26)
)],rD,L('")[AC] ±
0=
m(1?l211'L(»,-1)[AC]
are first-order operators we have in (3.26) a linear partial differential equation of first order for the scalar A satisfied in the points of S. This equation is of the form
Since AD"') and
D"'-1)
(3.27)
i=u
a,D,[A] + bA = 0
with certain scalars a b. Here
Ii a,D, represents differentiation in a direction tangential to S. To prove this it is sufficient to show that 0
or equivalently that A _ 0 is a solution of (3.27) on S. Substituting for A in (3.26) we find indeed that ?j,FD,L(m'[0C] ± m
_
?j?rL(m-1)[0C] rL(m)[C]
n,rD=L`m'[c] + 1
1
cL(m-1)[C]) = 0
+m
Hence (3.27) represents a first-order linear differential equation for i. along the surface S. If we introduce the "bi-characteristic" curves on S defined by
(3.28)' = a Ids
CAUCHY'S PROBLEM
59
(i.e., the integral curves of the direction field on S given by the a,) we obtain an ordinary differential equation (3.29)
ds+bA=0
for A along each of those curves. The value of ). for one particular
s, i.e., at one point of a bi-characteristic curve uniquely determines A all along the curve. Thus (3.29) or (3.27) represents a law of propagation along bi-characteristics for the magnitudes of the discontinuity of the mth derivatives of u.
As an illustration consider again the case of the scalar wave equation (3.30)
L[u]
= (Do - c2(D + D2 + D2))[u] = 0
One verifies easily that here 3
(3.31) (3.32)
L(1>[u] = L[O]u
-1-
2 (Do/)(Dou) - c2
L(2)[u] = 2 (D00) 1 - c2
=1
(DO)(D1u)
3
t-1
(D.0)2)u
= 2(r12 - C2(rri + r2 + 7)a))u
Without restriction of generality we can assume that the surface S is represented by an equation of the form (3.33)
O(x) = xo - '(x1,x2,x3) = 0
Then
-
A()2)
(x ,y7) = L(2) = 2 ( 1 - c2 iG 1
The surface S is characteristic if L(2) = 0 on S, i.e., if V(x1,x2ix3) is a solution of the partial differential equation (3.34)
1 - c2 1=1
axl,12
=0
identically in x1, x2i x3. Since x0 does not enter explicitly in the expression for A(x,i1(x)) it follows that (3.34) implies that Lc2> = 0 for all x0, x1, x2, x3, not only on S.
In the present case of a single equation the vectors C and r
FRITZ JOHN
60
reduce to the scalar 1. The jump in any second derivative Di'u of a solution across S is then given by (3.35)
D#u'
- D'''u" = 27/''2
Since for characteristic S the expression L(2) vanishes identically, the differential equation (3.26) for A reduces to 0
=
Lci)[A]
=
2L[ o]
+
ai + c 21s
2
aXo
i
aa
1 ax; ax=
= di + 1L[0] ds
Here the bi-characteristic curves are defined by (3.36)
dd °
x,
= 2,
= 2c2 x
for i = 1, 2, 3
The bi-characteristic curves can be determined from a system of ordinary differential equations without reference to the particular characteristic surface S on which they lie. We shall not give the general proof of this fact. But in our example of the wave equation, we have from (3.34) along a bi-characteristic d 2c2 ds
aIP
ax;
=
3
dxk
a (2c2
k =o ds dxk `
= 4c4
a aX,
k1 a aak ax ,
=
2c4
R
(\2 _ ax i
k.1-0
aX k
/O
Consequently the dx,lds for i = 0, 1, 2, 3 are constant along a bi-characteristic curve. This means the bi-characteristic curves are straight lines in xoxlx2x3 space. Since by (3.34, 3.36) ' . (dX \2 c=1
ds
f2
= c2 t ` ds
they are straight lines forming the angle arc tan c with the x0-axis.
We can give a geometric interpretation in xlx2x3-space. Here
x0 is identified with the time t and S is replaced by its level surfaces s,
,(xl,x2ix3) = const. = t progressing with time. We think of a solution u with discontinuities in the second derivatives which are located in xlx2x3 space on the surface st, the "wave front." The characteristic differential
CAUCHY'S PROBLEM
61
equation (3.34) expresses that the wave front spreads with normal velocity c (see page 24). The bi-characteristics in four-dimensional space correspond to the rays in three-space given by
d' ^c2az
fori = 1,2,3
{
The rays are straight lines orthogonal to all surfaces s, The amount by which a second derivative of u changes at a point in three-space when the wave front passes is given by (3.35). Since the 71, are constant along a ray the jump in any second derivative is a constant multiple of A. Along a ray, the quantity A satisfies the ordinary differential equation
2d
+Ac2Aip=0
This formula can be interpreted geometrically as stating that the "intensity" A2 of the discontinuity varies inversely as the area of corresponding surface elements on the st cut out by the same rays.
We observe that the considerations given are purely formal and do not show that there actually exist solutions with discontinuities of the type described. It can be shown23 that the initial value problem for a hyperbolic equation with discontinuous
initial values leads to solutions that are discontinuous across characteristic surfaces which pass through the loci containing the discontinuities of the initial data. 23 See Courant-Hilbert, [1], Vol. 2, pp. 629 et seq.
CHAPTER 4
Linear Hyperbolic Differential Equations
We restrict ourselves to the case of a single scalar equation of order m of the form (4.1)
L[u] = I A(x)Du(x)
= w (x)
121:5M
Here x = (xo,xli ... , xn). The variable xo plays the role of the time t. Combining the remaining "space variables" x1, x2, ... , x into a vector X we write x = (xo,X). We make the assumption that the coefficients of the highest order terms in L are constant, i.e.,
for jai = m
A« = const.
Cauchy data shall be prescribed on the hyper-plane xo = 0: (4.2)
Dou(x) =fk(X)
forxo = 0,
and k = 0, 1,
..., m -
1
The initial plane shall be noncharacteristic, i.e., A(m.O....,o) : 0
Dividing by this constant we can normalize our equation in such a way that (4.3)
A("1,o,...0) = 1
In general no solution of the Cauchy problem will exist unless the coefficients are restricted by certain inequalities which express
the hyperbolic character of L. Solution of the problem in the hyperbolic case will follow the lines developed in Chapter 2 for the special case of the wave equation. 62
LINEAR HYPERBOLIC EQUATIONS
63
It is possible to reduce the problem for general data w, ft to the standard Cauchy problem
L[u] = 0
(4.4) (4.5)
(Dou),.=0 =0
fork =0,...,m -2:
(Do-1)s,=o =f(X) The reduction of the general problem to the standard problem is achieved by introducing first of all m-1 1
u* = u(x)
- 1k, fk(X)xo
as new unknown function. This function will have vanishing initial data and satisfy an equation of the form
L[u*] = w*(x)
(4.6)
with known w*. To obtain u * (x) we make use of Duhamel's integral zo
u*(x) =
(4.7)
V(xo,X,s)
ds
0
where V(xo,X,s) = v(x,s) is a solution of the differential equation (4.4) satisfying the initial conditions (DoV)x,=8 =
0
fork=0,1,...,m-2
w*(s,X)
for k = m
-1
which depends on the parameter s. In verifying that u* satisfies (4.4) and has vanishing initial data we make use of (4.3) and of ,the observation that Dxou*(x)
=foz°D0V(xo,X,s)
Dou*(x) =
X0
ds
fork < m
Do(xo,X,s) ds +
0
=J ='DoV(xo,X,s) ds + w*(xo,X) 0
Obviously the Cauchy problem for V can be reduced to a standard
FRITZ JOHN
64
Cauchy problem by a translation in the xo direction. The differential equation will not be changed by this translation in case none of the A"(x) depends on xo. We have in that case V(xo,X,s) = U(xo - s,X,s)
(4.7a)
where u(x) = U(xo,X,s) is a solution of the standard problem (4.4, 4.5) with f (X) = w*(s,X). Hence it is only necessary to consider the standard problem.
Even Cauchy problems for more general "spacelike" initial surfaces can be reduced to the standard problem by the methods used for the wave equation. It will be seen that the initial value problem for equation (4.4) can be solved by iteration once the corresponding problem for the "homogeneous" equation
Lm [u] = I A"D"u = 0
(4.8)
Ix1=m
has been solved and proper estimates obtained for its solution. This equation only involves the principal part
Lm = I A"D"
(4.9)
I"1- m
of the operator L, which, by assumption, has constant coefficients.
The standard Cauchy problem consists in finding a solution of (4.8) subject to the initial conditions (4.5).
u
4.1. Solution of the Homogeneous Equation with Constant Coefficients by Fourier Transform24
The customary approach to the solution of (4.8, 4.5) consists in finding special functions f for which the problem can be solved
explicitly, and then obtaining the solution for general f by superposition. Assume that we can solve the problem for all f (X) of the form f (X) = O(X, Y) depending on parameters Y = (yl, . . . , yn)
and let the corresponding solutions be u = v(x,Y); assume also that a "general" f can be written in the form (4.10) 24 See Garding [1].
f(X) = Jf(Y)0(X,Y) dY
LINEAR HYPERBOLIC EQUATIONS
65
with a suitable function f (Y). Then a formal solution of (4.8, 4.5) for f given by (4.10) is represented by (4.11)
u(x)
=f f (Y)v(x,Y) dY
This formula defines an actual solution of our problem, if the integral converges and if formal differentiation under the integral sign can be justified. Uniqueness of the solution follows by application of the method of Holmgren (see Chapter 3).
The classical method of Cauchy [1] employs Fourier transformation. This corresponds to choosing (4.12)
O(X,Y) =
n
(2i)-n12 eiF."1 ,
X. Y =
x=3'c
The function f (X) is then the Fourier transform off (X). For f sufficiently regular and vanishing sufficiently strongly at infinity we have the reciprocal formulae25 (4.13)
f (X) = (27r) _nh2 fe.t 2'f (Y) dY J
A
(4.14)
Y (Y) = (2i) -9:12 e- i T 1'f (X) dX
and the identity of Parseval between the Hilbert norms off and f: (4.14a)
fIfxI2dx = f If(Y)I2dY
For our purposes it is sufficient to observe that for f (X) in Co (i.e., for f (X) of compact support and writh continuous derivatives
of orders < s) the function f (Y) defined by (4.14) will be entire analytic in Y and will be such that 11,13f (y)
is bounded uniformly for all real Y. This follows by integration by parts. For s > n the integral in (4.13) will then converge absolutely and represent f (X). as See Courant-Hilbert [1], Vol. I, p. 80. When no limits are indicated integrals are to be extended over the whole space.
FRITZ JOHN
66
The first step is to obtain the solution u = v(x,Y) of (4.8, 4.5) corresponding to f = O(X,Y). We write
L. = I
(4.15)
P(D)
Ix) =:n
where P is a form of degree m in D = (Do, ... , D,,,), the characteristic form of the operator L. We try to obtain v of the type v(x,Y) = v(xo,X,Y) _ 0(X,Y)Z(xo,Y)
(4.16)
Using the special form (4.12) of 0 we find that Z(xo,Y) has to be a solution of the ordinary differential equation P(Do,iY)Z(xo,Y) = 0
(4.17)
with initial conditions (4.18)
(D Z(xo,Y))zo=o =
10
fork=0,l,...,m-2
t1
fork=m-1
The solution of this ordinary differential equation problem is given by
Z(x0,Y) =
(4.19)
1.
e'A°
2 Jr P iA,iY ( )
dA
where the integration is extended over a path r containing all roots A of the denominator of the integrand: P(iA,iY) = imP(A,Y) = 0
(4.20)
To verify that the Z of (4.19) satisfies the differential equation (4.17) we only have to observe that P(DO,iY) Z =
2t
dA = 0
since the integrand is regular analytic inside F. Moreover (
kZ):,=o =
1
Tff
r P(i),iY)
dA
has the values given by (4.18), as can be seen by expanding r to infinity and using that by (4.3) the coefficient of Am in P(iA,iY) is im.
LINEAR HYPERBOLIC EQUATIONS
67
A formal solution of (4.8, 4.5) is then represented by the expression
u(x) = u(xo,X) =
(4.21)
Z(xo,Y)f(Y) dY
(2,r)-"I2
This formula gives an actual solution, if the integral and the integrals obtained by differentiations of order <m converge 71
al
x1
T2
A2
bl
2
3
73
b2
a3
b3 1\4
Fig. 8
sufficiently well. Since the integrand is analytic in Y, convergence
depends mainly on the order of magnitude of Z(xa, Y) for large
real Y. It turns out that in the special case where all roots A of (4.20) are real for every real Y the expression Z and its x0derivatives increase at most like a power of I YI for I YI - oo. The u of (4.21) will then represent a solution for all f of class Co with sufficiently high s.
Assume then that for real Y equation (4.20) has real roots A = Al, A2, ... , A. For the purpose of getting an estimate for Z we choose a particular type of path r, possibly consisting of several closed curves. Take for each Ak the closed interval defined 1 < A < Ak + 1. The union of these intervals conby Ak sists of a finite number of separate intervals 11, 12, . . . , I!, where
-
m. Suppose I,, is given by a, < A < b,; associate with I, the boundary y, of the rectangle in the complex A-plane given by
-1
a9 < Re A < b9
(see Fig. 8). Take now for r the union of all y,,. Then r is a path enclosing all roots Ak. The points of r have at least distance 1 from each Ak, so that on r m IP(A,Y)I=IIIA-AkI >1
k=1
FRITZ JOHN
68
Moreover F has at most length iT[2(bv
p=1
- an) + 4] < 8m
I t follows from (4.19) that (4.22)
I Z(xo, Y) I <
4m
e1xol
We derive similar estimates for the x0-derivatives of Z. By (4.20) ;./1 171 is a root of the equation
P().
Y)
=0
FYI' IYI
which has bounded coefficients for all Y, the term (A/I YI)'n having coefficient 1. It follows for A = Ak that All YI is bounded uniformly in Y: (4.23)
Af I YI
IAkl
Then for A on I' certainly 121 (4.23a)
l D Z(xo, Y) I
= <
M I YI ' V2, and hence Zk
27rim
4m
i
(M
ketxox
P(A,Y)
dA
I YI -- 1/2)kelxol
7T
Hence the integral in (4.21) and its derivatives of order <m will converge absolutely and uniformly in x for bounded x0, if
f(Y) =
0(IYI-n_m_l)
for large I YI
This is always the case for f in C I n+m. Thus we have proved: if the characteristic equation (4.20) has only real roots A for real Y,
then the initial value problem (4.5, 4.8) has a solution u(x) of class C"' given by 14.21) for all x provided f is of compact support
and in
C1-"+m.
One can show that if the reality condition is violated then Z(xo, Y) is unbounded for large I YI, and in fact increases faster than any power of I YI. In this case it is possible to prove26 that
the initial value problem has no solution for any f of compact 26 See John [6].
LINEAR HYPERBOLIC EQUATIONS
69
support except when f = 0. We are justified in calling homogeneous operators Lm of the form (4.15) hyperbolic (with respect to the plane x0 = 0) if and only if P(1,0) : 0 and the characteristic equation (4.20) has only real roots A for real Y. They belong to equations for which the Cauchy initial value problem with data on x0 = 0 is well posed. One can formulate the conditions f o r hyperbolicity geometrically. The n + 1 quantities A, y , . . . , y,, can be interpreted as homogeneous coordinates of a point in n-dimensional projective
space 7r,,. The points with P(A,Y) = 0 form an mth degree algebraic surface a in 7r,,, the "normal surface" corresponding to
the differential equation. The point (1,0,
. . .
,
0) in 7r is the
"origin." For varying A and fixed Y the points (),Y) describe the line joining the origin to the point (0, Y) at "infinity." Condition (4.3) which states that the plane x0 = 0 is not characteristic is equivalent to a not passing through the origin. Hyperbolicity of equation (4.8) (with respect to the plane x0 = 0) means that the
origin does not lie on a and that every line through the origin intersects a in m real points (which possibly coincide or lie at infinity). In the case of the wave equation
L[u] = (D2
- c2(Di y D? + Ds))u = 0
the normal surface is the sphere of radius l/c about the origin in projective three space: i.2
=
c2(y2T
+ +y3)
Since every line through the center intersects the sphere in two real points, the wave equation is hyperbolic. 4.2. Extension to Hyperbolic Systems of Homogeneous Equations with Constant Coefficients27
The preceding considerations can also be applied to a homogeneous system of equations (4.24)
Lm[u] = P(D)u =
A2D2u = 0
Ixl=m
27 See P. D. Lax [2] for the case of first-order systems.
FRITZ JOHN
70
where u now stands for a column vector with components u1,. .. , u`'`
and A. for a square matrix with constant elements. The term "homogeneous" refers to the fact that all derivatives D°` are of the same order m. We associate with this equation the characteristic matrix
(4.25)
A(y) = A(y0,Y) = m! I Aaya = m! P(y) ial = m
whose determinant is the characteristic form Q(y) of degree mN.
The matrix P(1,0) must be nonsingular, if the initial plane xo = 0 is to be noncharacteristic. Multiplying (4.24) from the left with (P(1,0)) -l we can bring about that (4.26)
P(1,0) = Acm,o....,o
becomes the unit matrix I. For the standard Cauchy problem for (4.24) we have initial data (4.5), where now f (X) stands for a vector. Again formulae (4.13, 4.14) apply, the vector f (Y) representing the Fourier transform off (X), formed component-wise. The solution Z(x0iY) of (4.17, 4.18) (with 1 replaced by I) is now a matrix given by the modified expression (4.19) : (4.27)
Z(x0iY)
=
2_.
, (P(iA,iY)-le''to dA
where the path r is to include all mN roots A = Ak of the characteristic equation (4.28)
Q(A,Y) = 0
i.e., all values A for which the matrix (P(iA,iY))-1 is not defined. Hyperbolicity of the system (4.24) now consists in the require-
ment that all mN roots A of (4.28) are real for every real Y. Constructing the path F as before we have on r IQ(iA,iY)I = IQ(A,Y)I z 1 Again we have an inequality 1Akl < -411 YI
for the roots Ak of (4.28) which leads to an inequality
IAI < M I YI + V2
for A on l'
LINEAR HYPERBOLIC EQUATIONS
71
Each element of P(iA,iY) for A on r and I YI large can then be estimated by I YIm. We observe that the elements of the matrix P-1 are rational functions of the elements of the matrix P whose denominator is the determinant of P and whose numerator is of degree N - 1. Consequently the elements of for A on r and I YI large become infinite at most like The same holds for the elements of Z(xo,Y). It follows as before that the expression u given by (4.21) represents an actual solution (P(iA,iY))-1
I
of (4.24, 4.5) of class C"` if the components off belong to Co N+n+1. 4.3. Method of Decomposition into Plane Waves
In order to obtain a more direct representation for the solution
u of the initial value problem in terms off, it is tempting to substitute for f in (4.21) its expression (4.14). Interchanging the integrations we obtain formally (4.29)
u(x)
=JK(xo,X - Y) f (Y) dY
where (4.30)
r K(xo,X) = (27r) -n I e'1'1' Z(xo,Y) dY
If the expression for K could be evaluated, we would obtain an explicit integral representation for u in terms of f. This procedure involves some difficulties. One has to justify the interchange of integrations, and the evaluation of K leads to considerable convergence troubles, even in the simple case of the wave equation, where (4.31)
P(A,Y) = A2
-
c2 I YI2,
Z(xo,Y) =
sin cxo I YI C l YJ
These difficulties can be overcome by reinterpreting the singular integrals which occur as distributions in the sense of L. Schwartz [1]-
We shall discuss a different approach leading to explicit solutions that uses decomposition off into plane waves rather than
into exponentials as in the Fourier
method.28 We shall restrict
28 See John [2], Courant-Hilbert [1], Vol. II, pp. 699 et seq., Courant and A. Lax [1].
FRITZ JOHN
72
ourselves for simplicity to the case of three space dimensions (n = 3) and of a scalar equation (4.8), in which the resulting formulae are particularly simple. (The method can again be extended to systems of equations.) We start with Poisson's formula for a function f (X) =f (x1ix2,x3) in C,':
f(X)=-O
ff(Y)dY
where r = I Y - X1, and &, represents the Laplacean with respect to X = (x1,x2ix3). Since Axr = 2/r, we have for f in Co
f f(Y±X) IYJdY
f(Y)rdY= -
.f (X) _
8-
f (Dy f (Y))r dY
This can be looked at as a decomposition of f into functions r with spherical symmetry. Now r itself can be decomposed into plane wave functions, i.e., functions which are constant along a family of parallel planes. We have indeed for X - Y = rZ and for
varying over the unit sphere with surface elements dSt
f
ICI=1
CI=1
since
f, C1=1
Z. I dSC
the "moment" of the unit sphere with respect to the plane = 0, is the same for all planes through the origin and easily computed. Hence Z-
(4.32) f(X)
_-
1fi1T2
fzx f (Y)
(jix - Y)
I dS) dY
This is the desired decomposition off into plane wave functions I (X of the form Y) 1. Let now L. be strictly hyperbolic in the sense that the roots A of (4.20) are not only real for real Y, but distinct for real Y 0. For any root A of
-
(4.33)
P(A, 4) = 0
LINEAR HYPERBOLIC EQUATIONS
73
and any sufficiently regular function g(s) of the scalar real vari+ Axo) is a solution of able s, the expression u = g((X - Y) Lm[u] = 0. If the roots Ak of (4.33) are all real and distinct, and if f is the plane wave
.f (X) = h((X - Y) ' then, choosing g(s) so that g""'-1>(s) = h(s), the solution of the initial value problem (4.5, 4.8) is g((X - Y()
(4.34)
-{- Akxo)
PA(Ak,
k=1
This follows from the well-known identities "`
_
All.
k'I1
A'
1
27ri
r P(A,
dA
=
j = 0, 1, ... , m - 2
f0
for
`
for j = m -1
1
(which also underlie the proof of (4.18)). In our case where h(s) = Isl we take g(s) = s"2(sign s)/m!. Combining (4.32, 4.34) we
get by superposition a solution of (4.5, 4.8) represented by the formula (4.35)
u (x)
16-0
f (Arf(Y) )
x 141=1 k=1
P,1(Ak,b)
dY
If m > 4 we can remove the Laplace operators by integration by parts, and obtain for u(x) an expression of the form (4.29), where
K(xo,X - Y) = ((X - Y)-.
)kx0)
sign ((X - Y)
16a2(m - 4)!P;.
Itl=1 k=1
C -!- Akxo)
dS t
here the Ak are the roots of (4.33). (It should be noted that in the strictly hyperbolic case P,(Ak, Q is bounded away from 0 uniformly for = 1.) Formulae of this type are due to Herglotz, Bureau, and Petrowskii.29 As an example30 we take the fourth-order equation satisfied I
29 See Herglotz [1], Bureau [2], Pctrowskii [3]. 30 See Bureau [1], Courant-Hilbert [1], Vol. I, p. 706.
FRITZ JOHN
74
by the components of the displacement in an isotropic elastic medium:
- c"2 0)u = 0
L[u] = (Do - c'2 A) (Do
where c', c" are distinct positive constants connected with the elastic moduli of the medium. Here on I I = 1 P(A,?) = (A2 - c'2)(A2 Ak = ±c', ±C" c112),
Hence
sign ((X -
1
K( xo X - Y)
T6_0 1
87r
k
Y)
C + Akx o)
t=i
4A - 2 (c'2 + c"2) Ak 11 + Akxo/rl - 11 - Axxo/rl 4A - 2 (C'2 + C"2) Ak
dS
1
c" I r - c'xol - c' j r - C"xol - (c" - c')r
87r
rC"C' (C"2 - C'2)
where r = IX - Yl. The two spheres r = c'xo and r = c"xo divide the space into three parts. If, say, c' < c", we have K = 0 for r > c"xo, K = const. for r < c'xo, and K variable in the intermediate spherical shell. This has as a consequence that u(x) = u(xo,X) depends only on the initial values f (Y) in the sphere I Y - X1 < c"xo. The dependence of u on the values off in the smaller sphere I Y - X1 < c'xo is different from that in the spherical shell between the two spheres. The effect of an initial disturbance at Y on a point X of distance r at the time x0 (represented by K(xo,X Y)) makes itself felt at the time r/c"; it is given by one expression up to the time r/c', and by a different analytic formula from there on. Physically, there are two wave fronts propagating from a disturbance, corresponding to dilation waves traveling with velocity c" and equivoluminous waves
-
traveling with velocity c'.31 4.4. A Priori Estimates
While the Fourier integral representation (4.21) for the solution of (4.5, 4.8) is not especially convenient for the purpose of finding explicit formulae, it constitutes a powerful tool for deriving a priori 31 The distinction between these two wave forms becomes apparent only if the full system of elastic equations for the displacement vector is considered, instead of the single scalar equation discussed here.
LINEAR HYPERBOLIC EQUATIONS
75
estimates. We shall only consider the case of a single equation (4.8), which is strictly hyperbolic. Better estimates than (4.22) can then be obtained for the auxiliary function Z(xo, Y) .
Since now by assumption the roots A = Ak of (4.20) are all distinct we can evaluate the integral (4.19) by the method of residues and obtain the simple expression eli
e
Z(X 0) ) = ii-mV
izoAk
k=1 PA(Vk,Y)
Putting 2k = I YI luk, Y = I YI 77, where 1711 = 1, and using that P(1, Y) is homogeneous of degree m in A, Y, we find g
Z(x0,Y) _ (Z
IYI)i-m
of 'I Mk
is k=1 P, (Yk,n)
varies Here the uk are the roots of P(,u,i) = 0. Since continuously with i and does not vanish for real 0, the expression IP,(,uk,71) I has a positive lower bound b on the unit sphere 1171 = 1. It follows that
IZ(xo,Y)I < b
IYI1-m
Similarly, using that the Iµkl have a common upper bound M for ICI = 1, we obtain ID0Z(x0, Y) I < b Mk I
Y11-m;k
Combining this estimate for I YI > 1 with the estimate (4.23a) for I YI < 1 we find that there exists a bound C only depending on P such that I DOZ(x0,X) I < Cexo(l + I
YI2)(1-m+k)/2
fork=0,...,m-1, x0 > 0, and all Y
For a = (ao,a1) D"u(xo,X) =
. .
we have for the solution u of (4.5, 4.8)
.,
f
(yri)a"Do
dY
FRITZ JOHN
76
By Parseval's equation it follows then that for lal < m - 1
J
ID"u(xo,X)I2 dX =
I(y1)",
...
(iy,.)"^D0 Z(xo,Y)f(Y)12 dY
s+ ^C2e'°(1 ± 1YI2) 1-m+x°If(Y)I 2dY < J(1 T IYI)2 `x +x...+x < C2e2--o
If
12 d Y
f.f 2 (X) dX
=
We introduce again the quadratic expressions (4.36a) (4.36b)
H[f (X)]
dX
K[u(xo,X)] = max H[Dxu(xo,X)] lxi<"&-1
= max J(D'u(xo,X))2 dX 1
Then for the solution of the initial value problem (4.5, 4.8) K[u(xo,X)] < C2e2''°H[ f (X)] For the wave equation this result follows from the energy identity. (4.37)
From here on we proceed as in the case of the wave equation.
We first of all obtain from (4.37) estimates for the operator L;1, where u = L,;,'[w(xo,X)] is defined as the solution of L,,,[u] = w fork =0,...,m - 1 (D'u)r°-o = 0 More precisely we assume that w(xo,X) is of class C°° and of bounded support in X, uniformly for bounded x0. Then u = shall be the function represented by Duhamel's integral (4.37a)
u(x) = u(xo,X) =
r °U(xo - s,X,s) ds o
where U(xo,X,s) = U(x,s) is the solution of (4.5, 4.8) corresponding to f (X) = w(s,X). By (4.37) C2e3°-2RH[w(s,X)] K[U(xo s,X,s)] S
-
The square roots of K or H satisfy the triangle inequality for norms so that K[u(xo,X)]
j°v'K[U(xo
- s,X,s)] ds
r ds 0
LINEAR HYPERBOLIC EQUATIONS
77
and
K[u(XO,X)]
<- CZfo
f T0H[w(s,X)]
x0e2x°-2s dS
ds
0
T
_
°H[w(s,X)] ds
;C2(e° - l) 0
We can write this inequality in the form K[Lm1[w(xo,X)]] <_ O(xo) f x°H[w(s,X)] ds 0
in analogy to (2.50) with O(x0) = IC2(e° - 1). Defining
Kk[u(x)] = max K[Dau] = max H[D'1u] lark
IBI_k+m-1
a°- 0
0°<-m-1
Hk[w] = max H[Daw] Ixl -:k
a°=0
we have more generally as in (2.52) H0
(4.38)
k[w(s,X)] ds
Kk[L,,1w(xo,X)] < O(xo) fox
For applications of this inequality we have to extend the inequality (2.53) to an arbitrary number of dimensions. It is easy to derive the proper extension, the lemma of Sobolev, by making
use of the Fourier transformation formulae (4.13, 4.14). We have for any integer s > n/2 and for f in Co
f(X) = (21T)
-n12 f
IYI2)-812ei.C'Y][(1
[(1 +
+ IYI2)`12.f(Y)] dY
and thus by Schwartz' inequality f2(X) < 21r)-" f (1 + I YI2)-8 dY f (1 + IYI2)" If(Y)I2 dY
=Cndf (1 + IY12)8If(Y)I2dY with C2
8=
(27T) -"
f
(1 ;--
1
Y12)-' d Y < o0
FRITZ JOHN
78
On the other hand fo r a = (ai, ... , (1 + I YI2) 8
= (1 + y1 +...
Y = (yi,
+.y,2)
8 =Ia1<8 I a
... , yn) s!
(s
y2«
- lal) .
(see 3.6b). By (4.13)
Daf (X) = (27r) -n12 fj(i y)«e:x r f(Y) dY
and hence by Parseval's identity
f(Df(X))2 dX = Thus
(4.39) f2(X) < "l.8
f y2x If(Y)12 dY
J
I a! (s - lal) ! (Dnf(X))2 dX f Iai<8
8(1a
s!
8 a,
(S
s! lal )
,}H8[f(X)] = y2,8H8[f(X)]
This is the required generalization of (2.53). It permits to estimate the value off (X) at any point in terms of the mean H8[ f ], where s can be any integer exceeding n/2. We can choose here (4.40)
S=
(n + 2) /2
for even n
(n + 1) /2
for odd n
4.5. The General Linear Strictly Hyperbolic Equation with Constant Principal Part
Having the necessary a priori estimates for the principal part
at our disposal we can solve the more general equation (4.1) by iteration, exactly as in the case of the wave operator. We make the assumption that the principal part L. of L is strictly hyperbolic, ie., that the roots A of the characteristic equation P(A,Y) = 0 for real Y 0 are not only real but also distinct. This condition is essential if one wants the full operator L to be hyperbolic. Indeed there are cases where the principal part Lm is nonstrictly hyperbolic and where the Cauchy problem for the operator L resulting from addition of suitable lower order terms is not well posed.32 32 See Girding [I]. A. Lax [1].
LINEAR HYPERBOLIC EQUATIONS
79
We write equation (4.1) in the form
Lm[u] = N[u] + w
(4.41)
where
N=- " <mA"(x)D" j
The A", fk, and w shall be of class Ca'. In order to give the coefficients of N compact support in X we replace (4.1) by
Lm[u] = (X) (N[u] + w) where (X) is an auxiliary function in Co with (4.42)
C(X) =
(4.42a)
1
forJXI
0
for JXI > 2r
Similarly we replace the initial conditions (4.2) by (4.43)
(Dau(xo,X))ro-o = (X) fk(X)
for k = 0,
... , m -
1
We define a sequence of functions u'(x) recursively by Lm(ur-1)
(4.44)
= (X) (N[u'] + w)
imposing the initial conditions (4.43) on each u', and choosing m-1 1
(4.44a)
uo = C(X)
ko ki fk(X)x0
Then
N[u' - u'-']] u'+1 - u' = Lm' Let s be the integer determined by (4.40). There exists a bound (4.45)
A(xo) such that for all u A(xo)Ks-.1[u(xo,X)]
(compare with (2.59)). Applying this inequality to u'+1 - u' we obtain from (4.38, 4.45) K$TI[u'+1 -
u'] S Cxo)
f
Zo
0
A(a)Ks+1[u'(a,X)
-
u1-'(a,X)] do
-
It follows again, as in Chapter 2, that the K,+1[u'+k u'] tend to 0 for j -- oo, uniformly for bounded x0 and k > 0. This means
that the H,[D"(u'Tk - u')] tend to zero for lal < m, ao < m
- 1.
FRITZ JOHN
80
-
Using Sobolev's lemma we can estimate D"(u'+k u') at any point in terms of H,[D"(u'+k u')], and hence also in terms of KS-l[u'+k - u']. It follows that for j--,. oo the D"u' converge uniformly for bounded x0 if lal < m, oco < m. The recursion formula (4.44) yields then convergence also for the remaining mth derivative D;.. u'. Thus the limit u of the u' exists, is of class C'" and represents a solution of (4.42, 4.43). Then u will also satisfy (4.1, 4.2) for IXI < r. We should like to have a solution of (4.1, 4.2) defined simul-
-
taneously for all X. It is sufficient to prove that the solution u(x) of (4.42, 4.43) that has been constructed does not depend on r for x restricted to a bounded set and r sufficiently large. Then
lim u(x) will be defined for all x with x0 > 0 and satisfy (4.1, 4.2).
The fact that u(x) is independent of r for large r follows from the existence of a bounded domain of dependence of the solution
of the Cauchy problem for the differential operator L. The Fourier representation (4.21) gives no indication that the solution
of (4.5, 4.8) only depends on the values of the data in a finite domain. We can however appeal to the uniqueness theorem of Holmgren since the operator L," has constant coefficients. Accord-
ing to that theorem a function u is determined uniquely in a region R covered by an analytic family of free hyper-surfaces SS with common boundary if we are given the values of Lm[u] in R and the Cauchy data for u on one of the surfaces S,. By assumption (4.3) the hyper-planes x0 = const. are free with respect to Lm. The same holds then for any surface if its normal incloses a
sufficiently small angle with the x0-axis. Thus there exists a positive constant c (only depending on the coefficients of Lm) such that a hyper-surface x0 = 0(x1,
. . .
,
¢(X) is free if
(0r)2
Let now y = (yo, Y) be a point with yo > 0 and let S2 denote the hyperboloidal surface
xo}a
-
a-IXYl2 1
',
Ace
X - Y<-yoc
LINEAR HYPERBOLIC EQUATIONS
81
The SS with 0 < A < co are free, have the common boundary IX
Xe = 0
- Yl=yoc
and cover the conical region (4.46)
0 < x0 < yo
- I ix - Yl,
IX
- Yl < yoc
with vertex (yo,Y). It follows that a function u is determined uniquely at any point x of the set (4.46) by the values of in the set (4.46) and by the Cauchy data of u on the set So given by (4.47)
x0 = 0,
IX - Yl
So represents a domain of dependence for the point (yo,Y) (not the smallest possible one). Let now for a given (yo, Y) the constant
r be so large that (X) = I for all x in the set (4.46). It follows by induction from (4.44, 4.44a) that all u'(x) for x in (4.46) are determined uniquely by the values of w(x) in (4.46) and of the fk(X) in (4.47). The same holds then also for u(x) = lim u'(x). J_ M
For r -* oo we obtain then a global solution of (4.1, 4.2) with the property that the value of u at a point (y,,, Y) depends only on the value of w(x) in (4.46) and the values of the fk(X) in (4.47). This still leaves open the possibility that there exist solutions u of (4.1, 4.2) different from the one constructed here. The unique-
ness theorem of Holmgren presupposed analyticity of all coefficients Aa whereas we assumed only that the Ax are of class C-. However the argument of Holmgren only requires existence
of solutions of the Cauchy problem for the adjoint equation M[v] = w for a dense set of w. Let u be a solution of L[u] = 0 with vanishing Cauchy data on x0 = 0. Let w(X) be a function of class C°° of compact support. For given t > 0 we can construct
a solution v(x) of M[v] = w with vanishing Cauchy data on xo = t (finding v is just a problem of the type (4.1, 4.2)). This v
will have compact support in X. We can then apply Green's identity to the slab 0 < x0 < t and find that the integral of uw over the slab must vanish. Since this is the case for every w in Co the function u itself must vanish. This proves uniqueness of the solution of the Cauchy problem (4.1, 4.2).
FRITZ JOHN
82
4.6. First-Order Systems with Constant Principal Part
In the preceding section we dealt with a single equation L[u] = w for a scalar u. It was assumed that the principal part L. of L had constant coefficients and was strictly hyperbolic. The arguments used there can be extended to systems of equations
with constant principal parts. As long as the principal part is strictly hyperbolic in the sense that all roots of the characteristic equation are real and distinct no new difficulties arise. Unfortunately strict hyperbolicity is a condition only rarely satisfied by first-order systems. Many of the common equations of mathematical physics lead to characteristic forms with multiple roots, though the equivalent single higher order equations obtained by elimination may have simple roots. In the case of a scalar operator L strict hyperbolicity of the principal part Lm yielded the a priori estimates for the solution of the standard Cauchy problem which enabled us to invert L by iteration. Essentially we needed that the solution of the initial value problem for L. behaved no worse
than the data, at least when measured in terms of a suitable integral norm. Then it could be shown that addition of lower order terms did not affect hyperbolicity. Estimates for the solution
of the standard Cauchy problem for L. were based on the behavior of the auxiliary function Z(xo,Y) for large I YI. The crude
estimates (4.22, 4.23a) which only used the fact that Lm was hyperbolic were not sufficient for that purpose. Similarly the analogous estimates derived on page 71 for homogeneous hyperbolic systems are insufficient. We shall now indicate for the case
of first-order systems how adequate a priori estimates can be obtained without imposing strict hyperbolicity.
Let u denote a column vector with N components u'(x) _ u'(xo)X). We consider the first-order homogeneous system of equations L1 [u] _
with initial conditions
j-U
A,Dju = 0
u(0,X) ==f(X) Here the Aj are square matrices with constant real coefficients. The characteristic matrix associated with Ll is n
P = P(77o,271)
I Aj779 ... , 77n) =j=o
LINEAR HYPERBOLIC EQUATIONS
83
Its determinant is the characteristic form Q(770,
n
... ,
determinant I An, i
0
of degree N (see (3.11, 3.13)). Assuming that the initial plane xo = 0 is free we have Q(1,0,
... , 0) = determinant (A0) 0 0
Thus Ao 1 exists. We shall not normalize L1 by multiplying with Ao 1 as had been done previously. We shall assume however that L1 is hyperbolic in the sense that all roots 710 of
Q(n0,n1, ... , nn) = 0 are real for real rili ... , 77,,.
Using Fourier transformation we can obtain the solution of our initial value problem in the form (4-21). Here Z(xo, Y) _ Z(xo,yli
. .
.
, y,)
is the solution matrix of the ordinary differential
equations Y)l = e'x'YP(Do,iY) Z(xo, Y) = 0
for which Z(0, Y) reduces to the unit matrix I. For Z we have the integral representation (see (4.27)) Z(x0,Y)
e'Ao(P(iA,iY))-'A,, dA
where the path r in the A-plane encloses all roots A of Q(iA,iY) = iNQ(A,Y) = 0
Indeed for the Z defined by this formula we have P(Do,aY)Z
'P(iA,iY)e1 o(P(iA,iY))-1A0 dA 21T
T.eis,A0 dA
=0
Putting n
B(Y) _ -A01 I Ajt j=1
FRITZ JOHN
84
we have moreover, expanding F to infinity, Z(O,Y)
=
1
(P(A,Y))-'A0 dA.
r
2Trz
7
27ri
2-ffi Jr
B(Y)}
(I
[ I -}-
A2
B
j
dA
A3
B2
A
... dA = I
Choosing for I' the special path used previously we find that Z in the hyperbolic case behaves for large I YI at worst like I YI'-'. In the case of a first-order system we can use a simpler representation for Z which permits more precise estimates. One verifies immediately that the expression Z(xo,Y) = e,2uI ') =a (ixoB(Y))'
P
converges and also satisfies the ordinary differential equation and initial conditions for Z. For example, P(D0,iY)Z(x0,Y) _ (AOD0 - iA0B(F))Z(x0,Y)
= AQ(Do - iB(Y))z
(ixoB(Y))A = 0
If we can prove that the absolute value of any element of the matrix Z(xo, Y) does not exceed a fixed quantity M for all real Y and bounded x then by (4.21) H[u(xo,X)] =J (u(xo,X)) *(u(xo,X)) dX
=J (Z(xo,Y)J(Y)) *(Z(xo,YJ(Y)) dY N2.V12
f(Y)*.f(Y) dY = N2'Af2H[f(X)]
Once an inequality of this type has been established one can solve the initial value problem for more general differential equations of the type
L[u] = Ll[u] , E(x)u = w for a general matrix E(x) with coefficients in C.
LINEAR HYPERBOLIC EQUATIONS
85
Hence what is needed are conditions that guarantee uniform boundedness of the matrix eixo R(U')
for real Y. x0. Let for a certain Y the matrix B(Y) have an eigenvalue A and corresponding eigenvector v : B(Y)v = Rv
Then
P(1.,Y)v = Ao(2I - B(Y))v = 0 Thus P(J., Y) must be a singular matrix and A must be a root of the characteristic equation Q(7.,Y) = 0
Moreover Z(xo, Y) v = eixo' ")v = eiAXov
Certainly necessary for uniform boundedness of Z is that all roots
of the characteristic equation are real for real Y, i.e., that Li is hyperbolic. For if for a certain real Y we have an imaginary 7 then there also exists one with negative imaginary part. Letting then Y vary along a ray from the origin J. will increase linearly with I YJ, v can be held fixed, and eiAzov
for positive xo will increase exponentially with J YI.
In the hyperbolic case all eigenvalues % of B(Y) are real and the corresponding eigenvectors v can also be taken as real. Let B(Y) have a full set of N linearly independent eigenvectors v. Let C = C(Y) be the matrix whose columns are the eigenvcctors. Then B(Y) = C(Y) i(C(Y))-1
where A _ A(Y) is the diagonal matrix formed from the eigenvalues A,, . , 2` of B(Y). In other words the transformation C diagonalizes B. We then have Z(xo,Y) = C(Y)e`o-(C(Y))_i
where eixo A
86
FRITZ JOHN
is the diagonal matrix with elements e'roak
This latter matrix is certainly bounded for real 4- It follows that Z(xo, Y) is bounded if C(Y) and its reciprocal (C(Y)) -1 are bounded. We can always choose the eigenvectors making up the matrix C(Y) in such a way that they all have length 1. Then C(Y) is automatically bounded. Its reciprocal C-1 will then be bounded also, if the determinant of C is bounded away from zero, i.e., if the volume spanned by the eigenvectors is bounded away from zero. Thus we can assert that Z(xo, Y) will be bounded uniformly if for every real Y the matrix B(Y) has N eigenvectors of length 1, which are uniformly independent in the sense that they span a
volume uniformly bounded away from zero. Since the eigenvectors do not change if Y varies along a ray from the origin it is
sufficient to have this kind of uniform independence just for FYI=1. There are two important special cases in which this condition can be verified. One is the case of strict hyperbolici y, in which all
roots A are real and distinct for real Y :/= 0. In that case B(Y) has N linearly independent eigenvectors of length 1 which vary
continuously with Y. For Y on the unit sphere J YJ = 1 the volume spanned by those vectors will also vary continuously with Y and have a positive minimum. The other case is that of a symmetric hyperbolic system, in which all A. for j = 0, 1, . . . , n are symmetric matrices and in addition
A0 is the matrix of a positive definite quadratic form. Then A. can be represented as Ao = aa* where a is a nonsingular matrix. If here B(Y)v = Av we have in a*v an eigenvector of the symmetric matrix a-1AoB(Y) (a*)-1
Since the eigenvectors of a symmetric matrix can be chosen so as to be orthonormal and hence uniformly independent, it follows
that the eigenvectors of B(Y) can also be taken as uniformly independent.
LINEAR HYPERBOLIC EQUATIONS
87
4.7. Symmetric Hyperbolic Systems with Variable Coefficients33
These are systems of equations of the form (4.49)
A;(x) D,u ±- E(x)u = w(x)
L(u) i=o
where u is a column vector with N components u'(x0,x1,
... ,
u'(x0,X), and the A, and E are square matrices, the At being symmetric and A0 positive definite for all x. Following K. O. Friedrichs [1] we shall derive a priori estimates for the solution of the initial value problem for such equations without any recourse to Fourier transformations. By (3.13) the characteristic form associated with the system is given by (4.49a)
Q(YJ0ii1,
. . . ,
determinant I A,(x)rii I (j=0 l
We notice that in the symmetric hyperbolic case the roots 770 of Q They are the eigenvalues of the are real for any real 271, . . . symmetric matrix n
- I Asr71 j=1
taken with respect to the positive definite symmetric matrix A0. Many of the differential equations of mathematical physics derived from variational principles are naturally symmetric hyperbolic. This is, for example, verified immediately for Maxwell's equations EE, - curl H = 0, u Ht + curl E = 0 identifying x0 with the time t and u with the six-vector (E1,E2,E3, Hl,H2,H3). Other equations can often be brought into symmetric hyperbolic form. Thus, if v is a solution of the wave equation (4.50) (Do - c2 (DI + D2 + D3)) v = 0 we can introduce the four-vector u with components ui = Div. For the ui we then have the system c-2DOU° - D1ul - D2U2 - D3u3 = 0 (451)
DOu' -D,u° = 0
forj = 1,2,3
33 For the solution of the Cauchy problem for more general hyperbolic systems see Petrowskii [1, 2, 3], Leray [1, 2], Girding [2, 3].
FRITZ JOHN
88
which, when written in matrix notation, is easily seen to be symmetric hyperbolic. More generally let there be given a second-order linear differential equation of the form n
)L
Z xoro = I ark (x) Ur,T, + I b(x)v. , -P w (x) z-0
l,k=1
with a,k = ak,. Introducing the vector u with components ut = ax,
for i =0,...,n,we have
- `L a,ku2k - i=0btu = w z
0 uro
n
i
£
i,k=1 n
n
Iajkuz k=1
forj=1,...,n
0 °
k=1
k
This again is a system of the form (4.49) with symmetric A. Moreover A. will be positive definite, if the matrix of the a,k is positive definite.
Returning to the general symmetric hyperbolic system (4.49) we have, because of the symmetry of the A1, the identity n
(4.52)
,
j=0
D,(u*A,u) n
it
IA,D,u)u ; u*(ID,A, u+u*( j=0 =u*(j(D1A))
- E - E*)u
=,
it
A1D,u)
w*u -}-u*w
Consider now a "lens-shaped" region bounded by a surface and the hyper-plane x0 = 0. Let . . , O(X) be a function which is positive for X in a domain R of X-space and vanishes on the boundary of that domain. We x0 = O(X) = O(x1, .
integrate identity (4.52) over the region of x-space described by
0<x0
XeR
and obtain (4.53)
J so-d
A, f u dX -
u* I A0 `
Jo_ 0 LER
XER
J
XER
0<x <4
f u*A0u dX
[U* ( jD,A, - E - E*l u + w*u ; u*w ]dx0 dX ,=0
/
J
LINEAR HYPERBOLIC EQUATIONS
89
For the special case of the system (4.51) this result is identical with the energy identity (2.45). Replacing 0 by A with a positive
constant A we can introduce the family of hyper-surfaces Sx defined by XeR
xo = AO(X),
All Sx have the same boundary mainfold. We define m
Ea = u*I A0(x) - 2
cz1(X)A,(x) u =1 Identity (4.53) with 0 replaced by A can then be written in the form \
(4.54)
fEadx = fo duJ c[u*(I"D,A) -E-E*lu `j =0 /
sI.
a
+ w*u + u*wl dX
+f E° dX o
By (4.49a) the determinant of the coefficients of the quadratic form Ex in u is given by Q(1,-AOz1)
... ,
Thus the form Ex will be nondegenerate as long as the hyper-surfaces Sx are free. We call Sa spacelike, if the form Ex is positive definite:
Ex>0
for all x on SA and all u 00
All spacelike surfaces are free but the converse does not hold. Since by assumption A° is a positive matrix we can be sure that the Sx are free for all sufficiently small A. Let now the Sx be free for 0 < A a. Then there exist positive
quantities m, M such that on SA for 0 < A < a Ea > mu*u
r OCu* \.1 D fA
f- E-
E*`
I u + w*u + u*wI
M(u*u + w*w)
Here n and M depend only on the coefficients A,, E and on a. Assume that the Cauchy data for u vanish, i.e., that u = 0 for xo = 0 and X in R. Then (4.54) yields (4.54a)
mu*u dX < MJ d,J (u*u ± w*w) dX fSA
0
S,
FRITZ JOHN
90
Writing this in the form
d
[me-MA1114f
dafs u*u dX] < Me-Mz'mfo du Js w*w dX
we obtain by integration and substitution into (4.54a) the a priori estimate M (4.55) u*u dX < - f -1)duf w*wdX
f
m
s ).
o
s
for a solution u of L[u] = w with vanishing initial data. This inequality contains the uniqueness theorem that a solution u of L[u] = w is determined uniquely in a region bounded by the plane x0 = 0 and by a spacelike surface, if w is given in that region in addition to the initial values u on the "base" of the region. In particular, there will always exist a finite domain of dependence. If u has compact support in X we can apply (4.55) to the case
where the S, are the planes x0 = const. _ A, corresponding to 0 = 1, provided the A;, B are such that bounds m, M can be found uniformly in a slab 0 S x0 < a. Defining H[u(A,X)] = fu*(A,X)u(A,X) dX
we can write (4.55) in the form (4.56)
H[u(A,X)] <
M
fo (e
l)H[w(µ,X)] du
for u = L-1[w].
Inequalities (4.55) or (4.56) can be made the basis for an existence proof. One can either attempt to represent the solution as limit of solutions of problems which can be solved (e.g., by finite difference approximations or by the theorem of CauchyKowalewski for analytic data and coefficients), and establish convergence with the help of the a priori estimates, or use a more abstract Hilbert-space argument to conclude existence from unique-
ness. More recently the method has been extended by Friedrichs (see [2]) to "symmetric positive systems," including cases of mixed problems for systems that are not necessarily hyperbolic.
LINEAR HYPERBOLIC EQUATIONS
91
For n = 1 the general linear system of first-order differential equations takes the form
Aoat
(4.57)
+A1 ax=Eu+F
where we write t for x0 and X for xi. Here A0, A1i E are square matrices whose elements are known functions of t and X, and F is a vector depending on t and X. Again A0 is assumed to be non-
singular (i.e., the initial line t = 0 is free), and the initial condition in the Cauchy problem is
u(0,X) = f (X)
(4.58)
where now X is a scalar variable.
The matrix B(Y) = -Ao 1A1y1 here reduces to a simple multiple of the matrix Ao 1A1 and is diagonalized by the same matrices C as the latter. We assume that for any X, t the matrix -A0 'A, can be diagonalized by a matrix C = C(t,X) -Ao 1A1 = C AC-1
(4.59)
where A is a real diagonal matrix formed from the eigenvalues of -Ao 1A1. We make the further assumption that C, C-1, and A have continuous derivatives with respect to X and t. Multiplying equation (4.57) from the left by C-IAO 1 we obtain the new differential equation
C-i at -
AC-1
TX
= C-1A0 1(Eu T F)
If we introduce v = C-1u
(4.60)
as new dependent variable, we have for v a system of differential equations of the form (4.61)
av
-A
at
au
ax
= Gv + H
where
G = C-1A0 'EC - C-1
ac
+ AC-1
ac
H = C-1Ao 1F
FRITZ JOHN
92
The initial values for v are v(0,X) = C-1(0,X) f (X)
(4.62)
Since A is a diagonal matrix, we have in (4.61) a symmetric hyperbolic system of differential equations. In the present twodimensional case we can solve it by reduction to a system of integral equations.34 We assume without restriction of generality that the initial data f of u and hence also the initial data of v vanish. Let A have the diagonal elements u , . . . , ,uv. For each k = 1, ... , N we introduce the family of curves C. defined by the ordinary differential equation dX
(4.63)
_
fik(t,X)
dt
Since the µk are the eigenvalues of the matrix -Ao-'A1, we have along each Ck
det (A0 dX - Al dt) = 0 so that the Ck are the characteristic curves of the differential (4.64)
equation (see 3.3). Along Ck the k-th differential equation of the system (4.61) can be written in the form dvk
auk
d7
at
avk
N
luk-IG,v5+Hk ax ;=1
Integrating this equation along Ck from a point Pk of the x-axis to a point P = (t,X) we find
!v
(4.65)
vk(P) _J I f Gk,v, + Hk) dt Ck `7 =1
/
for k = 1, ... , N. The system of integral equations (4.65) leads to an a priori estimate for the solutions vk. Introduce as norm of the vector v for given I > 0 the maximum norm defined by (4.66)
IIvII ` = max JVk(S,X) I X.x.a
u
s; The discussion given here is based on Courant and P. D. Lax [2, 3) and Courant [1], where also systems of quasi-linear, first-order hyperbolic systems in two independent variables are solved. See also Cinquini-Cibrario [1] and A. Robinson [1].
LINEAR HYPERBOLIC EQUATIONS
93
It follows from (4.65) that there exists a constant M such that Mt
(4.67)
livill <_
A, (Ilall8 + IIHOI`) ds
if t is restricted to an interval 0 < I < tl. This inequality makes it possible to solve the system (4.65) by iteration. Take v° = 0. Define for known vn the vector v>t-1 in a point P as follows: take the various characteristic curves Ck through P corresponding to the different values k = 1, 2, . . . , N. They may intersect t = 0 in points Pk. Define the vector v11+1 by 1
=J
f
Gk,vj
Hk)I dt
/ the integral being taken along Ck from Pk to P. The estimate (4.67) easily shows uniform convergence of the v" towards a Ck` 1-1
function v satisfying (4.65). Existence of first derivatives of v can be shown by proving that
the first derivatives of the vk satisfy similar recursion formulae and converge; v then satisfies the differential equations (4.61)
and u = Cv will be a solution of the original problem (4.57) with vanishing initial data.
CHAPTER 5
A Parabolic Equation: The Equation of Heat Conduction
5.1. Parabolic Equations in General
It is possible to define rather concisely what is meant by the terms "elliptic" (no real characteristics) or "hyperbolic" (possessing well-posed Cauchy problems). In addition to "equations of mixed type" which are either elliptic or hyperbolic depending on the special region or special solution considered, there remains a large class of equations encountered in applications which are
referred to as parabolic. It is rather difficult to give an allembracing concise characterization of parabolic equations, since they differ considerably in their properties.35 We can use as guide to a definition the description of parabolic equations given in Chapter 1: equations with a well-posed initial boundary value problem for which effects travel with infinite speed.
The parabolic equations or systems of equations encountered 35 See Petrowskii [2], Silov [1], Lax and Milgram [1]. 36 It should be mentioned that still other types of equations with well-posed boundary initial value problems exist that might also be included in the class of parabolic equations. Thus in some applications one is led to consider equations of the type of
Replacing this equation by the system AU= w,
aw = 0
it becomes obvious that we have a well-posed problem if we prescribe u for t > 0 on the boundary D of a domain in X-space and also prescribe Au for
t =OinD.
94
THE HEAT EQUATION
95
ordinarily are of the form k
L[u]
= atk -
N[u]
with a dependent scalar or vector u = u(t,X) = u(t,x1, ... , X.)N is a differential operator containing no t-differentiations of order higher than k - 1.36 The prescribed initial data consist of the values of u and its 1-derivatives of orders < k - 1 taken at t = 0. All derivatives of u for t = 0 are then formally expressible in terms of the data from the differential equation. We deal with infinite propagation speed only if the order m of N is higher than k, i.e., when the plane t = 0 is characteristic. Otherwise the initial data would be the full set of Cauchy data and then, by Holmgren's theorem (at least for linear L with analytic coefficients), the solution u at a point is determined uniquely by the initial data in a finite region. Whether there actually exist well-posed
initial boundary value problems can be checked easily only in special cases where a priori estimates for the solution -can be given.
This is the case for certain second-order parabolic equations for which a maximum principle can be proved and for certain types of
equations with constant coefficients which can be solved by Fourier transformations'
A typical example of a second-order parabolic equation is furnished by the equation of heat conduction discussed below. More generally, equations of the type au
ar
= N[] u
have been considered where N[u] is an elliptic operator of second or higher order in the space variables. An example of a parabolic equation involving higher t-derivatives is furnished by the equation of transverse vibrations of thin plates 2
a2u
acs
+
22
a + axe) u = 0 1
37 See Rosenbloom [ 1 ], Gelfand and Silov [ 1 ], and the lecture by A. Milgram in this volume.
FRITZ JOHN
96
5.2. The Heat Equation. Maximum Principle
We consider the equation of heat conduction (5.1)
L[u]
_ `at - kA) u = 0
where k is a positive constant and u = u(t,xi, ... , xn) = u(t,X). An important property of this equation is the maximum principle which can be given the following form. Let R denote a closed and bounded region in X-space with boundary B, and let T be the cylindrical region in t, X-space described by
XER,
to
Let u be continuous in T and satisfy (5.1) and be of class C2 in the interior of T. Then u assumes its maximum in T in a point (t, X) lying either in the set X E R, t = to, or in the set X-e B,
to < t < tl. Proof:
Let M be the largest value u assumes in either of the
sets T': XER, t = to and P: X e B, to < t < tl. Consider for any e > 0 the function v(t,X) = u(t,X) + e IX12
Then L[v] = -2nke < 0. Let t* be an arbitrary number between to and tl and let T* denote the set X e R, to < t < t*. The function v cannot assume its maximum for T* in a point with X E R B, to < t < t*, for then at that point av/at > 0,
-
LIv < 0 contrary to L[v] < 0. Hence v assumes its maximum for
T* either on T' or on T", i.e., at a point where u < M. Then in T* v < M + e max IX12 R
Since u S v and a is an arbitrary positive number, it follows that u < M in T*, and hence also in T.38 In the same way it is proved that u assumes its minimum for
T in T' + T. Thus, if u = 0 in T' + T", it follows that u = 0 in T. This proves uniqueness of the solution of the following 38 For a stronger form of the maximum principle for more general parabolic equations see Nirenberg [2].
THE HEAT EQUATION
97
initial boundary value problem: determine a function u(t,X) which is continuous for
to
XER,
belongs to C2 and satisfies L[u] = 0 for
XER -B,
to
and assumes prescribed values on T' and T". We can prove a similar maximum principle for the infinite slab.
Let u(t,X) be a solution of class C2 of (5.1) for all X and for 0 < t < 4, and let u be continuous for 0 < t < tl. Let M = sup u(t,X) N = sup u(t,X)
forallXand0
for all X and t = 0 Then M = N, provided M < oo Proof :
For any positive e we define
v(t,X) = u(t,X) - e(2nkt + X12)
Then L[v] = 0. Moreover, if M > N, v(0,X) = u(0,X) - E X12 < N for all X v(t,X) < M - E IX12 < N IXI2 > (M N)/E, for
-
0
v(t,X) < N
for
X12 < (M
- N)/e,
0 < t < tl
Hence
u(t,X) < N + e(2nkt1 ± 1X12)
for
1X12 < (M - N)/E,
0
For e - +0 and fixed X it follows that
u(t,X) < N for 0 < t < t, and all X i.e., M S N. By definition also M > N. This maximum principle implies the following uniqueness theorem for the pure initial value problem.
FRITZ JOHN
98
Let u[t,X] be a solution of (5.1) for 0 < t < tl and all X which is continuous and uniformly bounded for 0 < t < ti and all X. Then u(t,X) is determined uniquely by its initial values u(0,X). The a priori requirement on the solution u to be bounded is essential. One can construct nontrivial solutions u(t,X) of (5.1), which are in C°° for t > 0, and vanish for t= 0.59 These solutions are necessarily unbounded for large IXI; in fact they increase faster than any expression of the form e"I'. On the other hand we have the result of Widder [1] that if u is bounded from below
(e.g., u > 0 in the case where u is the absolute temperature), then u is determined uniquely by its initial values. 5.3. Solution of the Initial Value Problem
A formal solution of the initial value problem
L[u] = 0
(5.2)
for t > 0
u(0,X) =f(X) is obtained immediately from the Fourier integral formulae (5.3)
(4.13, 4.14) for f. Since e'x' Y-ktY-Y
is the solution with initial values e"', we find (5.4)
u(t,X) = (27r)-n12
f
f(Y) dY
= f K(t,Z - X) f (Z) dZ where (5.5)
K(t,Z
- X) = (2ir)-n f = (4irkt)
dY -n/2e-rs/4k
with r = IZ - XI. (The expression K represents the "fundamental solution" of the initial value problem (5.2, 5.3), with the initial value K(0,Z - X) = b(Z - X) where 6 is the Dirac function.) 39 See Tychonoff [ 1 ], Hormander [ 1 ], Hille and Phillips [ 11.
THE HEAT EQUATION
99
Instead of trying to justify the operations leading to the solution
f
- X) f (Z) dZ
u(t,X) = K(t,Z
(5.6)
we can verify directly that the u given by (5.6) satisfies (5.2, 5.3) by taking note of the following properties of K:
(a) K and its derivatives decrease exponentially for r - oo and fixed t > 0; (b) L[K] = 0 for t > 0 and any Z; (c) K(t,Z - X) > 0 for t > 0; (d) lim K(t,Z - X) = 0 uniformly for r z & > 0; t-.+o
fK(t,Z - K) dZ = 1 for t > 0 and all X;
(e)
lim f,>6 K(t,Z
(f)
- X) dZ = 0 for every d > 0.
Properties a-d are verified immediately. For the proof of property e we note that
JK(t,Z
- X) dZ = f K(t,Y) dY [(4rrkt)-112e-y,'/4kt] dy1 ... dyn =fJ ... IT i=1
IT[
" d ye]
=1
Property f follows from da,f u rn-1 f >a K(t,Z - X) dZ = fl,71=a r
dr
1
dcof fin, =1 -n/2 7T
f InI=1
rn-1(4nkt)-n/2e-rs/4td Go
an-1e-a' da
dco
-, 0
a/Vaxt
for
Let nowf(X) be continuous and uniformly bounded for all X. It is clear from properties a and b that (5.6) defines a u(t,X) for t > 0 which is in C°° and satisfies (5.2). Let M = 1.u.b. If (Z) 1. Given any e > 0 there exists a positive b = 6(e,X) such that
If(Z) -f(X)I <e/3
for r <6
FRITZ JOHN
100
and, by property f, there exists a positive r = T(E,X) such that
K(1,Z-X)dZ< r>a(e,X)
e
3M
for 0
Then for 0 < t < r, using properties c and e,
Iu(t,X) -f(X)I = I fK(IZ - X)(f(Z) -f(X)) dZI Jr
+f
K(t,Z - X) If(Z) --f(X)I dZ
E
K(t,Z - X) dZ
rr , 6
3
J r
+-2M
K(t,Z-X)dZ
r -J
This proves that (5.7)
lira u(t,X) =f (X) c-. -+ o
and hence that (5.3) is satisfied. Since f (X) is uniformly continuous in every bounded set, a(F,X) can be chosen independently
of X for bounded X. This shows that (5.7) holds uniformly for
bounded X and hence that u(1,X) is continuous for t > 0. It follows moreover from (5.6)r and properties c and e of K that Iu(t,X)I < MJ K(t,Z - X) dZ = M Thus the solution constructed is bounded and hence, by the preceding uniqueness theorem, is also the only bounded solution of (5.2, 5.3). Hence, for continuous and bounded f (X) formula (5.6) represents the unique solution u(t,X) of the equation of heat conduction for t > 0 which is continuous and bounded for 1 > 0 and reduces to f (X) for t = 0. It is evident from (5.6, 5.7) that u(I,X), for positive t however
small, depends on the values of the initial function f (X) at all points of space. This expresses the fact that here the speed of
THE HEAT EQUATION
101
propagation is infinite. A disturbance at any point is felt immediately afterwards everywhere. Of course, far away disturbances cause, in a short time interval, only minute changes. The bulk of the disturbance takes a finite amount of time to arrive at a distant point. Indeed by property f of K the contribution to u(t,X) of the initial values f (Z) at points Z with IZ - XI > b tends to 0 with t. 5.4. Smoothness of Solutions
We observe from (5.6, 5.5) that u(t,X' is defined and continuous for complex t, X as well, as long as Re (t) > 0. This follows from the fact that for complex
t=t'+it"
X=X'+iX", and real Z (5.8) K(t,Z - X) I =I
(47rkt)-nl2e-(X-z)-cX-z>,4ktI
= (I + 02)''4ex"-.t"'4
+ 02),X' + OX" - Z)
where 0 =
t"ItI
It follows that
Iu(t' + it",X' + ix") < (1 + x
f If(Z)I K(t'(1 + 02),X' + OX" - Z) dZ
< M(1 +
02)n/4 ex X"/4k '
In particular for real positive t (5.8a)
f
Iu(t,X' + iX")I < ea" 14k: If (Z) I K(t,X' - Z) dZ
< 4je-''-
'14kt
If moreover f (X) > 0 for all X, then (5.8b)
I u(t,X'
iX") I <
-14x1
f f (Z)K(t,X' - Z) dZ
= ex--X-14xt u(t X') Thus solutions u(t,X) of (5.2) that are uniformly bounded and
FRITZ JOHN
102
continuous for real X, t with 0 S t are necessarily analytic for complex X, t with Re (t) > 0.40 Let Sgt denote the mapping defined by (5.6) which associates with a function f (X), bounded and continuous in real X-space, the function u(t,X) depending on the parameter t. For real t > 0 the operator Sgt is highly smoothing; it converts bounded continuous functions of X into entire analytic functions of X which are bounded for real X.
For s > 0 we can consider u(s + t,X) as the solution of (5.1) with initial values u(s,X) for t = 0, since equation (5.1) does not contain t explicitly. It follows from the uniqueness of the initial
value problem for bounded u that the operators Sgt have the semigroup property
(5.9)
Sgt+8 = S242s
for s, t > 0
Since u(t + s,X) can be generated with the help of formula (5.6) either directly from the initial values u(0,X) or indirectly by way of u(s,X), we find by comparison the identity (5.10)
K(t ' s,X) = S)1K(s,X) = f K(t,Z - X)K(s,Z) dZ
which is the concrete counterpart of (5.9). The smoothing character of the operator Sgt makes it in general impossible to continue the solution of (5.2, 5.3) for negative t.
In order that u(t,X) could be continued backward in time as a uniformly bounded function the initial distribution f (X) would have to be analytic. A bounded continuous temperature distribution u given for t = 0 can always evolve in the future in accordance with the heat conduction equation (5.1) but cannot in general have originated from an earlier temperature distribution. The problem of determining the future is well posed, while that for the past is not. We can talk of "time's arrow" for equation (5.1).
The irreversibility of the operator Qt is connected with the fact that the differential equation (5.1) is not preserved if we '0 Analyticity of u follows from the uniform convergence of the integral (5.6) for bounded complex X, l with Re (t) bounded away from zero, and the analyticity of the integrand in X, t. As appears on page 107 a solution u for
fixed t > 0 is always analytic in X even without the assumption that u is bounded uniformly for all real X.
THE HEAT EQUATION
103
substitute -t for t. However with u(t,X) we have in v(t,X) = u(-t,iX) a formal solution of (5.1) with initial values v(O,X) = u(O,iX) =f (iX). This suggests the expression (5.11)
u(-t,X) = v(t,-iX) = fK(t,Z + iX) f (iZ) dZ ,1
= S2_tf(X) for extending u backwards in time provided f is analytic and such that the integral in (5.11) converges.41 It is easy to obtain necessary conditions which a function v(X) must satisfy in order that the operator SZ_t can be applied to it. Let v(X) be representable in the form v(X) = Sgt f (X) = u(t,X) where u is defined by (5.6) and f (X) is continuous and bounded. We have from (5.8a), for complex X, and real positive t (5.12)
Iu(t,X) I < Meh1"' (X )I E14kt
This inequality shows that v(X) is an entire function of X with a
certain growth restriction for complex arguments. Since, by Cauchy's formula, the derivative of a function in the center of a circle of radius r is at most the maximum of the function on the
circle divided by r, we find from (5.12) that, for real X, the gradient of u (i.e., the maximum of any directional derivative) is at most42
M ers1.ut
r Choosing the most favorable r, i.e., r2 = 2kt, we find that the gradient of a heat distribution v(X) which has existed for a time t and has been bounded by M is at most
41 See Widder [2], page 454. For numerical solution see John [I]. 42 A directional derivative of u(t,X) is given by (dl. u(t,X -{
34)/x=o
where 5 is a real vector with 1 = 1. We apply Cauchy's derivative estimate to u(t,X + ,34) as a function of the complex variable A. By (5.12) Ju(t,X + ).4)l < Me-r214
for 121 < r
FRITZ JOHN
104
Consequently an infinitely old heat distribution that has always been bounded by M, must be constant. We can obtain a still better result if we assume that f (X) z 0 for all X. Then from (5.8b), for complex X = X' + iX", eIIni ( 214ktu(t,Re (X)) = e1 '12I''s(X')
= I u(t,X) I < Applying the same arguments as before, we find that the gradient off (X) for real X is at most v(X) I
1 er2l4 t max v(Y) r 11'- XIsr
This inequality gives an estimate for the gradient of v(X) for positive u purely in terms of the age t of the distribution v and of
the maximum of v in a neighborhood of X, without a priori knowledge of the maximum M of u in the past. Thus observations of temperature gradient and temperature in the neighborhood of a point lead to an upper bound for the age t of a positive distribution if only heat conduction is involved.0 5.5. The Boundary Initial Value Problem for a Rectangle
So far we have dealt with the pure initial value problem for equation (5.1) in unbounded X-space. This is not quite a properly posed problem since, as was mentioned, there is no uniqueness
without some a priori restriction on the growth of the solution u(t,X) for large IXI. In this paragraph we shall solve the simplest initial boundary value problem, restricting ourselves to the case of a single space variable. Let u(s,z) be a solution of the equation 2
(5.13)
L[u] = as
- k aZ2 = 0
in the semi-infinite strip (5.14)
0
0<s
We want to express the value of u at an interior point (t,x) of the strip in terms of prescribed boundary initial data: (5 15)
u(0,z) =f(z)
for 0 < z
u(s,0) = g(s),
u(s,R) = h(s)
for s > 0
43 For early attempts at determining the age of the earth from temperature gradients see Carslaw [1], pages 57-60.
THE HEAT EQUATION
105
For this purpose we make use of Green's identity. Let v(s,z) be a regular solution of the adjoint equation (5.16)
av TS
M [V]
-k
2
avz2
=0
in the rectangle
0
(5.17)
Then
_ 0
JI0T
ds Rdz (vL[u] - uMM1[v]) l0
=f TdsJORdz( a'uv s (5.18)
+ k azaz
T
R
R
0uvdz- fo uvdz d=0
s=`L'
-k z-o
az -
v
- v az) l
ku z-R
' `t`az-vaZ)ds
az)ds
Let now v depend on two parameters t, x in addition to s, z : v = G(t,x,s,z)
and be such that (a) M[G]
- - k a1G = 0 aZ2
for0
0 <s
0 <x
(b) G(t,x,s,z) - K(t - s,z - x) 0 < z <_ R,
0 <x
0 <s
(c) G(t,x,s,0) = G(t,x,s,R) = 0
for 0 < x < R,
0<s
(d) G(t,x,t,z) = 0
is continuous for
for 0 < z < R,
0 <x
x0z
Assuming such a Green's function G has been constructed, we
can apply identity (5.18) with v = G as long as 0 < T < t.
FRITZ JOHN
106
Here by (5.7) lim
f
RuG dz
= lim
fR
u(T,z)G(t,x,T,z) dz
0
T
R
= lim f u(T,z)K(t - T,z - x) dz T-- o fR
= lim
T-.t_ J 0
u(t,z)K(t - T,z - x) dz = u(t,x)
This leads to the desired representation
f f (z)G(t,x,0,z) dz R
u(t,x) =
(5.19)
ds + kf 'g (s) (aG(;sz))dS - k f 'h (s) (aG(xsz)) az =R az It remains to produce a Green's function G. This can be achieved
by the reflection principle for the simple situation considered here. The function G(t,x,s,z) is a solution of M[G] = 0 which vanishes for z = 0, and hence can be continued across the line z = 0 as an odd function of z : G(t,x,s,z) = -G(t,x,s, -z) Similarly the vanishing of G for z = R leads to the continuation formula
G(t,x,s,z) _ -G(t,x,s,2R - z) Hence for any integer n
G(t,x,s,z) = G(t,x,s,z - 2nR) = -G(t,x,s, -z + 2nR)
- K(t - s,z - x) is continuous for 0 < z < R G(t,x,s,z) - K(t - s,z - x - 2nR)
Since G(t,x,s,z)
0 < x < R, we find that
must be continuous for 0 < x < R, 2nR < z < (2n + 1)R, and that G(t,x,s,z) + K(t - s, -x - z + 2nR) must be continuous
for (2n - I) R < z < 2nR, 0 < x < R. This suggests the expression +Co
(5.20)
G(t,x,s,z) _ I [K(t - s,z - x - 2nR) - K(t - s, -z - x + 2nR)]
THE HEAT EQUATION
107
which has the correct singularities for 0 < x < R and all z.44 It is easily verified that indeed the G defined by (5.20) has all the desired properties a, b, c, and d and can be used as Green's function in formula (5.19).
It is not difficult to show that G(t,x,s,z) is of class C°° in its arguments for
0<s
0
0<x
x-z
and vanishes with all its derivatives for s = t. This has as a consequence that the expression (5.19) for u can be differentiated
arbitrarily often with respect to x or t for 0 < x < R, 0 < t. More precisely the u given by (5.19) is even defined and analytic in x for complex x and real positive t, as long as
Re (z - x - 2nR) 2, Re (z + x - 2nR) 2 are positive for any integer n and for z near 0 or R. This is the case for
0
1Imxl
jImxl
Hence any solution u defined in the rectangle
0<s
0
is of class C'° in s and analytic in z for real z with
0 <s
0
The analogous statement can be made for solutions defined in any rectangle with sides parallel to the coordinate axes. Solutions of the equation of heat conduction share with solutions of analytic elliptic equations the property that they are of class C°° in interior points of their domain of existence.45 44 Substituting for K in (5.20) its expression from (5.5) for the case of one space dimension, one can write for G the closed formula I
G(t,x.s,z) = 2R Os
(x
-z
2R '
irr(t - s)k R2
- 03
x + z in (t - s)k1 2R '
R2
Ij
where O is the Theta function defined by +"0
03(v. int) = (nt)-112
e-(c+n)2/t n
o0
(see Magnus-Oberhettinger [1], page 99). 45 See Holmgren [2]. For more general equations with this property see Hdrmander [1], Mizohata [1].
CHAPTER 6
Approximation of Solutions of Partial Differential Equations by the Method of Finite Differences46
The aim of a numerical computation scheme is to obtain a sufficiently good approximation to the solution of a problem by a finite number of steps, each of which can be carried out with
sufficient accuracy by a person or a machine. Essentially the original continuous problem has to be approximated by a discrete one in a finite-dimensional space, like the problem of solving a system of linear equations in a finite number of unknowns. Among the methods used in this connection finite difference schemes play a prominent part. Other methods such as that of Raylcigh-Ritz
or expansions in orthogonal functions may be preferable in special situations and may give much more accurate results with less
computation. The advantage of the use of finite differences lies in the extreme generality of the method; replacing derivatives by difference quotients suggests itself as a natural approach in every conceivable situation involving differential equations. However, a general method is likely to be inferior in efficiency to special methods which, in a given situation, take advantage of special features of a problem.' From the theoretical point of view application of an approximation method is only justified if an estimate for the resulting error (or at least for the probable error) can be produced. Whether or not an error of one percent in the result is tolerable depends 46 The present chapter can onl) give a very incomplete glimpse of a vast and rapidly expanding field. See the books by R. D. Richtmyer [l J, Fors',the and Wasow [ I ], Gollatz [ I ], Ladv enskaya [2J. John [8]. 47 An important class of numerical approximations of solutions of hyperbolic equations are those making use of the characteristic manifolds of the differential equation. Sec, e.g., Courant-Isaacson-Rces [1]. 108
FINITE DIFFERENCES
101)
on the special purpose for which the result is intended. Moreover,
whether or not a sequence of 10,000 rational operations of a certain kind will result in a maximum error of one percent depends on the special arrangement of the computation, the number of significant figures kept, and the type of machine used. If we want to keep error estimates free of such special accidental
features, we have to permit operations involving limits and potentially an arbitrary number of steps. Such estimates will be of the form : for a given e > 0 there arc functions N(f) and 6(e) such that a certain scheme carried to N steps, each performed with a maximum error b, will result in a final error of size at most e. The difference between this kind of statement and a pure assertion of convergence is that we are not interested only in the existence of the functions '4'(e), 6(e) but also in explicit expressions for them,
which then, in a concrete application, enable us to calculate the number of steps needed to get one percent accuracy in the result while working with 10 significant figures in the computation.
This leads one to consider the stability or convergence of a difference scheme, even in cases where we only intend to use a finite sequence of operations. 6.1. Solution of Parabolic Equations
We begin by considering a single second-order linear parabolic equation of the form (6-D O
L[ul =
au
at
- AAu - 12137 ;_,
au ax,
- Cu - w
where u, w, .4, R C are functions of t and X = (x, ... ,
x,,,)
and .1 denotes the Laplace operator; u shall be determined from (6.1) and the initial condition (6.2)
u(0,X) - f (X)
AV e replace the derivatives in (6.1 ) by finite difference quotients.
For simplicity we think of a rectangular grid G in t,X-space with mesh width fi in the direction of the space-coordinate axes, and mesh width k in the t-direction. Let r(t,X; be the approximation to u considered, where r is defined in G. The operator Eh shall transform the function ri`t,_Y! , ... . . . ,r) , v ) into
-
FRITZ JOHN
110
v(t,xl,
... x,-,,x, + h,x,+i, ...
Similarly E; h shall mean
replacing x, by x, - h, i.e.,
E,h=(Eh)-1 We note that symbolically, by Taylor's formula
E;'v(t,X) =k0ki I 1
00
ak
hk
axi
v(t,X) _ 00 1ki hDz) v(t,X)
or
Eh =eh D,,
E,h=e-h''
We then replace (6.1) by the difference equation (6.3)
v(t + k,X) - v(t,X)
Ah,a[v) =
k
-
-
ih2
- A(t X) IE"
2B,(t,X) Eh
2
Ei
h
v(t X)
v(t,X) - C(t,X)v(t,X)
2hEi
1
= w(t,X) and the initial condition (6.2) by v(O,X) = f (X)
(6.4)
Here "central" difference expressions have been used to approximate derivatives in the space directions. Formally the difference equation goes over into the differential equation for h -- - 0, k --- 0. Indeed for a function u(t,X) in C4 we have, by the mean value theorem,
u(t -k,X) -u(t,X)
- au at
k h
E'
_2
'
ka2u 2 ate '
Eh
-
2h
h
h.,+ '
E,1
2s
u = Dtu -f- eh Dsu
L
u
D,u +
12
D'u
where the bars indicate intermediate arguments. Hence we find, for the truncation error r = Ah.L [u] - L[u] (6.5)
that (6.5a)
Irl < Af(k ± h2)
FINITE DIFFERENCES
111
where M involves bounds for the A, B, and for u and its derivatives up to order 4. Does this mean that v u for h, k -* 0 and
sufficiently regular u? This cannot be true in general, unless k/h , 0. For otherwise, in computing u at a point (t,X) only the values of the initial data in a bounded region would have been
used, while in general (see Chapter 5) u depends on the values off in the whole X-space. Thus some restriction on the manner in which k and h tend to 0 is necessary as was first noted in the classical paper of Courant, Friedrichs, and Lewy [1]. We shall find a sufficient condition for convergence by establishing an a priori estimate for the solutions of the difference equation (6.3). We have from (6.3) for A = k1h2 (6.6)
v(t + k,X) = (1 - 2n)AA + Ck)v(t,X) M
+,I [A(A + B,h)Ehv(t,X) + A(A - Bjh)E1 hv(t,X)] + kw(t,X) Let llvIIt = l.u.b. Iv(s,X)I
(6.7)
I
o_R
By (6.6)
I v(t + k,X) I <
[ii - 2nAA + CkI n
+A
4=1
(IA + hB,I + JA - hB,I) IlvIlt + k IIwIIt
If the conditions (6.8)
1 - 2nAA + Ck > 0,
IhB,I < A
are satisfied for all t, X the "absolute" signs on the right hand side can be omitted and it follows that (6.9)
Ilvllt+k < (1 + k IIClit) Ilvllt + k IIwIIt
Conditions (6.8) hold certainly if the coefficients A, B C are bounded, the h, k sufficiently small, and if there exist two constants q1, q2 such that for all t, X and the values of A = k/h2 in question (6.10)
0 < q1 < A,
2nAA < q2 < 1
FRITZ JOHN
112
Relations (6.10) can always be satisfied by suitable q1, q2, A if A
is positive and both A and I IA bounded. Let p(t) denote the solution of the ordinary differential equation (6.11)
dt = IICII[V + 11w11 [,
V(0) = 11111
It follows from (6.9) that v(t,X) is majorized by tp, (6.12)
I v(t,X) I = V(t)
provided the stability condition (6.10) is satisfied and h and k are sufficiently small. This can be proved by induction. Let 11v 11' < p(l). Then by (6.9, 6.11)
V(t + k) = V(t) +
f
[ rk (11C11' iv(T) + 11w11') dT
[
V(t) +
k(IICIII V(t) +
(1 + k IICII[)
IIUII`
1lw111)
k 11wll[ > IIUII[+k
We are now in a position to estimate the error of approximation,
v - u. Let us admit the possibility that in the computation of v(t ±- k,X) by (6.6) a certain additional inaccuracy has been committed at each step which is represented by an additional term, the "round-off error" p(t,X), on the right hand side of (6.6). The only information available shall be that Ipl < e, where
e depends, e.g., on the number of significant digits available. Put
1=V-U Then by (6.5a) I'1h,k[?1] I = I Ah,k[v] - Ah k[u] I = w + k - Ah k [U] I
=k
L[u] -
Ah.k[u]I
< k + M(k + h2)
Moreover, 11(0,X) = 0 if we assume that we start out with the correct initial data. Applying inequality (6.12) to 91 instead of v (with w replaced by k + A1(k + h2) and IICII [ by its largest value)
we find for the total error a at the time t the estimate (6.13)
1111
{=
E
tGh2 -i- Af(). + 1)h2)et c.,"
FINITE DIFFERENCES
113
The term with e/2h2 represents an estimate for the accumulated round-off error which, for fixed A, tends to oo as the mesh size decreases; the term with li-1h2 estimates the accumulated truncation
error which tends to 0 like h2 for h - 0. As far as formula (6.13) is concerned, the best choice for h is 114
( 6 . 14 )
h
=
1
( 2(2 + I )M )
It leads to a total error ( 6 . 14 a)
t I7il
le(2 T 1).11 etile '
u
-
A
= 0( V
)
Roughly speaking the result has then about half as many correct significant figures as the computation at each step. It follows that in the absence of round-off error, the solution v of the difference scheme converges to the solution u of the differential equation, provided the stability condition (6.10) is satisfied, and the coefficients of (6.1), as well as u and its derivatives of
order <4, are bounded uniformly for bounded t. The convergence of v to u implies that u also is majorized by V(t) : (6.15)
lu(t,X)I c ,(t)
This is a generalization of the maximum principle derived in the
preceding chapter. It holds for bounded coefficients and for solutions with bounded derivatives of order <4. This in turn implies unique determination of such solutions by the data w and f.
So far we have proved convergence of the solution of the difference scheme (in the stable case) under the assumption that there exist solutions u of the differential equation problem which have bounded derivatives of sufficiently high order. The existence of such solutions can be proved under sufficient regularity assumptions for the data, e.g., when all the A, B C, f, w are in C- and are such that each of their derivatives is bounded uniformly. In fact existence of a solution of the differential equation can even be proved by starting from the a priori inequality (6.12) for the solution v of the difference scheme, deriving analogous estimates for the difference quotients of v to establish their boundedness, and choosing a suitable subsequence of the (h,k) for which v and its difference quotients converge.
FRITZ JOHN
114
Having once established convergence of the solution of the difference scheme under rather generous regularity assumptions
on the coefficients and data, one can try to extend the convergence proof to less regular cases. It is easiest to relax the restrictions on the data. Indeed convergence must certainly hold if the data w and f are assumed to be only uniformly bounded
and uniformly continuous. For then we can find sequences of data, f,,, w converging uniformly to f respectively w, for n - o0 and such that all derivatives of an f or w for fixed n are bounded uniformly. (One such approximation to f would be represented by the formula (5.4) with t = 1 /n.) If u,,, v are the solutions of differential and difference equations corresponding to f,,, w,,, we have from (6.12, 6.15) that the u and v,, form Cauchy sequences converging respectively to functions u, v. There exists for every
e > 0 an N such that
< t,
F Iv in some given 1-interval. Moreover we have from (6.13) that
I u-U
NI
there exists a b(t,N(e)) such that IuN -- vNI < t
for h < b
Consequently
lu - vl < 3E
for h < b(e,N(t))
i.e., the solution v of the difference scheme corresponding to f, w converges towards the function u for h - 0. Proof of convergence for discontinuous data requires more refined estimates which we shall not give here."a
It is possible to extend the methods described to the case of initial boundary value problems for (6.1). That is easiest in the case where u is to satisfy (6.1) in a cylindrical region to 1,X--space
whose base is a rectangular parallelopiped R in X-space with
boundary B. We fit a suitable rectangular grid such that B consists of grid planes. The standard problem would be to obtain a solution of (6.1) for t > 0, X E R with prescribed initial values f for t = 0, X E R, where in addition u vanishes for X e B, t > 0. The proper difference scheme consists then in using the difference equation (6.3) for gridpoints (t,X), where t >- 0 and X is interior
to R, in addition to the requirements v =f for I = 0, X E R, 47a See John [7], Aronson [1].
FINITE DIFFERENCES
115
and v = 0 for X e B, t > 0. One again establishes the estimate (6.12) and obtains convergence for u of class C4 in the closed cylindrical region. For cylinders with bases of more complicated shape interpolation schemes have to be used to obtain values of v in grid points near B by a recursion formula.
The schemes discussed so far are of the nature of recursion formulae, giving v(t + k,X) explicitly in terms of values of v already computed. There are also so-called implicit schemes for initial boundary value problems." The prototype of such schemes is obtained by replacing in the difference equation (6.3), the term [v(t -}- k,X) - v(t,X)]/k by the "backward" time difference quotient
v(t,X) - v(t - k,X) k
leaving all other terms in (6.3) unchanged. In this case we obtain a whole simultaneous system of linear equations (one for each interior point of R, the values of v having been prescribed on B), for determining v(t,X) for X F R provided the values of v(t - k,X) are known already. It is clear that determination of the v(t,X) in terms of preceding values requires considerably more labor. This is offset to a certain extent by the fact that this implicit scheme
has a much higher degree of stability than the explicit ones, permitting h and k to tend to 0 in any manner whatsoever, and thus allowing much larger time steps than would be possible under the restriction (6.10). 6.2. Stability of Difference Schemes for Other Types of Equations
Turning to more general initial value problems for hyperbolic or parabolic equations we have to deal with the following situation. We want to solve a linear differential equation or a system of differential equations (6.16) L[u] _ A"(x) D"u = w I"I _,n
subject to certain initial conditions which we write symbolically
as M[u] =J'. We assume that this problem has a sufficiently 411 See,
e.g., Rose [ I ], Douglas [ 1 ]. Forsythe and Wasow [ 1 J.
FRITZ JOHN
116
regular solution u (either an existence proof is already available
or possibly will be derived by the difference approach to be described). We replace L and M by difference operators A and ,u corresponding to a certain grid whose mesh size is described by quantities h, k...... 'he difference operators shall be "compatible" with L, M in the sense that for all sufficiently regular u the truncation error tends to zero with the mesh size: (6.17)
line (A - L) [u] = 0,
lim (µ - M) [u] = 0
forh,k,...
0
We take the solution of the discrete problem (6.18)
A[v] = w,
,u[v] = f
and hope that v --> u for h, k, . . . -* 0. One plausible necessary condition for convergence (CourantFriedrichs-Lewy condition) states: in the limit, the domain of dependence of a point 1' for the function v must be at least as
large as the domain of dependence of P for the differential equation. Obviously the difference equation cannot, in general, give the correct answer on the basis of insufficient information needed for the answer. Thus propagation speed for the difference equation should in the limit not lie below that for the differential equations. But this mere qualitative condition is in general not sufficient. A sufficient condition for convergence is furnished by an a priori estimate for the solution of (6.18), of the form (6.19)
Ilulll = 0(I1f II + Ilwll)
where suitable norms are used for data and solution (not necessarily the same ones). Then (6.17, 6.19) will permit us to conclude that Ilv - ull 0 for h, k, . . . -- 0. In a sense (6.19) guarantees hyperbolic or parabolic character of the difference scheme, which
is needed in addition to the compatibility condition (6.17). An estimate of the form (6.19) at the same time yields stability of the solution v of the difference scheme under small errors of computation (round-off errors).-" In the case of parabolic equations of the form (6.1) we have 49 For a general discussion of "stability" and its relation to "convergence"
see Lax and Richtmyer [1].
FINITE DIFFERENCES
117
succeeded to establish a priori estimates of the type of (6.19) under the condition (6.10). The basic fact used was that the coefficients on the right hand side of (6.6) could all be made positive. Such difference schemes with the right positiveness property can be constructed for any second-order parabolic equation using a result of Motzkin and Wasow [1]. However, schemes of this type cannot exist in general in the case of hyperbolic equations, because they would imply continuous dependence of the solution of the differential equation on the data if merely the maximum value of a function is taken as norm. However, as we saw in Chapter 2, the maximum of a solution of a hyperbolic
equation in several space dimensions cannot be estimated by maximum of the data. Other methods of estimating have to be used in general. In the case of a symmetric hyperbolic first-order system
A°(t,X)
(6.20)
at
=aIA,(t,X) ax + Bu + w
(where as in Chapter 4 all A, are symmetric and A. is positive definite) we take as difference equations ,l
(6.21)
A.
k,X)
E; h) v(t,X
,I (Eh + ) _ I 1 A,(E;' - E_ '`)v(t,X) f + Bv(t,X) + w a= 2h
v(t +
2n
This can be written in the form A°v (t
; k,X) i
( ( 1 A0 2n
,
k A) E 2h
zh
2n
A°
- 2hk A) E
} L u( t,X)
+ kBv(t,X) + kw For sufficiently small klh the symmetric matrices (6.22)
1 A°
2 A,
will all be positive definite. It can be shown50 that then the solution of the difference scheme with initial data f satisfies an a 50 See Friedrichs [1].
FRITZ JOHN
118
priori estimate similar to the inequality (4.55), provided the coefficients are assumed sufficiently regular. This yields convergence of the solution v towards u in the L2 norm sense. Using analogues of Soboleff's lemma for functions defined in lattice points, we can
then establish that even pointwisc v , u.51 A very general stability test of great practical importance is the so-called v. Neumann condition.52 This condition, strictly speaking, applies to differential and difference operators with constant coefficients; in many other cases, the sufficiency of the
condition still has the status of a "principle" rather than of an established fact. We start with a hyperbolic or parabolic linear differential equation (6.16) where x = (t,X) = (xo,X). We introduce a rectangular grid with mesh length h, in the x,-direction (i = 01 ... , n). We replace L by a difference operator A compatible with L. This involves introducing translation operators Ez',
(E0,
corresponding to replacing x, by x, + h,. Putting E = . . . , En), h = (h....... h,,) we can commonly write A in
the form
A[v] _
Bp(x,h)EOhv(x) ij
where fi _ (fin, and
... , (3n) is a vector with integers as components, El' =
(Eoa )a
Ordinarily the B will depend rationally on the components of h. We now replace in the neighborhood of a point the B,y(x,h) by x-independent values, corresponding to an "average" x in that
neighborhood. This can be justified partly by appeal to the intuitive principle that equations with variable coefficients will not behave better than the corresponding equations where the coefficients have been "frozen" locally into constant values, and
the common experience that they often do not behave much worse. We obtain then an operator of the form
A'[v] _ For many purposes it is convenient to restrict oneself still to the
"principal part" of the operator, dropping from A' all terms 51 See John [8]. 52 See Richtmyer [1], v. Neumann and Richtmyer [1], O'Brien, Hyman
and Kaplan (1].
FINITE DIFFERENCES
119
B,(h)EO' in which the denominator (which is a monomial in h) is a proper factor of the denominator of another nonvanishing term. For the resulting operator A" one considers the exponential solutions of the form where C is a constant vector and
Y real. In order that this exponential expression solves the differential equation, A must be a root of a certain equation involving Y and h. The v. Neumann criterion consists in the requirement that all these exponential solutions are bounded for bounded positive xo and bounded C, i.e., that IM(A)
is bounded from below uniformly in Y, h for real Y and the admitted h. Stability depends on the fact that the difference scheme satisfies this criterion. The v. Neumann test generally is a necessary condition for stability; it is often also sufficient when modified into a slightly stronger requirement.53 To apply the test to the difference equation (6.3) we would have to consider the exponential solutions of the equation
v(t + k,X) - v(t,X) k
-
E4- 2 + E,7' v(t X) h2
T=1
'
with a constant A. The expression eu "'+" solves the above difference equation if A is a solution of eilb _ I n eiy,h - 2 _F- e-iy,h k = Aj h2 or eidk
= 1 - 4Ak n
h2 i=1
sin2 3'h 2
Then the exponential solution becomes eix
+ixc = eir 1 etxkx = ei x
I
k jnY
-h
sin2 }'2
1
In order that these solutions are bounded uniformly in Y for real Y, fixed k/h2, and for k --* 0, we must have
1 - 4nA h2 < 1 ss For the case of parabolic second-order equations see John [7], Green[ I].
FRITZ JOHN
120
0 < 2nA hz < 1
which is slightly weaker than the sufficient condition (6.10) obtained previously. Similarly we find for the difference equations (6.21) corresponding to the symmetric hyperbolic system (6.20) that the v. Neumann condition follows from positive definiteness or semidefiniteness of the matrices (6.22). The v. Neumann test constitutes a very handy tool for indicat-
ing stability in cases where rigorous a priori inequalities are difficult to derive. In the case of differential equations with constant coefficients, it usually can be verified by Fourier trans-
formation. As an illustration consider the case of the wave equation z
(6.23)
L[u] = at`L - c2 Au = 0
We take initial conditions of the special form (6.23a)
u = f (X)
at = 0
for t = 0
to which more general initial conditions are easily reduced. We replace (6.23) by the difference equation (6.24)
v(t ?-- k,X) - 2v(t,X) + v(t - k,X) k2
=
n
(E;'
c2h-2
- 2 + E; h)v(t,X)
s=1
The initial conditions (6.23a) are replaced by
v(k,X) = v(-k,X) so that v(t,X) is an even function of I. The expression (6.24a)
v(0,X) = f (X),
eix'Y =18
is a solution of the difference equation if (6.25)
h2 sin2 (sk/2) = c2k2 I sin2 (y,h/2) /=1
For solutions s of (6.25) the imaginary part of s vanishes and
FINITE DIFFERENCES
121
hence is bounded from below uniformly in Y, k for real Y and
k-o0if (6.26)
k
1
h
cti/n
This is the v. Neumann condition for (6.24). The fact alone that the domain of dependence of (6.24) must include that of (6.23) would yield the same estimate. We shall prove that under the condition (6.26) v converges to u for sufficiently regular f (see H. Lewy [1]). Without restricting the generality we can assume f to be of compact support. If in addition f is in C"±', then
f (X) _
(27r)-1112
f e7.1-: f(Y) dY
The solution of (6.24, 6.24a) isr (6.27)
v(1,X) = (27r) -"/2f eir-Y cos (st) j (Y) dY
where s = s(Y) is determined from (6.25). The solution of (6.23, 6.23a) on the other hand is (6.28)
u(t,X) = (2Tr)-"12 I e'.L-1 cos (c I YI t) f (Y) dY
Now by (6.25) we know that for h, k - 0 Ist - c I YI
uniformly in any bounded set of Y-space. Moreover, whenever the stability condition (6.26) is satisfied, the solution s of (6.25) is real, and hence Icos (St) I < 1
for all real Y
Hence the integrand in (6.27) tends towards that of (6.28) in every bounded Y-set for h, k
0, while the integrand is majorized
by the integrable function I f (Y) I uniformly for all Y and all h, k satisfying (6.26). It follows that v - u, if h and k tend to zero in such a way that (6.26) holds. Thus condition (6.26) is sufficient for convergence in case f is in Co ' i. Using (6.27) we obtain trivially the a priori estimate (6.29)
Jv2(t,X) dX < f f2(X) dX
122
FRITZ JOHN
valid for solutions v of the difference equation (6.24) with initial conditions (6.24a), when the stability condition (6.26) is satisfied. The solution v only depends on the values off in the grid points.
Hence, we can replace f (X) by a function which is piecewise constant in cubes of side h centered about grid points; we find that v(t,X) will be piecewise constant for all t in the same way. Equation (6.29) then goes over into
t
G v2(t,X) C Y f2(t,X ) 1
where the sum is extended over all grid points in X-space. This formula can be made the basis of a more detailed error analysis
(both for round-off and truncation error), since from it the Fourier transformation has been eliminated.
Bibliography
Aronson, D. G. [1] The stability of finite difference approximations to second order linear
parabolic differential equations, Duke Math. J., 30, 117-127 (1963). Asgeirsson, L.
[1] Ober eine Mittelwerteigenschaft von Losungen homogener partieller Diffcrcntialgleichungen 2. Ordnung mit konstanten Koeffizienten, Math. Ann., 113, 321-346 (1937). [2] Ober Mittelwertgleichungen, die mehreren partiellen Differentialgleichungen 2. Ordnung zugeornet sind, Courant Annitenary Volume, Interscience, New York, 1948, pp. 7-20. Baker, B. B. and Copson, E. T. [1] The Mathematical Theory of Huygens' Principle, Clarendon Press, Oxford, 1939.
Bureau, F. [1] La solution elementaire d'une equation lincaire aux derivees partielles,
decomposable totalement hyperbolique, d'ordre quatre et a quatre variables independantes, Acad. Roy. Belgique. Bull. Cl. Sci. (5), 34, 566 -592 (1948). [2] Esrai sur l'Intrs, ration des Equations Lineaires aux Deruees Partiellec, Marcel Hayez, Bruxelles, 1936.
[3] Divergent integrals and partial differential equations, Comm. Pu,e Appl. Math., 3, 143-202 (1955). Campbell, L. L. and Robinson, A.
[1) Mixed problems for hyperbolic partial differential equations, Proc. London Math. Soc., Ser. 3, 5, 129-147 (1955).
Carslaw. H. S. [I] Introduction to the Mathematical Theory of the Conduction of Heat in Solids,
reprinted by Dover Publ., New York, 1945. Cauchy, A. [1] Memoire sur l'integration des equations lineaires aux differentielles partielles et a coefficients constants, Oeuvres Completes, 2e serie, Vol. I, 275-357.
Gauthier-Villars, Paris (1905). 123
FRITZ JOHN
124
Cinquini-Cibrario, M. [1] Sui sistemi di equazioni alle derivate parziali di ordine superiore, Ann. Mat. Pura Appl. (4) 29, 147-161 (1949). Collatz, L. [1] Numerische Behandlung von Differentialgleichungen, Grundlehren d. math. Wiss., 2nd ed., Springer, Berlin, 60 (1955). Courant, R.
[1] Cauchy's problem for hyperbolic quasi-linear systems of first order partial differential equations in two independent variables, Comm. Pure Appl. Math., 14, 257-265 (1961).
Courant, R. and Friedrichs, K. O. [1] Supersonic Flow and Shock Waves, Interscience, New York, 1948.
Courant, R., Friedrichs, K. 0., and Lewy, H. [1] tJber die partiellen Differentialgleichungen der mathematischen Physik, Math. Ann., 100, 32-74 (1928).
Courant, R. and Hilbert, D. [1] Methods of Mathematical Physics, Interscience, New York, Vol. I. 1953, Vol. II, 1962.
Courant, R., Isaacson, E., and Rces, M.
[1] On the solution of nonlinear hyperbolic differential equations by finite differences, Comm. Pure Appl. Math., 5, 243-255 (1952).
Courant, R. and Lax, A.
[11 Remarks on Cauchy's problem for hyperbolic partial differential equations with constant coefficients in several independent variables, Comm. Pure Appl. Math., 8, 497-502 (1955).
Courant, R. and Lax, P. D. [1] The propagation of discontinuities in wave motion, Proc. Nat. Acad. Sci., U.S.A., 42, 872-876 (1956).
[2] On nonlinear partial differential equations with two independent variables, Comm. Pure Appl. Math., 2, 255-273 (1949).
[3] Cauchy's problem for non-linear hyperbolic differential equations in two independent variables, Scritti :Matenzatici offerti a Mauro Picone, Bologna, 571-576 (1955). Douglas, J.
[1] On the numerical integration of a2u
a2u
au
aX2 + ay2 - at
by implicit methods, J. Soc. Industr. App!. Math., 3, 42-65 (1955). Duff, G. F. D. [1] Mixed problems for linear systems of first order equations, Canadian J. Math., 10, 127-160 (1958). Forsythe, G. E. and Wasow, W. R. [ 1 ] Finite Difference Methods for Partial Differential Equations, Wiley, New York, 1960.
BIBLIOGRAPHY
125
Friedman, A. [1] A new proof and generalization of the Cauchy-Kowalewski theorem to non-analytic and to non-normal systems, Proc. Symposia Pure Math. IV, Am. Math. Soc., 115-119 (1961). Friedrichs, K. O. [1] Symmetric hyperbolic linear differential equations, Comm. Pure Appl. Math., 7, 345-392 (1954).
[2] Symmetric positive linear differential equations, Comm. Pure Appl. Math., 11, 333-418 (1958). Friedrichs, K. O. and Lewy, H. [1] Uber die Eindeutigkeit and das Abhangigkeitsgebiet der Losungen beim Anfangswertproblem linearer hyperbolischer Differentialgleichungen, Math. Ann., 98, 192-204 (1928). Gnrding, L. [1] Linear hyperbolic partial differential equations with constant coefficients, Acta Math., 85, 1-62 (1950).
[2] Solution directe du probleme de Cauchy pour les equations hyperboliques, Colloques Internationaux du Centre Nat. de la Recherche Scient., 71,
71-90 (1956).
[3] L'inegalite de Friedrichs et Lewy pour les equations hyperboliqucs lineaires d'ordre superieur, C. R. Acad. Sci. Paris, 239, 849-850 (1954).
Garnir, H. G. [I] Espaces de Hilbert et problemcs aux limites de la physique, Colloq. Internat. Centre Nat. de la Recherche Sci., 71, 91-101 (1956).
Gelfand, I. M. and Silov, G. E. [1] Fourier transforms of rapidly increasing functions and questions of the uniqueness of the solution of Cauchy's problem, Uspehi Mat. Nauk (N.S.), 8, No. 6(58), 3-54 (1953): Amer. Math. Soc. Translations, series 2, 5 (1957). Goursat, E. [1] A course in mathematical analysis, Differential equations, Vol. II, Part II,
translated by E. R. Hedrick and O. Dunkel, Ginn, New York, 1917.
Green, J. W. [1] An expansion method for parabolic partial differential equations, Nat. Bureau of Standards Report 1629 (1952).
Hadamard, J. [1] Lectures on Cauchy's problem in linear partial differential equations, reprinted by Dover Publ., New York, 1952. Hellwig. G. (1] Partielle Differentialgleichungen, Teubner, Stuttgart (1960). Herglotz, G.
[1] Uber die Integration linearer, parteller Differentialgleichungen mit konstanten Koeffizienten, Ber. math. phys. Kl. d. sdchs. Akademie, 78 (1926), 80 (1928) ; Abh. math. Sem. Hamburg, 6, 189-197 (1928).
FRITZ JOHN
126
Hille, E. and Phillips, R. [1] Functional analysis and semi-groups, Rev. ed. Am. Math. Soc. Colloquium Publ., 31 (1957).
Holmgren, E. [1] Uber Systeme von linearen partiellen Differentialgleichungen, Ofversigt of kongl. Vetenskaps-Akademien Forhandlinger, 58, 91-103 (1901).
[2] Sur 1'Equation de la propagation de la chaleur, Arkiv for Matematik, Astronomi och Fysik, 4, No. 14,18 (1908).
Hormander, L. [1] On the theory of general partial differential operators, Acta Math. 94, 161-248 (1955). [2] Linear Partial Differential Operators, Academic Press, New York, 1963.
John, F. [1] Numerical solution of the equation of heat conduction for preceding times, Annali di Matematica, serie IV, 40, 129-142 (1955). [2] Plane Waves and Spherical Means Applied to Partial Differential Equations,
Interscience, New York, 1955.
[3] Continuous dependence on data for solutions of partial differential equations with a prescribed bound, Comm. Pure Appl. Math., 13, 551-585 (1960).
[4] On linear partial differential equations with analytic coefficients. Unique continuation of data, Comm. Pure Appl. Math., 2, 209-253 (1949).
[5] Linear partial differential equations with analytic coefficients, Proc. Nat. Acad. Sci., U.S.A., 29, 98-104 (1943). [6] Non-admissible data for differential equations with constant coefficients, Comm. Pure Appl. Math., 10, 391-398 (1957).
[7] On integration of parabolic equations by difference methods, Comm. Pure Appl. Math., 5, 155-211 (1952). [8] Advanced numerical methods, Lecture Notes, Courant Institute of Math. Sciences, New York University (1956).
Jorgens, K. [1] Das Anfangswertproblem im Grossen fur eine Klasse nichtlinearer Wellengleichungen, Math. Z., 77, 295-308 (1961). Ladyzenskaya, O. A. [1] The mixed problem for a hyperbolic equation, Gosurdarsty. Izdat. Tehn.-Teor. Lit., Moscow (1953).
[2] The method of finite differences in the theory of partial differential equations, Uspahi Mat. Nauk (N.S.) 12 (1957), no. 5 (77), 123-148. Lax, A.
[1] On Cauchy's problem for partial differential equations with multiple characteristics, Comm. Pure Appl. lviath. 9, 135-170 (1956).
Lax, P. D. [1] Asymptotic solutions of oscillatory initial value problems, Duke Math. J., 24, 627-646 (1957). [2] Differential equations, difference equations and matrix theory, Comm. Pure Appl. Math., 11, 175-194 (1958).
BIBLIOGRAPHY
127
Lax, P. D. and Milgram, A. N. [1] Parabolic equations. Contributions to the theory of partial differential equations, Annals of Math. Studies, 33, 167-190 (1954).
Lax, P. D. and Phillips, R. S.
[1] The wave equation in exterior domains, Bull. Am. Math. Soc. (in press).
Lax, P. D. and Richtmyer, R. D. [1] Survey of stability of linear finite difference equations, Comm. Pure Appl. Math., 9, 267-293 (1956).
Leray, J. [1] Lectures on hyperbolic equations with variable coefficients, Inst. for Advanced Study, Princeton (1952).
[2] On linear hyperbolic differential equations with variable coefficients on a vector space, Annals of Math. Studies, 33, 201-210 (1954). Lewy, H. [1] On the convergence of the solution of difference equations, Studies and Essays presented to R. Courant, Interscience, New York, 1948, pp. 211-214.
Lions, J. L. [1] Operateurs de Delsarte et problemes mixtes, Bull. Soc. Math. France, 84, 9-95 (1956). Magnus, W. and Oberhettinger, P. [11 Formulas and Theorems for the Special Functions of Mathematical Physics, Chelsea, New York, 1949.
Miranker, J. W. [I] A well posed problem for the backwards heat equation, Proc. Am. Math. Soc., 12, 243-247 (1961). Mizohata, S. [1] Hypoellipticite des equations paraboliques, Bull. Soc. Math. de France, 85, 15-50 (1957). Morawetz, C. S. [1] The decay of solutions of the exterior initial-boundary value problem for the wave equation, Comm. Pure Appl. Math., 14, 561-568 (1961).
Motzkin, T. S. and Wasow, W. [1] On the approximation of linear elliptic differential equations by difference equations with positive coefficients, J. Math. Phys., 31, 253-259 (1953).
Muller, C. [1] Crundprobleme der mathematischen Theorie elektromagnetischer Schwingungen,
Springer, Berlin, 1957. Nircnberg, L. [1] Estimates and existence of solutions of elliptic equations, Comm. Pure Appl. Math., 9, 509-530 (1956). [2] A strong maximum principle for parabolic equations, Comm. Pure Appl. Math., 6, 167-177 (1953). O'Brien, G. G., Hyman, M. A., and Kaplan S.
[1] A study of the numerical solution of partial differential equations, J. Math. Phys., 29, 223-251 (1951).
FRITZ JOHN
128
Petrowskii, I. G. [1] Lectures on Partial Differential Equations, translated by A. Schenitzer, Interscience, New York, 1954.
[2] On some problems of the theory of partial differential equations, Uspehi Mat. Nauk (N.S.) I, No. 3-4 (13-14), 44-70 (1946), Amer. Math. Soc. Translation, No. 12. [3] Uber das Cauchysche Problem fur Systeme von partiellen Differentialgleiehungen, Recueil math. (Mat. Sb.), 2,44, 815-868 (1937). Poisson, D. [1] Memoire sur la theorie du son, Journal de 1'Ecole Polytechn., 7, 319-392 (1807).
Pucci, C. [1) Sui problemi de Cauchy non "ben posti," Rendiconli Accad. Nazionale dei Lincei; Cl. Sci. fis nut., Ser. 8, Vol. 18, 473-477 (1955). Richtmyer, R. D. [1] Difference Methods for Initial Value Problems, Interscience, New York, 1957.
Riesz, M.
[1] L'integrale de Riemann-Liouville et la probleme de Cauchy, Acta Math., 81, 1-223 (1951). Robinson, A. [I] On the integration of hyperbolic differential equations, J. London Math. Soc., 25, 209-217 (1950). Rose, M. E. [1] On the integration of non-linear parabolic equations by implicit difference methods, Quart. Appl. Math., 14, 237-248 (1956). Rosenbloom, P. C.
[1] Linear equations of parabolic type with constant coefficients. Contributions to partial differential equations, Annals of Afath. Studies, 33, 191-200 (1954). [2] The majorant method, Proc. Symposia Pure Math. IV, Amer. Math. Soc., 51-72 (1961). Schwartz, L. [1] Theorie des Distributions, Hermann, Paris, 1950. Silov, G. E. [1J On conditions of correctness of Cauchy's problem for systems of partial differential equations with constant coefficients, Uspehi !1'Iat. Nauk (N.S.), 10 (1955) No. 4 (66) 89-100. Sobolev, S. L.
[1] Sur tin theoreme d'analyse fonctionelle, A1at. Sb. (N.S.), 4, 471-497 (1938).
Stellmacher, K. L. [1] Ein Beispiel einer Huygensschen Differentialgleichung, Nachr. Akad. Wiss. Gottingen, Math. Piys. K1., 10, 133-138 (1953).
[2] Eine Klasse huygensscher Differentialgleichungen and ihre Integration, Math. Ann., 130, 219-233 (1955).
BIBLIOGRAPHY
129
Tychonoff, A. [1] Theoremes d'unicite pour 1'equation de la chaleur, Mat. Sb., 42, 199-216 (1935).
von Neumann, J. and Richtmyer, R. D. [1] A method for the numerical calculation of hyrodynamic shocks, J. Appl. Phys., 21, 232-237 (1950). Weinstein, A. [1] The Cauchy problem for the wave equation and the equation of EulerPoisson-Darboux, U. of Maryland, Technical Note Bn-22, 1954.
Widder, D. V. [1] Positive temperatures in an infinite rod, Trans. Am. Math. Soc., 55, 85-95 (1944). 12] The convolution transform, Bull. Am. Math. Soc., 60, 444-456 (1954). Wilcox, C. H.
[1] The domain of dependence inequality and initial-boundary value problems for symmetric hyperbolic systems, Univ. Wisconsin Math. Res. Center Technical Summary Report 333 (1962).
Zaremba, S. [1] Sopra un teorema d'unicita relativo alla equazione delle onde sferiche, Rend. Accad. Lincei, Ser. 5, 24, 904-908 (1915).
Zimmerman, J. M. [1] Band-limited functions and improper boundary value problems for a class of nonlinear partial differential equations, Proc. Am. Math. Soc., 12, 243-247 (1961).
PART II
Elliptic Equations By Lipman Bers and Martin Schechter
CHAPTER 1
Elliptic Equations and Their Solutions 1.1.
Introduction
The theory of elliptic partial differential equations is not applied mathematics; it is a discipline in pure mathematics. But it is a discipline which developed under the influence of, and in intimate connection with, mathematical physics. Elliptic differential equations occur in mathematical physics either in their
own right or in connection with hyperbolic and parabolic equations, since these equations usually involve an elliptic operator. If, however, one wants to solve a problem for an elliptic equation which arises in applications, it is seldom enough to consult a book or an expert on the theory of elliptic equations.
In practically all cases, a problem which is of genuine applied interest will be on the one hand more special than the problems considered in the general theory, and, on the other hand, such a problem will usually have peculiarities which will preclude an automatic and mechanical application of the general theory. It may happen for instance that the domain considered will be unbounded, or that the boundary will have corners, or that the coefficients will be singular, or that the boundary value problem itself will be of an unconventional type. Thus in order to solve such a problem some independent work is usually needed. The general theory, however, shows what methods one may use, what results one may expect, what is likely to happen, and what seems feasible in the present state of knowledge. It is from this point of view that the theory of elliptic equations belongs in the toolbox of an applied mathematician. The theory can be divided into two large chapters. The first deals with linear problems and is at present probably nearing completion. The second chapter, and in our opinion the more fascinating one, deals with nonlinear problems. Here we have 133
L. BERS AND M. SCHECHTER
134
many powerful and deep individual results, but we are as yet far removed from a general theory. It is obviously impossible to give an adequate presentation of elliptic equations in such short space. This is so not only because the field is very rich, but primarily because it has not yet reached
that state of perfection at which an axiomatic treatment is advisable. At present the theory of elliptic equations is not a rigid
structure but rather a collection of results, methods, ideas, and points of view. We feel, therefore, that we can best fulfill our task
by describing some of the methods which have proved fruitful and by applying them to examples as simple as possible. It goes without saying that many things will have to be stated without proof, many proofs will have to be only sketched, and many important topics will have to be omitted entirely. We will begin by discussing linear elliptic equations.
1.2.
Linear Elliptic Equations
The simplest partial differential equations of elliptic type are the familiar Laplace equation and its nonhomogeneous form, the Poisson equation, n
Au(X1i
. . .
, x,) =f (XI, .
. .
,
Xn),
0
a2
j=1 aX;
It is well known that many profound investigations have been carried out in connection with these seemingly simple equations and that the theory of the Laplace equation includes as a special case the theory of analytic functions of a complex variable. Elliptic equations can best be described as equations which have
many properties in common with the Laplace (or Poisson) equation. In order to give a definition rather than a description we consider first the most important case, a second-order equation' n
a2u
I a,k OX= aXk i.k=1
au a,+ au =f aX, 1=1
1 We shall assume throughout, unless otherwise stated, that all functions
are real valued. One can also consider elliptic equations with complex coefficients. See Appendix I to this chapter.
SOLUTIONS OF ELLIPTIC EQUATIONS
135
This equation is called elliptic in a region if at every point of this
region it can be transformed, by introducing new independent variables, into the form au -AU i=1+Ia;-+au=f axi "
(We shall see later in Chapter 6 that for elliptic equations in the plane such a transformation is possible not only at a point, but in
the whole domain.) A necessary and sufficient condition is obviously that the quadratic form E atk4;Ek be definite. This gives a hint as to the definition of ellipticity in the general case. A partial differential operator of order m can be written in the form aw
Lm
L
pi+...+ pn=/' aDl . a9(x)Dp
. .
Dn(x)
axi'
. . .
ax".
IpI Sm
n > 2. The multi-index p = (p1, ... , p,,) Here x = (x1, ... , is a "vector" with integral non-negative components. The norm of such a "vector" is defined by IpI = pi + . . . + p,,. For every vector symbolic "vector"
we write p = i , . . . cc". D is
D=(D1i...,D.),
the
D;=a
The characteristic form of L is by definition the form (of degree m)
I
The equation is called elliptic if this form
IpI =m
is definite for every point of the domain considered. In this case m is even. We set m = 2m' and normalize L by the requirement
(-1)mQ(x,) _ (-1)"
aD(x)D > 0
IPI =m
for
7- 0
1.3. Smoothness of Solutions
A good heuristic principle in investigating elliptic equations states that every property of the Laplace (or Poisson) equation
L. BERS AND M. SCHECHTER
136
remains true for general elliptic equations if properly reformulated. Now the Laplace equation in two dimensions describes harmonic functions, i.e., real parts of analytic functions of a complex variable, so that every solution of the Laplace equation is itself a real analytic function. The same is true for Laplace's equation in
higher dimensional spaces. A generalization of this property of the Laplace equation is the following ANALYTICITY THEOREM.
If the coefficients of the elliptic operator L
are real anal tic functions, and if the right-hand side f is real analytic, then all solutions of Lu =f are real analytic functions.
This remarkable theorem is true also for nonlinear elliptic equations and for elliptic systems (Bernstein [1,2], E. Hopf [1], Lcwy [11, Petrowsky [1], John [1], Morrey and Nirenberg [1], Morrey [1], and others). It is one of a series of theorems which assert, roughly speaking, that a solution of an elliptic equation is as smooth and as well behaved as the coefficients permit it to be. A proof of the analyticity theorem will be given in the Appendix to Chapter 4. In order to state another such theorem we recall the definition of Holder continui y. A function g(x) is said to satisfy a Holder condition with constant K and exponent x in a set if at any two points x' and x" in this set (g(x') - g(x") I < K Ix' - x"1" (Here and hereafter Ix12 = I (xl,
,x,+)12 = x`I +
...
+ X2.)
THEOREM ON HOLDER CONTINUITY OF DERIVATIVES (E. Hopf [1] page 34 and Douglis-Nirenberg [I ]). If the coefficients and the right
hand side of the elliptic equation Lu = f of order m satisfy Holder conditions with (positive) exponent x < 1 (belong to class C2), then the m-th derivative of every solution also satisfies a Holder condition with the same exponent. If the coefficients and the right hand side have derivatives up to the order s which satisfy Holder conditions with exponent x < I (belong to class CS Z), then every solution u has derivatives up to the order s m and these derivatives satisfy a Holder condition with the same exponent a. I
The Holder condition is essential. An equation with continuous but not Holder continuous coefficients need not possess solutions
SOLUTIONS OF ELLIPTIC EQUATIONS
137
which have continuous derivatives up to the order of the equation;
this is true even for the simplest elliptic equation, the Poisson equation Au =f. In the two preceding theorems, "solution" stood for what we will henceforth call a classical solution, that is a function having as many derivatives as the order of the equation indicates (a function of class C,H). It is useful, and sometimes necessary, to extend the concept of a solution.
In order to do this we first extend the concept of a derivative. Let u(x), v(x) be two measurable, locally integrable functions
defined in a domain 9. (From now on all functions will be assumed measurable unless otherwise stated and two measurable functions which differ only on a null set will be identified.) We say that v = D''u in the weak sense if dx = (-1)1 P1 f3, vs dx f,,UDPO
for every infinitely differentiable function O(x) with compact support2 in 9. Such functions will henceforth be called "test functions." It is clear that every continuous derivative is a weak derivative. We say that v - Dpu in the strong Lv sense (q > 1) if for every compact set )( C there exists a sequence of infinitely differentiable functions {0,(x)) such that Jxi
- ulvdx
0,
jIDP. - vivdx
0
These concepts are due to Sobolev [1] and Friedrichs [1] who also established the following THEOREM.
Let u(x), v(x), x e ?, be two functions which are
locally integrable to some power q > 1. Then v = Dpu in the weak sense if and only if v = Dpu in the strong Lq sense.
The "if" part of the theorem is of course obvious; the "only if"
part will be proved in Appendix TI to this chapter. Instead of "strong" or "weak" derivatives we shall henceforth say generalized 2 The support of a function is the smallest closed set such that the function vanishes on its complement. The functions considered are those which vanish outside bounded closed sets contained entirely in 9.
138
L. BERS AND M. SCHECHTER
derivatives and mean, in most cases, generalized L2 derivatives. A function having L2 derivatives up to the order m and satisfying
the differential equation Lu =f almost everywhere (these two words must necessarily be added on!) will be called a strong solution of the equation. THEOREM ON THE CONTINUOUS DIFFERENTIABILITY OF STRONG
Let the coefficients and the right hand side of the elliptic equation Lu =f be Holder continuous. Then every strong solution is a SOLUTIONS.
classical solution.
In principle, one could also define the concept of a strong solution in a different way, by requiring that there should exist a sequence of smooth, say infinitely differentiable, functions {(x)} such that 0,, -- u, Lay fin the LQ sense for every com-
pact _Y- c 9. It turns out, however, that for elliptic equations this would not lead to a new class of solutions (cf. Chapter 3). Another extension of the concept of a solution is possible if we assume sufficient differentiability of the coefficients of the equations
considered. More precisely, we require that we should be able to form the formally adjoint operator
L*u =IrI<_m I
(-l)ITID'(a,(x)u)
which should itself be a differential operator with continuous coefficients. It is clear that the adjoint operator is elliptic and that under sufficient smoothness conditions L* * = L. An integrable function u(x) defined in a domain 9 is called a weak solution of the
equation Lu =f if 5uL*Odx Jlsfodx
for every test function 0. (For classical and strong solutions this holds by integration by parts.) THEOREM ON THE STRONG DIFFERENTIABILITY OF WEAK SOLUTIONS
(Friedrichs [2], Browder [1], John [2], Lax [1], Schwartz [1], Niren berg [1], and others).
If the elliptic equation is such that a weak solution may be defined, then every weak solution is also a strong solution.
The last theorem can be extended further if we assume that the
coefficients in the operator L are infinitely differentiable. A distribution T in 9 is a linear functional which assigns a number
SOLUTIONS OF ELLIPTIC EQUATIONS
139
T[c6] to every test function O(x) such that T[clol + c202] =
c1T[O1] + c2T[02], c1 and c2 being constants. It is required that this functional be continuous in the following sense: if {0v} and all
derivatives of 0, converge uniformly to zero, and if all test functions 4, vanish outside the same compact set, then T[Oy] , 0. It is clear that if u(x) is an integrable function, a distribution T
may be defined by setting T[q] =
I
JI
uo dx. In this case one says
that the distribution T is the function u. A distribution has, by definition, derivatives of all orders which are also distributions. These are defined by the formula D P T [q] = (- l )1 '1 T [DP¢] . A distribution may be multiplied by a C,,, function co according to the formula co T [0] = T [coo] ; the definition of the sum of two
distributions is obvious: (T1 + T2) [0] = T1[q] + T2[0]. Thus it makes sense to form the distribution LT. THEOREM ON DISTRIBUTION SOLUTIONS (Schwartz [1]). Let the coefficients of the elliptic operator L be infinitely differentiable and let f be a
square integrable function. If the distribution T satisfies the equation L T = f, it is a square integrable function (and hence a weak solution of the equation considered).
It is more convenient, though less precise, to write ft(x)0(x) dx
instead of T[0], t(x) being a "symbolic function" representing the distribution. We shall do so in what follows. Of the theorems just stated, the proofs of some of which will be given later (cf. Chapters 3 and 4), the most striking one is certainly the analyticity theorem. It is also the one which characterizes the
elliptic equation. Thus an equation with constant coefficients which has only analytic solutions is certainly elliptic, while there are nonelliptic equations having solutions which are only infinitely differentiable. On the other hand, if we think of elliptic equations as arising from applications, an analyticity assumption is hardly natural. 1.4.
Unique Continuation
There are cases where the heuristic principle of the preceding section is not as reliable as had been expected. This is particularly true for properties which are inferred from the analytic character
140
L. BERS AND M. SCHECHTER
of solutions. It has recently been shown that such properties are
not always true for solutions of equations with nonanalytic coefficients.
We say that an elliptic equation possesses the weak unique continuation property if every solution of the homogeneous equation
Lu = 0 which vanishes in an open set vanishes identically. In view of the analyticity theorem, this property is obvious for elliptic equations with analytic coefficients. We say that an elliptic equation has the Runge property if the following is true. Let 917, c c 9/2 be domains homeomorphic
to the sphere such that the closure W. of 50, is contained in 9. Let u0 be a solution of the homogeneous equation Lu = 0 defined
in 1 and let e be any positive number. Then there exists a solution u of the same equation defined in g2 such that JU0 - ul < e
in 6,. THEOREM (Malgrange [11, Lax [2]).
An elliptic equation with sufficiently smooth coefficients has the Runge property i f and only if the adjoint equation has the weak unique continuation property.
The weak unique continuation property can also be interpreted in another way. For any partial differential equation of order m the Cauchy problem consists of finding a solution which takes given values together with its derivatives up to the order m - 1
on a given hyperplane (or, more generally, on an (n - 1) dimensional hypersurface). It is well known that this "initial value problem" is not a properly posed problem for an elliptic equation. In general, it will not have a solution, and the solution, if it exists, will not depend continuously on the prescribed data. For analytic elliptic equations, however, it is known that the Cauchy problem cannot have more than one solution. It is not difficult to verify that the uniqueness theorem for the Cauchy problem is equivalent to the weak unique continuation property. An elliptic equation is said to possess the strong unique continuation property' if every solution which vanishes at a point of infinite
order, that is, tends to zero faster than any power of the distance from this point, vanishes identically. The strong unique continuation theorem has been established
SOLUTIONS OF ELLIPTIC EQUATIONS
141
for second-order equations in the plane, under the most general hypotheses on the coefficients (Carleman [1], Morrey [1], Bers and Nirenberg (1]), and for second order equations in any number of dimensions, in the case of sufficiently smooth coefficients (Muller [1], Heinz [1], Aronszajn [1], Cordes [1], Hartman and Wintner [1]). The weak unique continuation theorem has also been proved for certain elliptic systems and equations of higher
order (Carleman [2], Douglis [1], Nirenberg [2], Landis [1], Pederson [1], Calderon [11, Mizohata [11, Hormander [1], Protter [1], and others). On the other hand, there exist very recent examples (due to Plis [1,2] and to P. Cohen [1]) of elliptic equations for which the weak unique continuation theorem does not hold. A proof of Carleman's theorem is given in Section 2.8. 1.5.
Boundary Conditions
We have up to now discussed general properties of solutions. One is, however, interested primarily in finding solutions satisfying conditions which determine the solutions uniquely. For the
Laplace equation the simplest and most important properly posed problem is the Dirichlet problem : to find a harmonic function
defined in a domain !f and taking on prescribed continuous values at the boundary 9 of the domain. The theory of the Dirichlet problem for the Laplace equation is well known and it is not necessary to recall how many results and methods in modern
analysis were developed just in order to treat this problem. For
the general linear elliptic equation of order m the Dirichlet problem consists of finding a solution defined in a domain W (which we will always assume to be bounded) such that the function and its first (m' - 1) normal derivatives take on given values at the boundary 9. The theory of this Dirichlet problem will occupy us on several occasions later on, but we may now state the main result. We consider the two formally "adjoint" Dirichlet problems
m'-1
Lu =f in V,
DPu=0onI
for JpJ
L*u =fin«i,
DPu =0on5
forfpJ <m' - 1
L. BERS AND M. SCHECHTER
142
(There is almost no loss of generality in restricting oneself to homogeneous boundary data.) We say that these two problems form a Fredholm pair if the following three theorems hold. (a) The nonhomogeneous problem Lu =f has a solution for every right-hand side f if and only if the homogeneous problem Lu = 0 has only the trivial solution u = 0. (b) The homogeneous problem
Lu = 0 has at most a finite number of linearly independent nontrivial solutions and the adjoint homogeneous problem L*u = 0 has the same number of such "eigen solutions." (c) The nonhomogeneous problem Lu =f has a solution if and only if the right hand side f is orthogonal (in the sense of Hilbert space) to every solution of the homogeneous adjoint problem. We shall see later (Chapter 4) that upon proper definition of the terms and under "reasonable" assumptions, these theorems always hold for elliptic Dirichlet problems. Also the Dirichlet problem is always solvable if the domain is sufficiently small. If one considers the operator L + A involving a parameter A, then the set of values of this parameter for which there exist eigen solutions is discrete.
In addition to the Dirichlet problem, one can consider more general boundary value problems for elliptic equations. For instance, let B;
_ 1P:Sne5b, (x) D',
j = 1,
. . .
,
m'
be linear partial differential operators of order m; < m with coefficients defined on !. If the set {B5}71 is normal in the sense
that the m, are distinct and none of the B, are tangential to one can find another normal set
B* = I b3(x)D',
j = I.... , m'
IP! :5m1
such that the problems
Lu=find,
B,u=0onT
forjl,..., m'
L*u =fin T,
B*u = 0 on 4
forj = 1,...,m'
are adjoint. It has been shown that, under suitable assumptions, properties (b) and (c) above hold for such problems (Schechter [5,2], Browder [4]). When the set {B;}; `_ 1 is not "normal," the adjoint boundary conditions are no longer differential; they are
SOLUTIONS OF ELLIPTIC EQUATIONS
143
integro-differential. However, it can be shown that properties (b) and (c) hold also for such boundary conditions (Schechter [3], Peetre [2]). Another type of boundary problem is the so-called "mixed" problem in which different boundary conditions are given on adjacent portions of the boundary. For second-order equations, such problems have been studied extensively by many authors (Giraud [1], Fichera [1], Lions [1], Magenes [1], Miranda [3], Stampacchia [3], and others). Recently, extensions to higher order equations have been given (Schechter [4], Peetre [2], Shamir [1]). Appendix I. Elliptic versus Strongly Elliptic
We have defined an operator
L=
a9(x)D'
with real coefficients to be elliptic in a domain 9 if I a,(x)E" is definite in E at each point of 1. If one wants to Ipl ='n
admit complex coefficients this definition cannot be carried over. In such cases L is defined to be elliptic if Q (x,E) is positive definite. This coincides with the original definition in the case of real coefficients. The broader definition has the disadvantage of admitting some queer specimens. In fact, in two dimensions one can have elliptic operators of odd order. Thus the Cauchy-Riemann operator a
1( a
a
a =2ax +Z ay) is elliptic. Moreover, operators of even order can have disturbing properties. For instance, the Dirichlet problem is not "properly posed" for the equation of Bitsadze [1] (1)
u,z + 2iuxv + u., = 0
(If 9 is the unit disc, u =f(z) (1
-
JZ12)
is a solution, which
vanishes on the boundary, for any complex analytic function.)
L. BERS AND M. SCHECHTER
144
In order to avoid such cases, many authors (Vi"sik [1], Girding
[1], and others) have found it convenient to consider those is definite in 9, where y(x) is
operators for which Re y(x)
an appropriate continuous, complex valued function. Such operators are called strongly elliptic. For them the so-called Girding
inequality holds and the methods of Chapters 3 and 4 can be applied. Clearly, elliptic operators with real coefficients are strongly elliptic.
It was later discovered that the Dirichlet problem can be solved for equations which are not strongly elliptic (Schechter [1]). It was shown that a necessary and sufficient condition for the Dirichlet problem (or many other boundary value problems for that matter) to be well posed is that L be properly elliptic (Schechter [5,2], Browder [4], Agmon-Douglis-Nirenberg [1]). An operator L is properly elliptic if m _ 2m' and the polynomial in 7. Q(x,
+ ),4')
has m' roots with positive imaginary part and m' roots with negative imaginary part for every pair of linearly independent real vectors } and 4:'. Every strongly elliptic operator is properly elliptic, but the converse is true only for second-order equations. An example of a fourth-order operator which is properly elliptic but not strongly elliptic is
L = Di -;- D - D4
t (Di + D2) D3
in three dimensions.
It is easy to show (Lopatinskii [1]) that in dimensions higher than the second every elliptic operator is properly elliptic. Thus restrictions need only be made in two dimensions in order to avoid the pathological occurrences mentioned above. Appendix lI.
"Weak Equals Strong"
We shall prove that if u(x) and v(x) are locally in L, in q > 1, and fuDPdx = (-l)1"l I vO dx (1) s s
IV,
SOLUTIONS OF ELLIPTIC EQUATIONS
145
for everv test function 0, then for every compact set it" c 9) there is a sequence of 'C,, functions fu-,(x) } such that
(2)
1w, - ui" dx
JlDw. - vI4 dx - 0
0,
as i,
> oo
For this purpose it is convenient to use mollifiers (Friedrichs [1]). Letj(x) be an infinitely differentiable function in n-space with
support in the unit sphere xj < 1 such that j(x) > 0 and
JJ(x) dx = 1
(3)
For x in a compact set Y c 9P with distance >E from 9, define the C., function fs j (x
JEu(x) = e
F
)u(y) dy =1 .lj(z)u(x - ez) dz
A very important property of the operator J, is that (4)
ul" dx
f-) J,u
0
as e - 0
For by (3),
j(z) [u (x - Fz) - u(x)] dz
JFu(x) - u(x) _ and by Holder's inequality3 IJEu - uI
s,X2
;1
j(z) dz
!
xf whence lieu
- ul°
j(z) Iu(x - ez) - u(x) I' dz
I
f
j(z) Iu(x - ez) - u(x) IQ dz
2 1
This gives
5 IJu - ulq dx < sup
f Iu(x - ez) - u(x)lQ dx
3 If q = 1 we have I JEu - u,
:5J.'
j(z) lu(Y - Fz) - u(x)l dz
which is equivalent to the following step.
L. BERS AND M. SCHECHTER
146
If u is continuous in 9, this obviously approaches zero as e
0.
In order to prove it for u only locally in Lq in 9, let 91 be a domain such that X c 9J1, 91 c T. Then for any e' > 0, there is a continuous function g in 91 such that
- g(x) I', dx) q<
(fo'i l u(x)
Hence
(JIu(x - ez) -
1/q
u(x) I q dx)
<
1/q
(u(x - ez) - g(x - ez)Iq dx) \1/q
+ (
Ig(x - ez) - g(x) I q dx
if Ig(x) - u(x) fir
±
Iq
dx)1/q
/
.y
< e'
for I zI < 1 and e sufficiently small. Thus (4) is established.
In order to prove (2) we note that for fixed xo in .'' and sufficiently small j (x°
fg
e
x)
is a test function in T. Therefore by (1)
x f dx =
u (x) DPj (x°
e
(-1) IPI
f,v(x)j (x0
e
e
x) dx
Since this is true for any point xo in )Y, D'Jeu = J6v
in if'. Thus vIq dx =J IJFv - vlq dx , 0 as e -1 0 and the proof is complete. The same method can be used to show that in compact subdomains difference quotients converge, in the L sense, to weak derivatives. For instance, suppose that (5)
f
vc6dx
SOLUTIONS OF ELLIPTIC EQUATIONS
147
for all test functions 0. Let bh,u = jt [u(xl, .. . , x, + h,
.. .
... , xn)]
u(xl,
,
be a difference quotient in the direction of x=. We claim that for any compact set J( c eF and any q > 1
f
(6)
I b,hu - vj? dx
0
as h - 0. In order to prove this, let 9l be any domain such that . c 91 and ik1 c <20. We note that for any test function 0 with support in X", the function
T=
1
fx=
h
c(x1,...,xz-1) S,xt11,...,x,i) 4
is a test function in 91 for h sufficiently small. It satisfies
LTvdx Cfr I0IP d x =f
(7)
I01p
dx
for any p > 1. Moreover, D=To =
1
{ (xl,
... , x,, ... , -O(xl,...,x=-h,..
Therefore by (5)
jbuc6 dx = -
ff
ub; ho dx
r 1
ff uATO dx = ffvTgdx and by Holder's inequality and (7), 5buc6 dx
(J'ivi? dx )""(f
11?'
JT0j? dx )
dxll/y
11?
i
I vI?
dx1(f-,,, I01? fl
L. BERS AND M. SCHECHTER
148
where q' = q f (q - 1). Since4 d;'uO dx
IIq
fr ul° dr) = l.u.b.
10'
dx)
where the 1.u.b. is taken over all test functions 0 with support in
K, we have (8)
I vl ° dx
Now let e > 0 be given. By the Friedrichs-Sobolev theorem proved above, there is a C,, function w such that
(I la-D w1°dx
(9)
1,'q
1
<
.
1
E
3
Moreover, since
J(u - w)D, dx = -J (v - D,w)tp dx for all test functions (10)
V
in 1', we have by (8)
f,r,k5'(u - w) I" dx
5Iv - D,wj° dx
Also, by taking h sufficiently small we can make (11)
(f;r- I o'w
- D,w)° dx) < 3
4 Here we have made use of the well-known fact that 1/°
Jul° dx)
- l.u.b. (fr IrI° dxl1
where the l.u.b. is taken over all v in L°, in .*. This follows from Holder's inequality and the fact that v = IuI° u is in L. in .;f' and lv14'dx - J fIul'1 dx J
We now merely note that the l.u.b. need only be taken over a dense set in e.g., over all test functions.
SOLUTIONS OF ELLIPTIC EQUATIONS
149
Combining (9), (10), and (11) we have
(j'icu
- vdxl/
'
`
< (J'io'(u
r
- w) I ° dx)
\t/a D,wl dY ` --
11.1
I D,w - vI" dxl < E
and (6) is proved.
It should be noted that we needed 1' to be bounded away from 9 in order for To to be a test function in W and for (9) to hold. But T operates only in the direction of x,. Hence we have the following REMARK.
If there is a sequence {w,.} of Cx functions in 9 such
that
£ 1a -D,w,I"dx---0
asv - >oo
then (6) holds for any compact subset 'f' such that small displacements parallel to the x, axis do not take t- outside '32.
CHAPTER 2
The Maximum Principle 2.1.
Second-Order Equations
Until recently, the familiar maximum principle seemed to be a counterexample to the heuristic principle stated in Section 1.3. It had been extended to general second-order elliptic equations, but a reformulation which would remain valid for equations of
higher order was not discovered until the work of Miranda [2] and Agmon [1], who found important results which may be considered as analogues of this principle. We shall not describe these results here, but rather shall be concerned only with secondorder elliptic operators, which we write in the form (1) Lu = Mu -F au where
M
n
a2 i.k _ la:x
ax: axe
i
a
at ax i
Here aik, ai and a are functions of x = (x1, ... , which we assume to be continuous. Once and for all we assume that the operator is uniformly elliptic, more precisely that is, that there exist constants m > 0 and K such that n
(2)
n
11 a,kE,
i,k= 1
k ? M j 2, 1=1
1a,I < K,
Tail < K,
We shall be concerned with functions u defined in a bounded domain T. Unless otherwise stated, we assume the functions to be = T U of of class C2 in 9 and of class Co on the closure
( being the set theoretical boundary). 2.2.
Statement and Proof of the Maximum Principle
The maximum principle in its strong form is formulated in the following four theorems (E. Hopf [2], Giraud [2]). 150
THE MAXIMUM PRINCIPLE
151
I. If Mu z 0 in 9 and u reaches its maximum at an interior point, then u is a constant. 11.
If a < 0, L (u) z 0, u reaches its maximum at an interior point
and the maximum is positive, then u is a constant. III. Let Mu z 0 and let the maximum of u be reached at a boundary point R on 9. Assume that there is a solid sphere (closed ball) lying entirely in which has R as a boundary point. Then either u is a constant or the outward normal derivative du/dN at R is positive.
IV. Let a S 0, Lu > 0, and let the maximum of u be reached at a boundary point R on W. Let this point satisfy the condition formulated in
III. If the maximum of u is positive, then either u is constant or the outward normal derivative du/dN at R is positive.
In Theorems III and IV the existence of du/dN is assumed; but if du/dN does not exist, the same conclusion holds for the lower outward normal derivate, as will be seen from the proof. We shall first prove III under the additional assumption that
u < 01 in 1. We may draw two concentric spheres So and Sl such that the outer sphere So passes through R and such that the closure of the domain go enclosed by the two spheres lies entirely
in T except for the point R. Suppose that we can find a C2 function g(x), x e fo, such that Mg > 0 in gro, g = 0 on the outer sphere, dg/dN < 0 at R. Set v = u + Eg, where s is positive and so small that v < u(R) on the inner sphere S1. Then My > 0 in go, v < u(R) on S1. Assume first that v achieves its maximum at a point Ro in go. We may assume, since this can be achieved by a rotation, that at Ro we have a,, = 0 for i :f-j, a=i > 0. Since the first derivatives of v vanish at Ro and Mv(R0) > 0, at least one second derivative is positive, which is impossible at an interior maximum point. Hence, the maximum of v on Vo is achieved on So and thus at R (since v = u on So). Therefore the outward normal
derivative of v is non-negative at this point: 0 < (du/dN) + e(dg/dN). Thus du/dN > 0. The desired function g can be written down explicitly: g = e_2 - e-2x02
where a > 0 is a constant, r is the distance from the center Rl of the two spheres, and ro is the radius of the outer sphere. We
L. BERS AND M. SCHECHTER
152
have (assuming xi to be the origin and denoting the radius of Si by r,) Mg = e 2r2(4x21 a,,x;x, - 2aj a,x{ - 2al a,1) > e-"r2(4xzmr, - 2nxKro - 2nxK) so that Mg > 0 in ,, if x is sufficiently large.
Now we can prove I. Assume that the maximum ,u of u is achieved at an interior point x0 and let 6 > 0 be so small that the ball Ix - x01 < 26 lies in U. We claim that u(x) = u for Ix - x I < 6. In fact, assume that u(xl) < u, Ix1 - x01 < 6.
Then there exists a number p, 0 < p < 6 such that u < µ in the ball Ixl - xl < p and u = u at some boundary point X of this ball (IX - x1 I = p). By the part of III already proved some directional derivative of u is positive at z. This is impossible, since
all first derivatives vanish at an interior maximum point. Now let A c
since 9 is connected, A = 9 and u = u. Once I is proved, it follows that the additional hypothesis under which we proved III was superfluous. In order to prove II assume that u achieves a positive maximum at a point x0 in W. Then au < 0 in a neighborhood of this point and in this neighborhood Mu = Lu - au > 0. By I, u = u(x0) in a neighborhood of x9. Thus the set of points at which u = u(xo) is open. Since it is closed by continuity, it must coincide with the whole domain 9.
For the proof of IV, assume that k is a point on I with = max u = u(.x) > 0. Then Mu = Lu - au > 0 near z and, by III, either du/d1V > 0 at X or u - µ at some interior points close to X. In the latter case u is constant by II. 2.3.
Applications to the Dirichlet Problem
We shall now discuss some applications of the maximum principle to boundary value problems. Consider first the Dirichlet problem u = O on (3) Lu =f in Vt, where f, 0 are given bounded functions.
THE MAXIMUM PRINCIPLE THEOREM.
153
If a < 0 then the solution u of the Dirichlet problem (3)
satisfies the inequality
Jul < max 101 + (e" - 1) max If I
(4)
where a = (11m) [K -{- (K2 + 4m)112] and d is the diameter of 9.
The interesting thing about this inequality is that it does not depend upon the size of a, or on the continuity properties of the coefficients, or on the shape of the domain.
An immediate corollary of this estimate is the uniqueness theorem for the Dirichlet problem. If f = 0 = 0, a < 0 and (3) holds, then u = 0. Of course, this also follows directly from II. In order to prove the estimate we may assume, without loss of
-
generality, that 9 lies in the strip 0 < x1 < d. Set g(x) = e' eu'. In the strip considered, we have that e"d - 1 > g > 0 and Lg = (a11a2- a1x)e'zl + ag < -mat ; Kx = -1. Now set
-
h =maxI0I +g(x)maxIfI.Then Lh =maxIf(Lg+amaxI0l -max If I and h > max 101. Hence v = u - h < 0 on 9 and Lv=f - Lh >f -}- maxIfI >0 in W. By II, v :!5: 0 in 1.
Similarly, u ! -h. Hence Jul < max 101 - max If I g(x) which implies the assertion. The proof shows that the theorem can be considerably strengthened.
If a is not nonpositive, we cannot expect in general that the Dirichlet problem will have a unique solution. Nevertheless, the
uniqueness theorem can be proved and an estimate can be obtained if the domain is sufficiently small. THEOREM. Assume that a < k where k is a positive number, and that the diameter d of <9 is so small that e'"1 - 1 < 1 /k. Then every
solution of problem (3) satisfies the inequality (5)
IuI
c maw ICI ; (e«a - 1) max If 1 -k(e""-1)
This implies that under the hypotheses of the theorem the Dirichlet problem has at most one solution.
For the proof of the theorem we set a- = (a + Ial)/2, a- _ (a - Ial)/2 and observe that the function u may be considered as
a solution of the Dirichlet problem for the equation Mu - a-u = ,
,
f =f - a+u. Since a- < 0 and 0 < a" < k we have that
L. BERS AND M. SCHECHTER
154
max If I < max If I + k max Jul and (4) yields max Jul < max 101 +
(e«a
- 1) (max if I + k max Jul)
which implies the assertion. For a < 0 the maximum principle can also be used in order to prove the existence of a solution of the Dirichlet problem under the hypothesis that we know how to solve this problem for "nice" domains, say for small discs (Perron's method of upper and lower functions). But we shall not discuss this here.
2.4.
Application to the Generalized Neumann Problem
As another application we consider the oblique derivative problem. We assume that the boundary of 9 is sufficiently smooth and we are looking for a solution. of the differential equation Lit =f satisfying the boundary condition on I
Here a/aa indicates differentiation in a direction depending on the point considered. We require that this direction should be nowhere tangent to the boundary surface ik. A solution is assumed to be of class Cl in c. THEOREM.
If a < 0 and ul and u2 are two solutions of the oblique
derivative problem formulated above, then the difference u1
- u2 is a
constant.
Set u = u1 - u2. Then Lu = 0 in V, au f as = 0 on 9. Since u is continuous in 9 it achieves its maximum at some point £. Proof.
If u is not a constant, we may assume that max u > 0. By II, z
must be a boundary point. Near this point Mu = -au > 0 so that the normal derivative of u must be positive at f by III. But
all tangential derivatives vanish at X. This implies that the prescribed derivative au/aa = 0 at x, which is impossible.
THE MAXIMUM PRINCIPLE
155
2.5. Solution of the Dirichlet Problem by Finite Differences
As the next application of the maximum principle, we shall discuss the effective numerical computation of a solution of the Dirichlet problem (3) by the method of finite differences. For the sake of simplicity, we restrict ourselves to two dimensions and to a differential equation of the form
Lu = Du + alu, + a2u,, + au =f
(6)
The functions al, a2 and a are assumed to satisfy the conditions
aS0 We approximate the differential operator L by the finite $a11 + 1a21 < K,
difference operator
Lhu=Mhu+au
(7)
= h-2[u(x + h, y) + u(x - h,y) -{- u(x,y + h) + u(x, y - h) - 4u(x,y)] u(x + h,y) u(x h,y)
-
+ a1(xy)
2h
u x, y
u x, y + a2 (xy)
-
+ a (xy) u (x,y)
2h
the mesh width h being a small positive number. Note that Lhu
Lu as h
0 for every C2 function u.
In order to formulate a boundary value problem for this difference equation, we introduce the following notations. The points
(x + h, y), (x, y + h), (x - h, y), (x, y - h) will be called the h-neighbors of PO = (x, y) and will be denoted by P01, P021 P03, P. Py situated in 9, A lattice domain'h is a set of points Pi, P2,
....
having coordinates which are integral multiples of h, and such that for every point in the set its h-neighbors belong to K We also require that given any two points Qo and Q00 in 'h there should exist points Qo = Q1, Q2, ... , Q, = Q00 in Wh with Qj
an h-neighbor of Q,_1. Neighbors of points of Fh which are not
themselves points of Th form the boundaryh of !Yh; these points will be denoted by PV+1, PV+21 ... , P' ,-sr. The union of
L. BERS AND M. SCHECHTER
156
yh
and 'h is the "closure" 1h of !Ye,,. We assume that h is so
small that a lattice domain 65h exists and that
hK < 0 < 1
(8)
For the difference equation we now pose the following Dirichlet problem (9)
Lhu
=fatP1,P2,...,P.V;
u =0 atP.,+1,.
.
, PV+.11
The solution u is of course to be defined only at the N -i- M points P,.... , Pv h,,, or as we say, on the lattice 7h.
We claim that the following maximum principle holds for functions u defined on the lattice domain h. If Mu > 0 in 1,
then either u is a constant or it achieves its maximum on the boundary The proof is trivial. For a point Pi in F,, the condition 0 reads Mhu
u(Pi) <
4
j
1.,,u(P,,) 1
.i2 = 3
1 (1 41`
417a ) 1
-t
L
- Ilal 2
where we denoted by a1, a2 the values of a1(x,y), a2(x,),) at the point P,. The assertion now follows by observing that the ).,, are positive and their sum is 1. Thus if u(P,) = max u, then u = max u at all h-neighbors of P at all h-neighbors of the h-neighbors, etc. Hence u --- max u. From this maximum principle corresponding to Theorem I
above we conclude, just as before in the case of differential equations, that a function defined on the lattice domain and satisfying the inequality Lhu > 0 in,, cannot have an interior positive maximum unless it is a constant. Furthermore, the same reasoning which led to the estimate (4) leads now to the following estimate for the solution of the Dirichlet problem (9) : (10)
Jul < max 101 ; C max If I
THE MAXIMUM PRINCIPLE
157
where C depends only on K, the domain T, and 0. In fact, we may use the same comparison function as before since we have that Lh(ead - exz) _ -a2 e,,x(sinh (ah/2))2 ah/2
-blot e"x sinh (ah) + a(e"d - e"s) /2 )12l(1
(sin
- ot2 a
ah/2
J
- K ah coth (ah/2) J < - 1
if a > 0 is large enough, provided that 9 lies in the strip 0 <
x
system, and we have just found that the homogeneous system possesses only the trivial solution. Therefore, the Dirichlet problem for the difference equation is always uniquely solvable.
Next we want to show that the solution of the difference problem approximates the solution of the differential problem as the mesh length h
tends to zero. To do this we shall assume, merely for the sake of simplicity, that for a sequence of values of h tending to 0, all boundary points of W. lie on the boundary 1.
We shall assume also that the solution u of the Dirichlet problem (3) for the differential equation Lu =f exists and has bounded derivatives of order three. An easy application of Taylor's theorem shows that for any C3 function u
JLhu - Lul < Ch the constant depending on the maximum modulus of the third derivatives of u. (For a C4 function Lhu
- Lu =
0(h2) .) Now the
estimate (10) shows that every function U defined on the lattice domain 1h satisfies the inequality
maxIUt <maxIUj =CmaxILhUj yA
'A
A
L. BERS AND M. SCHECHTER
158
Applying this to the difference u- u(h), u"' being the solution of (9) we obtain the uniform estimate
u - u(h) = 0(h) which shows that the solution of the difference equation converges to that of the differential equation. It should be noted that the assumption made on the shape of the domain is unessential. As a matter of fact, without assuming anything about the boundedness of the third derivatives of the solution u of (3) on the closure !k, but merely the continuity of
these derivatives in 9, one could say that the solution of the Dirichlet problem for the differential equation can be approximated by properly chosen solutions u(h) of the difference equation
on every compact subset of W. As a matter of fact, this same method works for uniformly elliptic differential equations of the
form (1) and even for semilinear equations of the form Mu =
f
(Bers [2]). It is only necessary to be able to approxi-
mate the differential operator L by a finite difference operator L. for which one can prove the maximum principle. This is always possible, though it might happen that one needs a difference operator involving many more neighboring points than the simplest difference approximation used above (Motzkin and
Wasow [1]). Also, the restriction to two dimensions is completely unessential. Finally, a refinement of the argument would show that under proper conditions the first and second difference quotients of the
solutions of the difference equation converge to the first and second derivatives of the solutions of the differential equation. As a matter of fact, the argument could be reworked so as to give an
existence proof of the Dirichlet problem for the differential equation. 2.6. Solution of the Difference Equation by Iterations
The problem of solving the Dirichlet problem for the difference equation numerically, say on an automatic computing machine has been the subject of intensive investigation for which we refer to the literature. We shall only show how the maximum principle
THE MAXIMUM PRINCIPLE
159
can be used in order to prove the convergence of a simple iteration scheme (Liebmann's method).
It is now slightly more convenient to consider the Dirichlet problem for the homogeneous difference equation :
L,,u=0inIi.,
u=0on4,,
Liebmann's procedure is as follows. We assume some trial values
for the unknown quantities u(P1), . uo(Pi),
. . ,
u(P,;) and call them
... , ua(P.,-). We order the points of the lattice domain in
such a way that P1 has as an h-neighbor a point on the boundary 4r,,, P2 has as one of its h-neighbors Pi, and so on. Now we correct the trial value u,,(P1) so that the difference equation should be satisfied at this point. We then correct, in a similar fashion, the trial value uo(P2), making use of the new value u1(P1). After having corrected the values at all points P1, ... , Pv we obtain the next approximation u1(PI), ... , u1(PN). We now repeat the procedure in order to obtain approximations u2, us, We want to show, following Lowner (cf. Frank-v. Mises [1]), that v = 1, 2, ... , N lim u,(P,) = u(P,),
....
5-. ao
where u is the solution of the boundary value problem (9). To do this we observe that the k-th approximation uk depends linearly
on the (k - 1)-st approximation and on the given boundary values 0; thus there exist matrices (a,,,), (#,.,) such that N
M+N
uk(P,) _14si I a,Puk-1(PH) ,,=,v+1 + I f,,,4(P,,) It is easy to see that every correction consists in replacing the value uk_1(P,) by a linear combination, with positive coefficients, of the values of uk_1 at neighboring points. Hence the elements a,µ > 0. If we had luck enough to choose as our first approximation the exact solution, the corrections would be unnecessary. Hence and therefore
u(P,) = I a"Hu(P,,) + I #,,,O(P,.)
max, I uk(P,) - u(P,) I = Max, 11 a,,,(uk-1(P,.) - u(P,.)) I max
a,,, Iuk-1(P,,) - u(P,,) I <_ 0 max Iuk-1(P,)
- u(P,) I
L. BERS AND M. SCHECHTER
160
where 0 = max,, E a,,,,. Now assume that we choose as our first approximation the function u = 1 and that the given boundary values are ¢ 0. Then each correction consists of diminishing
the trial value; and the first approximation would consist of positive values less than 1. But the value of the first approximation at P. is now Z OCR,,. Thus 0 < 1, and since
max tux - uj < 0 max IUk_1 - uI < ... < 0' max Iuo -- uJ we see that the Liebmann method converges.
The maximum-principle approach to the method of finite differences is limited to second-order equations. An alternative approach (Courant-Friedrichs-Lewy [1] and others) uses L2 estimates and works also for equations of higher order. 2.7.
A Maximum Principle for Gradients
First derivatives of solutions of linear elliptic equations do not, in general, obey the maximum principle. An exception is a second-
order equation in the plane of the form (11)
A(x,y)uzx + 2B(x,y)ux , C(x,y)uvv = 0
(AC - B2 > 0). If u is a solution defined in a domain 1, then ux and u cannot achieve their maxima at interior points, unless they are constant.
We shall prove this in Chapter 7. At the moment we note that the assertion is obvious if A, B, C are of class C1 and u of class C3. Indeed, dividing the equation by C (or by A) and differentiating with respect to x (or toy) we see that u,, (or satisfies a secondorder linear elliptic equation. For an equation of the form (12)
"
Au
i-1
a,
au
ax;
-}- au = 0
with bounded coefficients we can state the following THEOREM.
Let K be a bound for the moduli of the coefficients in (12).
There exist positive numbers Ro and k depending only on K and n such that for every solution u of (12) defined in a convex domain A of diameter
THE MAXIMUM PRINCIPLE
161
R < Ro, of class C2 on the boundary I' of A, and vanishing somewhere in A,
sup, (grad uj < k maxr (grad uI Set ,u = supA Igrad uI, v = maxr (grad uI. Since u = 0 somewhere in Jul < Rµ, and by (12), assuming that R < 1, Proof.
zul < K(Vn -r R),u Set v (x)
= - I Au(y) dy
v(x) =
2
(n > 2)
_2
1
J' Au (y) log
x
dy
(n = 2)
yI
where uw,, is the area of the unit sphere in n-space. Computing the gradient of v by differentiating under the integral sign we obtain easily the estimate sup, Igrad vi < cKR,u where c depends only on n. The function v(x) is of class C1 everywhere, and by Green's third identity, fi(x) = u(x) - v(x) is harmonic in A. Now, the derivatives of 0 are harmonic and obey the maximum principle. Hence
sup, (grad I < maxr (grad 01 < v + cKR4u so that
it < sup, Igrad 01 + cKRµ < v -!- 2cKRy If R < Ro < 1 /2cK, we see that 1u < kv, with k = 1/(l - 2cKRo). The convexity of A was used only in asserting that max., Jul < R max, (grad ul ; it would suffice to assume 0 star-shaped with
respect to the point where u vanishes. For nonconvex ., with sufficiently smooth boundary the theorem remains valid, with k depending on A. No convexity is needed if a = 0. In this case the condition that u vanish somewhere in A is also superfluous: it can be achieved by adding a constant to u. The same method yields a maximum principle for an elliptic system in the plane Ox - tVv = x111 - a12V (13)
= x210
x22tV
L. BERS AND M. SCHECHTER
162
with bounded a;;. This is not surprising since for the special case a21 = a22 = 0, system (13) shows that 4 = u,,, tp = -uv where u is a solution of E u + a11um + a12u = 0 Setting 0 + i v = w we may write (13) in the complex form
(14)
w,, =aw±bw
(15)
where a and b are bounded complex-valued functions. This equation implies, of course, the inequality Iw2I < K IwI
(16)
and for such an inequality we have the following THEOREM.
Let w(z) be a function of class C1 in a domain A of diameter
R < R0, satisfying inequality (16), and assume that w is continuous on the boundary 1' of A. Then (17)
supA IwI < k maxi, IwI
Ro > 0 and k > 0 being constants depending only on K.
The proof consists of applying the maximum modulus theorem to the holomorphic function w(z) - r(z), z e A, where (18)
r(z)
iff
w,( z) dz dry
+ i?l
0
and noticing that sup_A Irl < 2R sup,, I wtl < 2RK sup,, IwI. 2.8.
Carleman's Unique Continuation Theorem
The maximum principle just stated implies a celebrated theorem of Carleman. THEOREM. A function w(z) of class C1 which satisfies inequality (16) in a domain V vanishes identically if it has in 9 a zero of infinite order.
Assume that w(z) = 0 (Izl`), z --) 0, for all N > 0. It suffices to show that w(z) = 0 for Izl < R, R > 0 and small Proof.
THE MAXIMUM PRINCIPLE
163
enough. The functions w(z) n = 1 , 2, ... , vanish at z = 0 and satisfy (16). Applying (17) to the domain 0 < I zI < R, we see that max I z-"w(z) I< kR-" max I w(z) I Izj
for n = 1, 2,
Izl=R
.... This is impossible unless w = 0.
In view of the connection between system (13) and equation (14), Carleman's theorem implies at once that solutions of (14) have the weak unique continuation property. In order to obtain
the strong unique continuation property we must be able to conclude from (14) and the relation u = 0 (Izl `), for all N, that U. and u also vanish of infinite order. This is indeed possible. The strong unique continuation theorem for the equation (19)
Au+au,, +buy±cu=0
is obtained by noting that this equation can be reduced to the form (14) by introducing the new unknown function U = u/uo, uo being some positive solution of (19). Finally the general linear
elliptic second-order equation in the plane is reducible to the form (19) under certain smoothness conditions on the coefliciens. A unique continuation proof not tied to our smoothness assumptions will be sketched later (cf. Chapter 6).
CHAPTER 3
Hilbert Space Approach. Periodic Solutions 3.1.
Periodic Solutions
The Hilbert space approach to boundary value problems is the end result of a development which goes back to Riemann. As is
well known, Riemann asserted the existence of a harmonic function with prescribed boundary values on the basis of the Dirichlet principle, that is, by characterizing harmonic functions as extremals of a variational problem. The criticism by Weicrstrass led to the abandonment of this method of attack, but around the turn of the century Hilbert showed that the Dirichlet principle can be used for rigorous existence proofs. The direct methods of
calculus of variations which developed from this result are described in several books (Weyl [1], Courant-Hilbert [1], Courant [11) and we shall not discuss them here. It was realized later that
the natural framework for the Dirichlet principle approach to boundary value problems is the theory of Hilbert space (Zaremba [ 1,2], Nikodym [1], Friedrichs [2], Sobolcv [2] and others) and that Hilbert space methods are not limited to second-order equations or to equations arising from variational problems (Visik [1],
Garding [1], Browder [1], Friedrichs [2], Ladyzenskaya [1], Nirenberg [1], Lax-Milgram [1], and others). As a matter of fact this approach to boundary value problems uses no deep results from the theory of Hilbert space proper. The main tools (described fully in Appendix I to this chapter) are the projection theorem and the representation theorem. Note that the representation theorem is an easy consequence of the projection theorem and the latter result is proved by minimizing the distance between an clement in the space and a closed linear subspace. In this sense, calculus of variations is still present in the Hilbert space approach. On the other hand, one must also use some deeper results: the Riesz-Schaudcr theory of linear equations in a Banach space (see 164
HILBERT SPACE APPROACH, I
165
Appendix II to this chapter). This is an abstract analogue of the Fredholm theory of integral equations. It is worth noting that the Fredholm theory itself was developed mainly for solving the Dirichlet problem without using calculus of variations. We shall describe the Hilbert space methods first for a somewhat
artificially oversimplified case. Instead of treating a boundary value problem we shall be looking for a periodic solution of an elliptic differential equation. In the periodic case certain technical difficulties disappear (as was observed by Lax [1]) and a partic-
ularly elegant treatment is possible. This treatment also makes essential use of the theory of distributions. More precisely, we consider in this chapter an elliptic differential operator of order m, L = a p(x)D" and assume that the Irl-»:
coefficients of L are periodic in all variables. Without loss of generality we may set all periods equal to 27r. We also assume, primarily for the sake of simplicity, that the coefficients of L are
infinitely differentiable. From now on all functions will be required to have period 27r in each variable, and all integrals will be performed over a period cube: 0 < x, < 2ir, i = 1, 2,
. . .
, n.
The Hilbert Spaces Ht
3.2.
First we consider real-valued trigonometric polynomials, that is, finite sums of the form u = u(x) = I ocle`f -. Here e = (lr, ... , are n-tuples of integers (positive, negative, or zero), I - x = fix, + . . . + and the (complex-valued) coefficients ap are
subject to the condition a_t = al. For t = 0, + 1, ±2.... we define the scalar product of u =
age'f'r and v =
ifte'f- by
the formula (1)
(u,v)t = (2ir)" I (1 + f. t)t at#-f
2(where
I = B; ... + f) . The axioms of a (real) scalar product are obviously satisfied and we note that }
(2)
(3)
(u,v)o = fu(x)v(x) dx
Oct = (u,e-'fs) o
L. BERS AND M. SCHECHTER
166
The product (1) induces the norm II
II
(1 + t
Ilullt = (u,u) t = (27T)"
(4)
t,
l) t l atl2
We note the Schwarz inequality I (u,U)sl S (lulls IlUlls
(5)
and the generalization (sometimes called the Schwartz inequality) (6)
l (u,U)sI C llulls+t IlUlls-t
For u = v this yields (7)
IIuIIB < llulls+t llulls_t
Also, (8)
Ilui18 S IIuII t
for s < t
From the inequality (1 + e- 1)8 < 8(1 + e. 1) '1 + e-(s-tp)1(t,-s) (1 + `. f)t:
we obtain (9)
E-(3-csu(t,-s)Ilullts
IIuIIB < ellullt,
for ti >S > t2,
e>0
For every partial derivative DP, (10)
DP
ate't'x = I (if).-
where, according to our conventions, (ie)P = (i11)Pl(i4)P2
...
(i(,)P.
Hence (11)
IIDPull: < Ilullt+Inl
and (12)
Ilullt < (const. depending on t) I IIDPullo
Now let K denote the elliptic differential operator K = 1 - A.
Then K I ate't'Z = (13)
KtI
(1 + t l) ate't'Z, so that for t = 0, ± 1, ateit-'
= I (1 + !' I)
tatefttz
HILBERT SPACE APPROACH, I
167
Clearly, (Ktu,v) 8 = (u,K`v) d = (u,v) a+t
(14) (15)
IIKtull8 =
Ilulle+2t
Completing the space of trigonometric polynomials with respect
to the norm Il t we obtain the (separable) Hilbert space H. The elements of Ht may be represented as (infinite) trigonometric octe't' series (with a_g = 61) for which the norm (4) is finite. II
Clearly
Ht cH,
(16)
We set H. = n Ht, H
fort>s
. = U Ht.
All relations developed above remain valid for elements of H. In particular: D9, as defined by (10), is a bounded linear transformation of Ht into Ht_,,,, and the scalar product (u,v), may be formed if u c- H,,,, v c- H.,-,. 3.3.
Structure of the Spaces Ht
A formal trigonometric series E age"' is the Fourier series of a periodic square-integrable function if and only if E I atl2 converges (Riesz-Fischer theorem). Hence there is a one-to-one correspondence between such functions and elements of Ho. We state this, somewhat imprecisely, as LEMMA 1.
Ho is the space of periodic L2 functions.
Noting (11) we also have LEMMA 2.
For t > 0, Ht is the space of periodic functions having
generalized L2 derivatives up to the order t.
In particular, a periodic function of class Ct belongs to H. A partial converse of this statement is also true. LEMMA 3.
(Sobolev [1]). If u(x) E H, and t > [n/2] + k + 1,
then u is of class Ck, and (17)
max IDDul < const. Ilullt
for IpI < k
L. BERS AND M. SCHECHTER
168
It will suffice to prove this for k = 0 (cf. (10)). Now if u = E afe'f r c Ht, t > [n/2] + 1, then
Ixfl)I
so
max Jul < E lxfl < (E (1 + 1. LEMMA 4.
1)-t)112
IluII1.
H., is the space of Cc functions.
This follows at once from Lemmas 2 and 3. The spaces H, and H_ t are dual via the product ( , ) o. More precisely: if v c- H t, then cw(u) _ (u,v)o is a bounded linear functional5 on Ht, and Il v II _t = II of ll, that is, LEMMA 5.
IIvII_t = sup (u,v)o,
(18)
Ilullt = 1
Conversely, every bounded linear functional on H, is of this form, v being uniquely determined.
Proof.
If co(u) is given, the representation theorem implies
the existence of a uniquely determined element i E H, such that (,,(u) = (u,b), and 11(011 = IIUIit Set v = Ktu. Then v c H_,, Ilvll_t = lI'llt
and (u,v)o = (u,v)t = cv(u). If v1 e H_, and co(u)
(u,v1)0, then
(u, v - v1)0 = 0 for u E Ht, and for u = K-'(v - v1) we have that Ilv - v1Ib_t = 0 or v = v1. If v e H_, is given, (u,v)o is a bounded linear functional on H, by (16). A periodic distribution T is a linear functional defined on C.,
periodic functions 0, such that T[s6k] - 0 if D"Ok , 0, k , oo, uniformly, for every p. By Lemmas 3, 4, and 5 this means:
T[ck]-- 0ifII¢kll,--*0, k- oo, for all t>0. IfueH_t, t>0,
we may define the linear functional T[¢] = (u,¢)0. Since C. functions are dense in H, we have LEMMA 6.
H_t (t = 1, 2,
. .
.)
is
the space of those periodic oo. 0, k
distributions T for which T [Ok] --- 0 if IlOkllt
We also have LEMMA 7.
(L. Schwartz [1]). H_ x is the space of all periodic
distributions. 5 See Appendix I to this chapter.
HILBERT SPACE APPROACH, I
169
Suppose that T is a distribution not in H_,,: T[¢] 0 (u,O)o for all u e H_t, t = 1, 2, .... By Lemma 6 there exist C., functions 0k, k = 1, 2, ... , with I T[tk] I > k IIOkIIk Set zVk = Ok/k Il0kilk For every fixed s = 1, 2, ... , Ilwkll, - 0, since Proof.
Iiv'kll, < 1/k for k > s. But I ThA] I > 1, which is impossible. For periodic distributions we can define differentiation (which coincides with the operation defined by (10)) and multiplication by a periodic Cx function. They are similar to the corresponding definitions for general distributions given in Section 1.3. If 0 is a C,,, function and u e H, then Ou E Ht and
LEMMA 8.
< (const depending on 0) II u II r Proof. It suffices to verify (19) for u E Ham. If t >_ 0 the
(19)
II Ou II i
inequality follows at once from (11,12). If t = -s < 0 we have by (18) and (6) that I* II-, = sup {(0u,v)0/ Ilv II,} = sup {(u,iv)o/ Ilv II,} Ilull -, II/v1I,/Ilvll, < const. Ilull -,
As a corollary we obtain LEMMA 9. If L is a (not necessarily elliptic) partial differential operator of order m with periodic C,, coefficients and u E Ht, then
Lu e H,-. and IILu li t-m < const. Ilu ll,
(20)
We conclude by a lemma which shows that the embedding of Ht into H87 s < t, is a completely continuous mapping. LEMMA 10.
(Sobolev [3]). For s < I every bounded set in H,
is conditionally compact in H3. Proof. Ilu(')11
Let {0) = {E a` >e'f'z} be a bounded sequence in Ht,
M. We must show that a subsequence is a Cauchy
sequence in Ha. Since I at ) I < M(1 ,- f i) -tl2 we may assume, selecting subsequences if need be, and using the diagonal procedure,
that for every t the sequence {m(P) converges. For every integer U) N > 0 set u(') = u(j)N + VU) N , where uN = E x( Pe'f z, the sum-
mation being extended only over those l for which t° l < N2. Clearly Iluv)
- u(' ) II3 will be arbitrarily small, for every fixed N,
L. BERS AND M. SCHECHTER
170
if j and k are large enough. On the other hand Il vY' - v1; II2 S const. (1 + N2)s-tM2 is arbitrarily small, for all j and k, if N is large enough. Hence II u(')
- ue4l , --i 0 for j, k -+ oo.
3.4.
Basic Inequalities
We consider now our elliptic operator L = E a,(x)D' and establish the basic a priori estimates. THEOREM
1.
(Garding's Inequality). For every periodic C.
function u (21) (u,Lu)0 . cl Iluiln,f2 - C2 Ilullo where ci and c2 are positive constants depending on L.
Inequality (21) holds also for u e Hm/2 Indeed, if u c- Hmi2i then Lu E H mi2, so that the scalar product (u,Lu) 0 makes sense, and (21) may be proved by a limiting process. COROLLARY.
THEOREM 2.
For e v e r y t = 0, ± 1 , ±2,
... ,
and every periodic
C,,, function u
(22)
Ilullt < c3(t) IILu + Aull:-m
for A > A
where c3(t) depends on t and L and A > 0 on L only.
Again we have a COROLLARY.
Inequality (22) holds also for u e H.
Proof of Theorem 1.
We remark first that by hypothesis
(-1)m12 I
c>0
c IEIm,
IpI =m
for every vector . Case (a). L has constant coefficients and only terms of highest degree m. Then L = I aDDD and for u = E a`e'e'z 1P1=m
(u)Lu)o = ( aeetlz
G
(I
Ip1 -m
(-1)m12Q(I) Iafi2 > C c
[1 + (t'
C' IIUI12
where c' = c sup {(1
2rm)(1
av(ie)v)a(eatz)o
t')m/2] (xfl2
C IIullo
+
52)-m/2}.
(l. 1)m12 I0CfI2
-C
Ixll2
HILBERT SPACE APPROACH, I
171
Case (b). L has only terms of highest degree m and is of the form
L = Lo + L1 where Lo has constant coefficients while Ll = b,(x)D9 with max lb,(x) I
n
W =M
(u,Lou) o
(23)
z
sufficiently small. By (a)
c' Il u lI1/2 - c
Il u ll o
Integrating by parts and noting that boundary terms disappear because of periodicity we have that (u)Llu)o
=f(
where
Ip
Imbp(x)u(x)D3'u(x)} dx = Ii + I2
Il = l bDDr'uD".u dx summed over Ipl = m, IP'I = Ip"I = m/2, $b3,qD2UD1U dx
I2 =
summed over Ipl + IQI < m - 1, Ip1 < m/2, I QI < m/2, the
being C. functions. Noting the inequalities in Section 3.2 we conclude that I1I = I 1121
(b,,D'° u, D?u) o l < 71 const.
II U II m f2
= Ej (b,,Dpu, DPu)0I < const. IIu U IIu II(m/2)-1 COIISt. + (e-m/2) Ilullo] IIuI1m12{EIIulIm/2
< e const. IIu II
For small a and (24)
n12
±
(e-m/2)
const. IIu III,
e>0
these inequalities, combined with (23), yield
(u, Lou + Liu) > ci
11'
t
- cl
(ci > 0)
1 1u1 1u
Case (c). L = Lo + Li + L2 where Lo and Ll are as in Case (b)
and L2 =
aD(x)DP. By integration by parts and by the Ipl <m
argument used before (u,L2u)o = 1J a,qD3uD°u dx
IPI + IQI < m,
I pI < m/2,
IQI < m/2
L. BERS AND M. SCHECHTER
172
so that (25)
(u,L.2u)01
I
const. IIu ll,r,/2 Ilu II (rn/2)-1
>0 For small a this together with (24) yields Garding's inequality. General Case. For a sufficiently small n > 0 we construct e const. Ilu
(e-"'/2) const. Ilu 112,
periodic C. functions aq(x), (u2(x), . . properties. (i) On the support of
.
,
(o).\-(x) having the following
each w, the oscillation of IpI = m, of L is less than 71.
each leading coefficient (ii) E (i)1(X)2 - 1. By Case (c) (26)
((u,u, Lo);u) o > pos. const. II oo;u Il ,,,2 - const.
II (,),u II 2
but (u,Lu) 0
(27)
f
= (I (1 )uLu dx =
((o,u, L(o;u)o ± R
R = -1 f(f))u {L(oju - (u,Lu) dx Integration by parts yields
R=J
dx,
IpI + 191 < m,
with some fixed C,, functions
141 < m/2
and as before (e-m/2)
IRI < e const. Ilu IIn,/l +
(28)
I pI < m/2,
const. Ilu Il0
Clearly II(,),u Ilo <- Ilu Ilo
(29)
and it is not difficult to verify, using for instance (11,12), that (30)
Il(,.),u II,,,/., > pos. const.
II U
II71t/2 - const. Ilu 112
Combining relations (26-30) we get Garding's inequality. The use of Fourier series (or Fourier transforms) seems indispensable for the proof of Theorem 1. But in the classical case m = 2 integration by parts suffices. Indeed, assume that
L-L1 L1
= -5 alk(x)
a2
a.r,ax; '
,
L2 a
L2 = Y a,(x) ax, - a(x)
HILBERT SPACE APPROACH, I
with E aik(x)
173
c E 6rj, c > 0. Then
au au
= SJ a,k ax. axk dx +
(u)Liu)o
a
2
>cI
If ax
au ku
t
dx
x
-con st. Ilullo Ilulll
11TIUxillo
pos. const.
1 1 U1 1i
- const.
11U 11
o
and since I (u,L2u) I is easily seen not to exceed e const. IIu II
(1/e) const. IIu Ilo for every e > 0, Garding's inequality follows.
-
At the moment we shall prove Theorem 2 only in a weakened form. More precisely, we shall show that (22) holds for A > A0(t), A0(t) depending on t and L.
Let s be a fixed non-negative integer. The differential operators K"L and LK" of order m + 2s are elliptic; hence, by Theorem 1, there are positive constants c;, c., depending on s such that Proof.
(u,K"Lu)o
:2! c'l Ilu ll8+,,,/2
(u)LKsu)o ? c1
IIu
112
- C2 Ilullo - c2' IIu Ilo
Using the first inequality together with (15, 6, 14, 4, 3) we have that flu II 8 I m/2
II Lu + Au II 8-m/2 =
IIuII "+m/2
II K3Lu + AK"u ll -s-m/2
> (u, K"Lu + ).K"u)o >- ci IIUII8+m/2 - c2 Ilullo + A(u,Ksu)o
=
ci flu Iii+m/2
- cz IIu Iln + )'11U112 ? ci
IIu IIS+m/2
+ (2 - c2) Ilullo ? ci Il u ll9 i m/2 for A > c2 and dividing by 11U 113. m/2 we obtain (22) for t = s + m/2. Similarly, IIuII-s-m/2
IILu + Aull_s-m/2 = IIK-'ulls-,,,/2 IILu + Aull-s-m/2
> (K-"u, Lu + Au) o = (K-"u, LK"K-"u + Au) IIK_"u112
ci
= ci
11K-"ull8+m/2 - c2
IIUI12
8+m/2 - c2
IIUII
+ A IIuI12-8
-2" + . IIuII? ? ci Ilullo 8+m/2
for A > c? and dividing by IIu II -s 1 m/2 we obtain (22) for t = -s + m/2.
A complete proof of Theorem 2 will be given later.
L. BERS AND M. SCHECHTER
174
3.5.
Differentiability Theorem
Theorem 2 (in its weakened form) implies LEMMA 11.
For every t and for A > 0 sufficiently large the operator
L + A is a bounded linear one-to-one mapping of H, onto the whole of H,-,n and the inverse mapping (L + 2) -1 of H,_m onto Ht is also bounded, independently of A.
Proof.
By Lemma 9 the linear mapping L + A: Ht -* H,-,. is
bounded. If 2 is so large that (22) holds (at the moment this choice may depend on t), then Lu + Au = 0 implies u = 0, so that the mapping is one-to-one. The inverse mapping is defined on
the range R, = (L + A) (H,) and is bounded by virtue of (22), the bound being independent of A for A > A0(t). Next, R, is closed. Indeed, assume that vi E R, and II v, - v 0. Then v! = Lu, + Au,, u, e H. By (22) we have that II u, - uk II t S c3(t) II Vi - Vk II t-m , 0 for j, k ---+ oo. Hence there is a u-E Ht such
that Ilu, - u II, -> 0. By Lemma 9, IILu1 + Au, - (Lu + Au) II t-m
0. Hence Lu + Au = v and v e R. In order to prove that R, = H,_,n we use the adjoint operator L* (cf. Chapter 1). We note that (31)
(u,Lv)o
- (L*u,v)o = 0
for periodic C,, functions (proof by integration by parts) and hence
(by an obvious limiting argument) also for u c- Hs, v e Ht with
s+t>m.
Assume that A is so large that (22) holds also for L* and that R, 0 H,_r. Then there is a w e H,_,n such that w 0 0, (w, Lit + Au),-,. = 0 for all u in Ht (projection theorem, cf. Appendix I to this chapter). Hence 0 = (Kt-mw, Lu + Au)o = (L*Kt-mw + AKt-mw, u)o for all u in H,, and in particular for all C,,, functions u. This implies
(for instance by (31)) that L*Kt-mw + AKt-mw = 0. Now, Kt-mw e Hm_, and if A is large enough, (22) applied to L* shows
that K'-mw = 0. Then w = 0, contrary to our assumption. THEOREM 3. If u is a periodic distribution and Lu a H u EHs+m
then
HILBERT SPACE APPROACH, I
175
This is a differentiability theorem. It implies that every (periodic) solution of Lu =f is a function if f E H_,,,, has L2 derivatives up to the order k if f e Hk_,,,, is of class C, if f e Hs with s >_ [n/2] + r + 1 - m (Lemma 3), and is of class C,,,, iff is. In the next chapter we shall extend this result to nonperiodic solutions.
Set Lu ==f. By Lemma 7 the distribution u belongs to Hk for some k. Hence f + Au E Hm1n (k,8) and if A is large enough u = (L + A)-' (f + Au) belongs to Hmin (k+m,s+m) Proof of Theorem 3.
Repeating the argument we see that u c- Hmin (k+im,s+m) for j = 1, 2,.... Thus u c- H,,+).,,. 3.6.
Solution of the Equation Lu = f
We shall show now that the equations (32)
Lu =f
(33)
L*v =f
form a Fredholm pair.° Here f is a given periodic distribution;
we already know that for f EH, every (periodic) solution belongs to He. In particular, all solutions of the homogeneous equations
Lu = 0 L*v = 0
(34)
(35)
are Cr,, functions.
Let A0 > 0 be so large that the bounded linear mappings M = (L + AO) -1: Ho H,,, and M* = (L* + An)-1: Ho --* H. exist (Lemma 11). Since H,,, c Ho we may, and shall, consider
M and M* as mappings of Ho into itself. By Lemma 10 these mappings are completely continuous. The homogeneous equations (34, 35) may be written in the form
u-k0Mu=0, 6 See Section 1.5.
u-A0M*u=0
L. BERS AND M. SCHECHTER
176
Furthermore, the operators M and M* are conjugate in Ho. Indeed, by (31), we have that (Mu,v)o = (Mu, L*M*v + AOM*v)o
= (LMu + ).0Mu, M*v)o = (u,M*v)o
Hence the Fredholm-Riesz-Schauder theory is applicable (cf. Appendix II to this chapter). We obtain THEOREM 4.
The homogeneous equations (34, 35) have the same
finite number of linearly independent solutions. THEOREM 5.
The equation Lu + Au = 0 has nontrivial solutions
only for a denumerable set of values ) with no finite accumulation point.
Indeed, the equation considered may be written in the form
u-A'Mu=0,with 2'=io-2. THEOREM 6.
Equation (32) is solvable if and only if (f,v) o = 0
for every solution v of (35).
Assume first that f e H, for some s -,,>- 0. Then equation (32) is equivalent to the following equation in Ho: Proof.
u - AOMu = Mf By the general theory, it is solvable if and only if (Mf,v)o = 0 whenever v - AoM*v = 0. But this is the same as saying that (f,v)o = A0(f,M*v)o = 2o(Mf,v)o = 0 whenever v - ,10M*v = 0. Since v - A0M*v = 0 is equivalent to L*v = 0, the result follows in this case.
Assume now that f e H , for s > 0. Let t be an integer such
that m + 21 > s. Since LK' and K'L* are adjoint elliptic operators of order m + 2t, there is a Al such that LK' + ,11 and K'L + Al are one-to-one linear mappings of Hm±l,_, onto H ,. It is easily Set M' _ (LK' + 21) -1, M'* = (K'L* + shown, as in the discussion preceding Theorem 4, that M` and M'* are conjugate in Ho. We now note that (32) is solvable if and only if (36)
LK'w = f
has a solution. Now (36) is equivalent to (37)
w - a1M'w = M !f
HILBERT SPACE APPROACH, I
177
and My E Hm+2t-s c Ho. Hence we may again apply the abstract theory to conclude that (37) has a solution if (M f v)o = 0 for all v E Ho such that v - AIM(*v = 0. This is the same as saying that (32) has a solution if (f,v)o = A1(f,Mt*v)o = A1(M f,v) = 0 for all solutions v of (35). Thus the theorem is proved in this case as well.
Combining Theorems 4 and 6 (or reasoning directly from the general theory) we get Equation (32) is solvable for every f c- H ,, if and only if (34) has only the trivial solution u = 0. THEOREM 7.
We can now complete the proof of Theorem 2. We first note that from Garding's inequality (21) it follows that for some A > 0, Lu + Au = 0 implies u = 0 whenever A > A [e.g., take A = c2 in (21) ].' Now assume that Theorem 2 is false. Then there exists, for some fixed t, a sequence {u5} in Ht and a sequence of numbers A; > A such that II u; IIt = 1, IILu, + A,uf llt_m , 0. Let Ao be so
large that the bounded mapping (L + A)-1: Ht_.,,, , Ht exists for 2 >_ Ao (Lemma 11). Clearly A, < A. and we may assume, A >_ A. By Lemma 10 selecting a subsequence if need be, that A, we may also assume that {u,} is a Cauchy sequence in Ht_,,,. Now
ui = (L + Ao)-1(Lu, -- A,u,) + (A0 - 2,)(L + Ao)-lu1 so that 0 for j, k -- oo. Hence there is a u in Ht with II us - Uk Il t 0 and we have : Il u ll = 1, Lu + Au = 0 which is 11u1 - u II t impossible.
The same method shows that whenever A is not an eigenvalue
(i.e., whenever Lu + Au = 0 implies that u = 0) (L + A) -': Ht is a bounded operator, the bound being uniform Ht_m on every closed set of A's not containing eigenvalues. Appendix I. The Projection Theorem
We shall consider some simple theorems in a real Hilbert space H. (H consists of elements u,v, ... , for which the operations of addition and multiplication by real numbers are defined and these 7 Recall that Lu + ..u = 0 implies, by Theorem 3, that u is a C,, function. Hence the GArding inequality can be applied.
L. BERS AND M. SCHECHTER
178
operations obey the usual rules of a vector space. Moreover, for every pair of elements u,v there is defined a real number (u,v) called the scalar product in such a way that (u,v) = (v,u), (u + v, w) =
(u,w) + (v,w), (Au,v) = A(u,v), (u,u) > 0 for u ; 0 for all u, v, w c- H and real number A. In such a case we can define a norm 11u II = 1/(u,u), and it is easily shown that (u)v)I S IIuII IIuII 11u + v11 <- IIuII + IIuII (1)
IIu+vII2+ Ilu -v112=2(IIulI2+ IIvII2)
for all u, v c- H. These statements are known as the Schwarz inequality, triangle inequality, and parallelogram law, respec-
c H which tively. Lastly, H is complete: for any sequence satisfies ll u - um 11 -> 0 as m, n -- co, there is a u E H such that 11 u,, -u11 -*0asn- oo.) A subset S of H is called a subspace if Au + ,uv is in S for all real numbers A, a whenever u and v are in S. It is closed if ueHand Ifu, -u11 ->0asn- imply that u E S. First we prove
c S,
Let M be a closed linear subspace of H. Then for every u c- H not in M there is a v e M such that LEMMA 1.
IIu - v 11 = g.1.b. 11u - w II WEM
Let d = g.l.b. flu - w 1j, w c- M. Then there is a oo. From (1) sequence c M such that IIu - w, II , d as n Proof.
we see that 4 Ilu - (wm + wn) III + IIWm - wnII2
=
2( IIu
- w. 112 + Ilu -wn 112) -' 4d2
as m,n -> oo. Since-(w,. + w,,) c- W, 4 IIu
-
(wm + wn) II2 > 4d2
- wn II -' 0 as m, n -> or,. Since H is complete, there is a v c- M such that 11 wf, - v 11 - 0. This means that and hence
11 wm
IIu - v 11 = lim flu - wn 11 = d.
HILBERT SPACE APPROACH, I
179
(Projection Theorem). Let M be a closed linear subspace of H. Then for every u e H, u = v + w, where v e M and (w,M) = 0 [i.e., (w,h) = O for all h e M]. THEOREM 1
Proof.
If u c- M, set v = u and w = 0. If u is not in Al, then
by Lemma 1 there is a v e Msuch that Ilu - v II = d, the "distance" from u to M. Now if f is any element of M
IIu-vll2-22(u-v,f)+22Ilfii2
IIu - L'IIZC llu - U - ?/ II2= for all real 2, in particular, for
2=(u-v,f) Ilf112
Thus
Ilu - vl12 < Ilu - vll2 -
2(u
IIf112
f)2 +
(u
f
II,f)2
and hence (u - v, f )2 < 0, which means that (u - v, f) = 0. Since f was any element of M, w = u - v meets the requirements of the conclusion of the theorem. COROLLARY 1.
If M 0 H, there is a nonzero element w of H such
that (w,M) = 0. Proof.
Take u not in M. Then by Theorem 1, u = v + w,
where v e M and (w,M) = 0. Clearly w
0.
A bounded linear functional Fu on H is a real valued function on H
which satisfies the following conditions: F(ui + u2) = Ful + Fu2; F(2ul) = 2Fui, 2 a real number; IF(u) I < K IIu II for all u e H and some K > 0. The smallest K for which the last statement holds is denoted by IIFII
THEOREM 2 (Representation Theorem; Frechet, Riesz). For every bounded linear functional Fu on H there is a uniquely determined element f e H such that Fu = (u, f) for all u e H and IIFII = IIf1I Proof.
Let N be the set of all v e H such that Fv = 0. N is a
linear subspace of H. For if vl and v2 are in N and Al and 22 are any real numbers, F(2ivl + 22v2) = ).1Fvi + 22Fy2 = 0. Moreover, Nis closed in H. For if {v,,} is a sequence in N and II v,, - v II -0, theniFvl and V EN.
180
L. BERS AND M. SCHECHTER
Now if N = H, the theorem is easily proved by setting f = 0. Otherwise there is a w 0: 0 in H such that (w,N) = 0 (Corollary 1). Therefore Fw - 0 and for any u E H
FIu
-Fwwl =Fu -FwFw =0
and hence u - Fw w is in N. This means that
Iu -F-tww,wJ
=
0
i.e., that (u, ') =Iw 11w112
Therefore
Fu=
(u
w Fw)
' IIwII2
1
and (w/III' 112)Fw is the element f in the conclusion of the theorem. If f' also represents F, then Il f _f'112 = (f - f ', f -fl) _
(f -f',f) - (f - f',.f ') = F(f -f') - F(f -f') = 0. Next, Ilf III = (,f f) = F(f) <- IIFII
Il,f 11,
so that
11f 11
<_ IIFII.
For every e > 0 therc is a g E H with IIg Il = 1 andF(g) > IIFII Hence IIFII - E < (g,f) < Il f ll. Thus IIFII = 11f 11.
- E.
The following theorems will be found useful later. THEOREM 3 (Lax-Milgram [1]). Let [u,v]
real valued function defined for pairs of elements in H which is linear in both u and v. Suppose it satisfies (2)
I[u,v]I < lull
(3)
Ilu 112
be a
IIvII
< K[u,u]
for all u, v in H. Then for every bounded linear functional Fu on H there is an f c H such that
Fu = [u,f] For fixed v e H, we have, by (2), 1 [u,v] I < const. II u II Hence [u,v] is a bounded linear functional in H. Thus (by Theorem 2) there is an element Sv in H such that Proof.
(4)
[u,v] = (u,Sv)
HILBERT SPACE APPROACH, I
181
Obviously, Sv is a linear mapping of H into itself. We observe that by (2) IISuII 2 = (Su,Su) = [Su,u] S IISuII
.
Hull
so that 11sull
<- Hull
Furthermore, by (3), Ilu 112
<_ K[u,u] = K(u,Su) s K Ilu ll
IISuII
giving 1lull
This shows that S maps H in a one-to-one way onto a closed subspace of itself. Moreover, this subspace is H itself. Indeed, if this were not so, there would be an element w in H which would
be orthogonal to every element of the form Sv (Corollary 1). This would imply that (w,Sw) = 0 so that [w,w] = 0 by (4). But this implies that w = 0 (by (3)). Now let Fu be any bounded linear functional on H. By Theorem 2 there is an f c- H such that
Fu = (u,f) By what we have just proved, f = Sw for some w e H. Hence
Fu = (u,Sw) = [u,w] for all u e H and the proof is complete. THEOREM 4 (Banach-Saks). Let be a sequence of elements in H such that II v II < K. Then one can find a subsequence lvn,} and an element v such that
vn1 + .
.
. + Unr
- v
31 0
v
asv -oo. Proof.
We shall show that there is a subsequence {vn,} such
that (vn, - vnk, g) -- 0 as j, k - o for each g e H. It suffices to do this for g in the closure S of the linear subspace S spanned by the vn. For every g E H can be written in the form g = g1 + g2
L. BERS AND M. SCHECHTER
182
where g1 E 9 and (g2,S) = 0 (projection theorem). Thus (vn,,g) = (vn,,gl) + (v8,,g2) = (vn,,g1) and (vnr,g1) converges.
Now for fixed m, the real numbers (Un,vm) are bounded, and
hence there is a subsequence which converges. By the usual diagonalization process, we can find a subsequence {vn, } such that (Vn,) converges for each fixed m. Thus (v,1, f) converges if f is
in S. That it also converges for f in 9 follows from the inequality (vn, -- vfk,f) I
<-
I (Un),f - f1) I + I (vn, - Unk7fl)
+ I(vnk,fl -f)I < 2K Of -fill + I (vn, - UnL,fl)I
where f1 e S. For any E > 0 we can take f1 so close to f that 2K II f - f1 II < e/2 and then take j and k so large that (vn, - Unk, fl) I < E/2.
Thus (vn, - Unk, g)
0
for each g E H.
If we now set Fg = lim (g,vn), we see that F is a bounded linear functional on H. Thus there is an element v e H such that (vn! - v, g) -> 0 for each g e H (representation theorem). Now set u f = v.n, - v. Then II u, II S K' for some constant K' and (u,,g) -+ 0 for each g E H. It is easily seen that we can now pick a subsequence {u, } of {u,} such that
I c t, ... , I (uj,, u,,+1) I c t
(uJ,'
Then
u;1+...+ur1 t
2
< [tK'2 +
3.1 and the proof is complete.
2(1
1,-1
+2.2 1
e-
K12
+2 - U
HILBERT SPACE APPROACH, 1
Appendix II.
183
The Fredholm-Riesz-Schauder Theory
We shall now prove some well-known theorems for completely continuous operators. It will be sufficient for our applications to
restrict ourselves to a real Hilbert space. However, all of the
theorems hold in any Banach space (with the appropriate definition of the adjoint of an operator).
As before we let H denote a real Hilbert space with scalar product ( , ) and norm II . We shall make use of the following elementary properties which are easily proved. II
LEMMA 1.
Every finite dimensional subspace of H is closed.
LEMMA 2.
A subspace Hl of H is finite dimensional if and only if
every bounded sequence in Hi contains a convergent subsequence.
We now consider a linear operator T which maps H into itself. Thus
T (au + ,8v) = xTu + pTv
for u, u e H and scalars a, j9. We assume that T is completely continuous. This means that whenever {u,} is a bounded sequence in H, the sequence {Tu,} has a convergent subsequence. We set L = I T, where I is the identity operator, and let N be the set of all u E H which satisfy Lu = 0.
-
LEMMA 3.
N is a finite dimensional subspace of H.
That N is a subspace of H follows from the linearity of L. To show that it is finite dimensional, let {u,} is a bounded sequence in N. Then Tu, = u,. Since T is completely continuous, {Tu;} has a convergent subsequence, and therefore the same is Proof.
true for {u,}. Hence N is finite dimensional by Lemma 2. Let M denote the set of those u e H which satisfy (u,N) = 0 [i.e., (u,v) = 0 for all v e N]. Clearly, Af is a closed subspace of H. LEMMA 4.
for all u e H.
There is a constant co > 0 such that lITull <_ co Ilull
L. BERS AND M. SCHECHTER
184
Proof. If this were not so, there would be a sequence {ui} in H such that 11ui II = 1 and 11Tu, II -1- oo. This is impossible since {Tui}
has a convergent subsequence. LEMMA 5.
There is a constant cl > 0 such that
cl' Ilull <_ IILull <- cl (lull
for all uEM. Proof. The second inequality follows immediately from Lemma 4. If the first were not true, there would be a sequence {u, j in M such that 11u, II = 1 and Il Lug ll -> 0. Since {Tui } has a
convergent subsequence, so does us = Tu, + Lui. Denoting this subsequence again by {u;}, we see that there is a u e M such that
u; , U. By Lemma 4, Tu, -> Tu and hence u = Tu. But this means that u e N. Since u is also in M, u = 0. But lull =
= 1, giving a contradiction. For any fixed v e H, Lemma 4 shows that
lira llu1II
11v 1111 Tu 11
<_ const.
11u II
I (v,Tu) I
for all u c H. Hence (v,Tu) is a bounded
linear functional on H. By the Frechet-Riesz representation theorem (Theorem 2 of the preceding appendix) there is a g e H such that (v,Tu) = (g,u). We set g - T*v. Clearly T* is a linear transformation of H into itself and satisfies (v,Tu) = (T*v,u)
(1)
Thus T** = T. LEMMA 6.
Proof.
T* is completely continuous.
Let {ui} be a sequence satisfying Ilui 11 < M. Then IIT*ui 112 = (T*ui,T*ui) = (ui,TT*u,) Ilui II IITT*ui II <_ coM IIT*ui II
giving IIT*ui 11 < c0M. Hence {TT*u,} has a convergent sub-
sequence. Denoting this subsequence again by {TT*u,}, we have
IIT*(ui - ut)112 = (T*(ui - u,), T*(ui - u,)) = (us - u,, TT*(u, - u,)) < 2M IITT*(ui - u5) II 0 as i, j -> oo. Hence T* is completely continuous. THEOREM 1. The equation v - T*v =f has a solution if and only iffEM.
HILBERT SPACE APPROACH, I Proof.
If v
185
- T*v =f, then (f,w) = (v - T *v, w) = (v,Lw) = 0
if w c- N. Conversely, suppose that f e M. By Lemma 5, 11 Lull and
are equivalent norms in M. For u e M, I (u, f) I c const. and hence (u, f) is a bounded linear functional in M. Since M is a closed subspace of H, it is itself a Hilbert space. Thus by the Frechet-Riesz representation theorem, there is a w e M such that 11 u 11
11 u 11
(Lu,Lw) = (u, f )
(2)
for all u e M. Now we claim that (2) holds also for all u e H. This follows from the fact that N is finite dimensional and hence closed. Therefore by the projection theorem (Theorem 1 of the
preceding appendix) every u E H can be written in the form u = u' + u", where u' E M and u" E N. Thus (Lu,Lw) = (Lu',Lw) = (u', f) = (u, f) since (u", f) = 0. Set v = Lw. Then (Lu,v') = (u, f )
for all u E H. This means that (u, v
- T*v) =
(u, f )
for all u e H. Thus v - T*v =f and the proof is complete. Set L* = I - T* and let N* and M* be the spaces corresponding to N and Al, respectively. We then have COROLLARY 1.
The equation Lu ==f has a solution if and only if
f E M*. Proof.
This follows from the fact that T* is completely
continuous and that T** = T. Define the operators L", n = 1, 2,
by the recurrence relations Li = L, L"'1 = LL" and let N" be the set of u e H ,
. . .
satisfying L'u = 0. LEMMA 7.
L' is of the form L" = I - T where T" is a completely
continuous operator.
Proof. We have 1 0(
') (-
= I - (nT - (n)T2
-f-
irr
... ± T")
L. BERS AND M. SCHECHTER
186
and the operator in parentheses is easily seen to be completely continuous. COROLLARY 2.
Each N is a finite dimensional subspace of H.
This follows immediately from Lemmas 3 and 7. N,, C Nn,1.
LEMMA 8.
Proof.
Obvious.
There is a positive integer k such that N,, 0 Nn,, for
LEMMA 9.
n k. Proof.
We first show that N,, = Ni+1 implies N,,12= N,,. Then 0 = L"+2u = L"+1Lu. Hence
Let u be any element in Lu E
N,,.
This means that L"Lu = 0 showing that
u c- N"+1 = N". Now suppose that the lemma is false. Then N. for all n. By the projection theorem (Theorem 1 of the preceding appendix) there is a sequence {un} in H such that un E Nn+1,
Ilu,, II = 1
and (u,,,N,,) = 0. We shall arrive at a
contradiction by showing that {Tu"} contains no convergent subsequence. To see this we note that T(u" - urn) = u,, - (Lu,, + Turn) Now for n > m, L"(Lu,, + Turn) = L"+1u,, + TL"urn = 0 Hence Lu" + Turn E N,, and IIT(u"
- urn)112 = Ilun 112 + IILu" + Turn II2 z 1
Hence {Tu"} has no convergent subsequence and the lemma is proved.
Let R (resp. R*) denote the range of T (resp. T*), i.e., the set of all v for which there is a u c- H satisfying Lu = v (resp. L *u = v). LEMMA 10.
Proof.
If R = H, then N = 0.
Assume that R = H and that there is a uo : 0 such
...
be solutions of the equations Lu1 = uo, that Luo = 0. Let u1i u2, (which we can solve since R = H). Then L"un = Lug = u1i N,, for all n, uo 0 while Ln+1u,, = Luo = 0. Thus Nn+1
...
contradicting Lemma 9.
HILBERT SPACE APPROACH, I THEOREM 2.
Proof.
187
R = H if and only if N = 0.
By Lemma 10, it is sufficient to prove that N = 0
implies R = H. If N = 0, M = H and hence R* = H by Theorem 1. Since L* is completely continuous (Lemma 6), we see from this that N* = 0 (Lemma 10). This means that M* = H and R = H by Corollary 1. THEOREM 3. N and N* are of the same finite dimension.
That they are finite dimensional follows from Lemmas 3 and 6. Assume that N is of dimension n and N* is of dimension v. We shall show that v > n leads to a contradiction. By the GramSchmidt orthogonalization process there are elements u1, u2, ... , u in N and v1i v2, ... , v, in N* such that Proof.
(vi,vi) = big (u:)u,) = 6-J, where 8i, is the Kronecker delta. Now define the operators n
Wu =u -Su
Su = Tu - I z=1
It is easily checked that S is completely continuous.8 Now if Wu = 0, then it
0 = (v,,Wu) = (v,,Lu) + 1 (u=,u) i=1
for each j = 1, 2, ... ,
= (L*v,,u) + (u,,u) = (u,,u) n. But from the definition of W, this
implies that Lu = 0, i.e., u e N. But if u e N and satisfies (u,,u) = 0
for all j, then u = 0. Hence Wu = 0 has only the solution u = 0. Applying Theorem 2, we see that we can solve the equation Wu0 = Vn+1. But n``
(Vn+1)Wu0) = (vn+1,Lu0) +
i=1
(ui,u0) (Vn+1,vi) = 0
contradicting the fact that II vn+1 II = 1. Hence v > n is not possible. Considering the operator V
S*v = T*v + I (v,,v)ui s=1
8 In fact, S - T maps any bounded sequence into a bounded sequence in N*, a finite dimensional subspace. We then apply Lemma 2.
L. BERS AND M. SCHECHTER
188
and applying the same reason as above, we see that n > v cannot be, and the theorem is proved. THEOREM 4.
The equation
u - ATu = 0 has nontrivial solutions only for a denumerable set of real A having no finite accumulation point. Proof.
sequence
We shall prove that there does not exist a bounded of distinct values such that each equation
u-ATu=0
(3)
has a nonvanishing solution. For assume there were such a sequence. Let u 0 be any solution of (3). Then for each n, u1, ... , u, are linearly independent. This follows by induction. If the first k - 1 are linearly independent and k
Ic,u, = 0 J
1
then
kA.c
0='k c,Tu;=I A-
u,
j=1 F,
1
Subtracting, we have 0 j=t
showing that all the c; = 0 for j 0 k. Since uk 0, C. = 0 also and the u1i . . . , u, are linearly independent. Let En be the subspace spanned by ul,
. . .
, u,,. Then by the projection theorem
there are elements vn E En such that 11vn II = 1 and (v,,,Ei_1) = 0.
Moreover, if w E E,,, then w - ).,,Tw is in En-1, for
and »
nxu
w-).,,Tw=Ix,u;-AnI j=1
J=1 A;
n-1 j--
1
-.2 x;u5
HILBERT SPACE APPROACH, I
Now if the sequence {A,,) is bounded, then {T a convergent subsequence. But for n > m
189
must have
Amvm) = v - (v - A,,Tv,, + AmTvm) and
Amvm)II2 = IIvn112 + IIv - A,,Tv + AmTv,,, II2 > 1
since both v - A Tv and Tum = L- are in Ei_1. This shows that m
has no convergent subsequence. Hence there does not exist a bounded sequence of distinct 1.,, and the theorem is proved.
CHAPTER 4
Hilbert Space Approach. Dirichlet Problem 4.1.
Introduction
We saw in the preceding chapter that the study of periodic solutions of elliptic equations lends itself to a particularly elegant treatment. Some of this machinery can be converted to give weak (or even "semistrong") solutions of boundary value problems and to establish interior regularity. But when it comes to proving that such solutions are smooth up to the boundary, new methods must
be devised. Several authors have studied this problem in great detail (Nirenberg [1], Browder [2], Aronszajn-Smith (cf. Lions [2] ), Schechter [5], Agmon [2], Peetre [1], and others). In this chapter we return to nonperiodic operators
L = I a, (x) D9 IPI <m
defined in a bounded domain V. We assume that L is elliptic and (for the sake of simplicity) that the coefficients a,(x) are in C., in the closure ci of W. We shall study the interior differentiability of weak solutions, formulate the Dirichlet problem within
the framework of Hilbert space, prove the existence of weak solutions and finally take up the problem of regularity at the boundary. 4.2. Interior Regularity
One property which follows easily from the results for periodic equations is interior regularity. THEOREM 1.
Let T be a distribution solution of L T = f in W. If f
is locally square integrable, then T is a function having L2 derivatives of order m. If f has L2 derivatives up to order t, then T has L2 derivatives up to order m + t. 1f f is infinitely dierentiable, so is T. 190
HILBERT SPACE APPROACH, II
191
First assume that T = u, a locally square integrable function. Then u is a weak solution of Lu =f. Since differentiability is a local property, it is sufficient to prove that in each subdomain 91 such that 91 c 9, u has L2 derivatives up to Proof.
order m + t, whenever f has L2 derivatives up to order t, If 91 is such a domain, let cc2 and 93 be such that cc1 92 C 93, cc3 C9. Let be any test function in g3 with in g2. Since u is a weak solution of Lu = f, we have f9, fo dx
t >_ 0.
q2,
=1
uLi O] dx
dx
for all test functions 0 in 9, where Li is a partial differential operator of order < m with C,,, coefficients having support in cJ3. Since all the integrands have support in c3, (1)
fiifw dx
=12
uLi w] dx
for all C., functions w which are periodic in some cube 2 containing T. We can alter u, f and the coefficients of L outside T. so that they become periodic in 2 in such a way that u and f remain in L2 while L remains elliptic having C., coefficients. and the coefficients of Li may be extended without Next, alteration. Denote the periodic extensions by u, f, L, etc. We have A
by (1)
A
A
A
f'=Lv+L,u where v = tic and L, is the formal adjoint of Li . Now by the theory of the preceding chapter f - L1u is in H,_,,,. Hence, by Theorem 3 of Chapter 3, v e H1. This means that u has L2 derivatives of the first order in cc2 '*,'Interpolating a domain between Y1 and cc2 and repeating the process in we see that , f - L,u is in H2_,,, and hence u has L2 derivatives of the second
order in a domain containing .11. This may be continued until L2 derivatives of order m + t are reached.
Assume next that T is a distribution. We shall show later (Corollary 2) that there is a positive integer k and a square integrable function u such that (2)
T[4]
=5
uAk4
dx
L. BERS AND M. SCHECHTER
192
for all test functions 0 in 4j. Assuming this for the present, we have
Jfc6 dr = LT[t¢] = T[L*¢]-J uYL*0 dt
for all such j. Since the operator LA's _ (AkL*) * is elliptic and of order m - 2k, we have by that part of the theorem just proved that u has L2 derivatives up to order m + 2k ; t. Hence by (2)
T[0] =
J
z
uq dx
for all test functions 0. Hence T = AAu and has L2 derivatives of order up to m ± t. The last statement of the theorem follows from Sobolev's lemma (Lemma 3 below). 4.3.
The Spaces H' and H'
We shall now describe some Hilbert spaces which will take the
places of the spaces H, of Chapter 3. For integers t > 0 and for C. , functions in ( we shall employ the scalar products (U) V) t = I J
dx
API
and the corresponding norms 11ul1t
We also set
=
(u,L)
=
(u,v)0
Completing with respect to these norms we obtain Hilbert spaces H° D Hl H2 As in the case of the H, spaces, functions in H' have L. derivatives in P up to order t. For each t, let Ho' be the closure in H' of the set of test functions
....
in f. Thus H is a linear subspace of H', and u e Ho if there exists a sequence {0J of test functions in qi such that 11 0, - uli e -'
0. Clearly Ho' = H°. On the other hand, Hot is a proper subspace
of H' for t > 0. If a function u belongs to H;, one says that it vanishes at the boundary together with its derivatives up to order
HILBERT SPACE APPROACH. II
193
t - I in the L2 sense. The terminology is justified by the fact that if u has continuous derivatives up to order t - 1 in and belongs to Ho, then these derivatives actually vanish on the boundary 'P, provided that 9 is sufficiently smooth. To see this, let {0Y} be a sequence of test functions such that III, - ull t -- 0. If w is in C1
in'° ff
[wD5q + &,,D,w] dx = 0,
.1=1,
n
Hence in the limit, j[wDiu
j = 1, ... , n
uD,u] dx = 0,
But if 0 is sufficiently smooth, this is equivalent to (3)
uw cos (n,x,) dS = 0,
j = 1, ... , n
fo where cos (n,x,) is the direction cosine of the normal to I and dS is the element of area. Since (3) holds for all w, u - 0 in i. The argument holds for all derivatives of u up to order t - 1. Some Lemmas in Ho'
4.4.
In many ways functions in Ho 'behave like the periodic functions
in H,. This can be seen from LEMMA 1.
Let 2 be a cube containing '9 in its interior. For u E Hnt,
set u in q A
U
0 i n 2 -Then u, considered as periodic in 2, is in Ht and' !IuIIT = Ilulli,
be a sequence of test functions such that II0r - u li s -- 0. Extend each ¢,, to be The proof of Lemma 1 is obvious. Let
When necessary, we indicate the domain over which a norm is taken by a superscript.
L. BERS AND M. SCHECHTER
194
identically zero outside 9 and consider it periodic in 2. Then
III" - uli = II - ulle
0.
By employing Lemma 1, we can immediately carry over some of the results of Chapter 3. For instance, by (9) and Lemmas 3 and 10 of that chapter, we have
If 0 S t2 < s < tl and e > 0, then
LEMMA 2.
Ilull
<e
Ilullel
+
6-`s-ts)J(ti-s)
IIu112
for all u c- Hot. LEMMA 3.
Ck in
If u c- Ha and t > [n/2] + k + 1, then u is of class
and max I Dpu l < cont. II u Il t Ipl
LEMMA 4.
For 0 < s < t every bounded set in Ho is conditionally
compact in H.
For functions in Ho we shall find it convenient to introduce the following scalar products and norms:
(u,v): _ 2 (t)JDPDPd v Ipi =t p
lute = (u,v)'12
where (PI I is the multinomial coefficient t !/p1 !
... p,,!
.
Integration
by parts gives
(u,v)e = (u,Aty) = (Atu,v) when u and v are test functions. LEMMA 5.
If 0 < s < t there is a constant c,e such that 1u18
for all u e H. Proof.
< c8e Iutt
It suffices to prove that II
for test functions W. Then
II o < const. 101,
in W. Extend 0 to be identically zero outside
41.
dx(x) =
-axi -
HILBERT SPACE APPROACH, II
195
Applying Schwarz's inequality and integrating over T, we have IIdII 0
ao (1
< const *
11
axi 11o
which implies the desired inequality. COROLLARY 1.
In Ho' the norms Ilullt and Iult are equivalent.
This means that the expression Iult
Ilulla
lult
is bounded for all u LEMMA 6.
+
Ilullt
0 in Ho.
Let T be a distribution. Then there is a non-negative
integer t such that (4)
I T[*]
cont. II
II
t
for all test functions ¢ in 9. Proof.
Following the reasoning of the proof of Lemma 7 of
Chapter 3, we note that if the assertion were false, there would be a sequence {0v} of test functions such that I T[O,] I > v 11 0, II Set Vv = 0v/v II 0Y II Then for every fixed s, II ip, II s = I10, II 8/v 11 0, II v - 0
as v -> cc. By Lemma 3, this implies that Dip - 0 uniformly in Tr for each p. Hence T [v,v] -- 0. But I T [V,] 1, > 1, a contradiction. COROLLARY 2.
For every distribution T there is a non-negative
integer t and a function u e Ho such that
f
T[¢] = guL1/ dx for all test functions 0 in 9. Proof.
By Lemma 6, there is a t ;.>_ 0 such that (4) holds.
Therefore T is a bounded linear functional on the Hilbert space H. Moreover, by Corollary 1, we may consider I It as the norm in Ho. Thus by the Frechet-Riesz representation theorem (Theorem 2 of Appendix 1 to Chapter 3) there is a function u in Ho such that T[0] = (u,o)t
196
L. BERS AND M. SCHECHTER
for all test functions 0. Integration by parts gives
T[O] =J uL's6 dx the required representation. Note that Corollary 2 was employed in the proof of Theorem I. We end the section by proving a simple consequence of the Banach-Saks theorem (Theorem 4 of Appendix I to Chapter 3). LEMMA 7. If ' v is a function in H°, and there is a sequence vn of functions in H' such that
II U. -V 110 - 0,
!IUnii, <_ C
then v is in H' and II v II, < C.
Since H' is a Hilbert space, by the Banach-Saks theorem there is a subsequence {'v,} of whose arithmetic Proof.
means converge to some element u e H'. But the arithmetic means
converge to v in H. Hence v= u c- H'. 4.5.
The Generalized Dirichlet Problem
We are now in a position to formulate the Dirichlet problem within the framework of Hilbert space, an approach going back to Friedrichs. If u is a classical solution of the Dirichlet problem with homogeneous boundary values in a sufficiently smooth domain 5' (cf. Section 1.5), then u is in H,,"12 and satisfies (5)
(u,L*O) = (fo)
for all test functions in ', where L* is the formal adjoint of L.10 Now if V' and 0 are test functions, (6)
(LW,O) = (tV,L* ,) = [ju',o]
where (7)
NA =',, :_,»is I fsh,,.,,.(x)DP'pDr
dx
IP1I _
In That u e Ho %s is not obvious, but it is easily proved. Compare footnote 11.
HILBERT SPACE APPROACH, II
197
is a bilinear form involving derivatives of order < m/2 in either t, or 0. Thus (8)
< const. II1P II m/2 I
II
on/2
By completion it follows that [y,,o] is defined for W and 4 in H,)"12 and (8) holds for such functions. Thus our solution of the Dirichlet problem satisfies (9)
for all test functions 0. We now take (9) as the basis of our definition of
The Generalized Dirichlet Problem. Given f c- H°; to find a u e H"/'-' such that (9) holds. We know that a classical solution of the Dirichlet problem is a
solution of the generalized Dirichlet problem. Conversely, a solution u of the generalized problem which has continuous derivatives in SO of order m is a classical solution of Lu =f; this
follows from (5) by integration by parts. Moreover, if it has continuous derivatives up to order m/2 in jp and P is sufficiently smooth, we see that all derivatives of order <(m/2) -- 1 vanish on Thus u is a classical solution of the Dirichlet problem. Thus we may solve the Dirichlet problem by first solving the generalized Dirichlet problem and then showing that the solution has the required smoothness properties. While the problem of differentiability on the boundary is in itself very interesting, one should nevertheless remember that the generalized way of formulating boundary conditions is in many
ways natural. Thus it is well known that, even for Laplace's equation, the Dirichlet problem may not have a classical solution, that is, a solution which is continuous up to the boundary, for the
most general domain. The Hilbert space formulation of the Dirichlet problem, however, is applicable without any restrictions.
In a certain sense, this formulation of the boundary condition does the thinking which would otherwise have to be performed by the mathematician posing the problem. Consider, for instance,
the Dirichlet problem for the Laplace equation Au = 0 in 3 dimensions. We assume that the boundary 47 consists of smooth manifolds of dimensions 2, 1, and 0 (surfaces, curves, points).
L. BERS AND M. SCHECHTER
198
The proper way to prescribe boundary data is to give them only on manifolds of dimension 2 and to require the solution to be bounded on other boundary components. On the other hand, in
the Hilbert space approach, it seems that one prescribes the values of the unknown function on all parts of the boundary. But these values are prescribed not in the classical, but in the generalized sense, and it turns out that the solution obtained "forgets" the superfluous conditions and satisfies exactly the conditions indicated before. 4.6. Existence of Weak Solutions
In this section we shall solve the generalized Dirichlet problem.
Our main tool will be the Girding inequality which we state as THEOREM 2.
There exist positive constants cl and c2 depending only
on L and T such that (10)
(0, Lc6) >_ ci
121 - C2 II¢II2
for all test functions 0 in I.F.
The proof of (10) is very similar to the corresponding periodic inequality. In fact an inspection of our proof of (21) of Chapter 3 will convince the reader that it can be carried over word for word to the present case. Employing (10) we can prove Let f be a function in H°. If .° > c2, there is a unique function u e Ho /2 such that THEOREM 3.
(11)
00) =(ff0)
for all test functions 0. Moreover, u has L2 derivatives up to order m. Proof.
Set
By Theorem 2, (12)
hPA, = [W,0] , 4(14) [u,u]' > ci
II U 11L/2
for all u e H0'7`/2. Moreover, by (8), (13)
[u,v]' < const. IIuII,02 IIvIlm/2
HILBERT SPACE APPROACH, II
199
Inequalities (12) and (13) allow us to apply the Lax-Milgram lemma (Theorem 3 of Appendix I to Chapter 3). Since f E H°, I (f v) I < 11f 110
Il v II O
<_ const. Il v 11 m/2
and (f,v) is a bounded linear functional on
H0-12. Hence by the
lemma, there is a u c- Ho'12 such that
[u,v]' = (f,v)
(14)
for all v c- H,-/2. By (6), (14) implies (11). The uniqueness follows from the fact that any other solution must satisfy (14) also. Hence, the difference w satisfies [w,v]' = 0 for all v e Ho /2. In particular,
this holds for v = w. Thus [w,w]' = 0 which implies w = 0 by (12).
Let us call the solution in Theorem 3 the generalized solution of the Dirichlet problem for Lu 4- ).Ou =f By (12) and (14) we have I1ull2nf/2
< const. If II0 Ilulj0
Setting u = (L + A0) -'f we see that II (L + )0) -'.f 11 m/2 = const. If 110
Thus by Lemma 4, (L + A0) -1 is completely continuous. This allows us to make statements about generalized solutions of the Dirichlet problem for the equations
Lu =f L*v =f
(15)
(16)
Lu = 0 L*v = 0
(17) (18) THEOREM 4.
Equations (17) and (18) have the same finite number of
linearly independent solutions. THEOREM 5.
The equation Lu + Au = 0 has nontrivial solutions
only for a denumerable set of values A having no finite accumulation point. THEOREM 6.
Equation (15) has a solution if and only if (f,v) = 0
or every solution v of (18).
Equation (15) has a solution for every f E H° if and only if u = 0 is the only solution of (17). THEOREM 7.
200
L. BERS AND M. SCHECHTER
The proofs of Theorems 4-7 are identical to those of the corresponding theorems for periodic equations (see the end of Chapter 3). 4.7.
Regularity at the Boundary
We are going to show that under further regularity assumptions a generalized solution u of the Dirichlct problem is a on f and
smooth function in
satisfying the equation and boundary
conditions. Assume that is of class C,,,, and that f is in C,o in Let u be a solution of the generalized Dirichlet problem with homogeneous THEOREM 8.
boundary conditions for Lu - f. Then u is in C,,, in 1. Since U E Ho'g' it will follow from the discussion in Section 4.3 that u and all its derivatives up to order (m/2) - 1 vanish on ik.
Hence u is a solution of the Dirichlct problem in the classical sense.
The proof of Theorem 8 is based on a special result referring to a domain S2, which is the intersection of the ball Ixi < r and the half-space x > 0. We consider SD, as contained in a half-cube E, the intersection of x > 0 and a cube 2 with edges parallel to the coordinate axes. Let V be the set of C. (not necessarily periodic) functions in E which vanish identically near x, - 0, and let V' be the set of C,,, functions which vanish identically near the remaining faces. In
addition to the norms of the spaces H' in Id' we shall find it convenient to employ the following "negative" norms: (19)
IIUII_8 =
i(v,w)
I
Ilwlls
which we shall consider as the counterparts of the periodic negative norms over a. We now have THEOREM 9. _Let L be an elliptic operator of order m = 2m' with C, coefficients in E. Let t be an}' integer > m' and set s = max (m, t
Ij v is a function in Ht n H."', f l Cs in S2, which vanishes near I xI = r and such that IILv II,_,,, , 1 is finite, then v e H'-1 in 52,.
HILBERT SPACE APPROACH, 11
201
The proof of Theorem 9 will be given in the next section. Here we shall make use of it in the proof of Theorem 8. We already know that u has derivatives of all orders at every interior point of V (cf. Theorem 1 ). I t remains to show that u is in C,, up to the boundary. Since regularity is a local property, it suffices to prove that for any boundary point x0 there is a neighborhood .:1 such that u is in C, in n -,1,(x0). Since is of class C,,, we can find a neighborhood .14''(x0) so small that it can be mapped
in a one-to-one C., way onto a ball Ixi < R in such a way that n A *(x0) maps into the hyperplane x,, = 0 with x0 going into the origin. Under such a mapping the ellipticity of L is preserved. Assume that the image of 1i l ;1 "'(x(,) is Q,,. Now by Sobolev's theorem (see the remark following the proof of Lemma 9) it suffices to prove that in some S2 r < R, u is in H' for all 1. We do
this by showing that for any r < R and e > 0, u c- H' in SZ, implies u e H' I' in S2,_F.
This may be done as follows. Let be a C,, function in f2n which is identically one in SZ,_, and vanishes near Ixi = r. Since u c H' in S2 so is v = u. We wish to apply Theorem 9 to v (since uEH
we need only consider t > m'). In order to do so we
must verify that II Lv II t-m+i is finite. But
Lv=Cf+L'u where L' is an operator of order <m. Hence IILvIIt-m-,, < const. (Ill IIt-m+i +
is obtained by
I
integration by parts in the usual way. Since O u II t is assumed finite, the same is true for IILvII: -,,, i Hence Theorem 9 applies giving v E H'-' in Q,.. Since is identically one in S2,_ this means that u E H'-' in S2,_E and the proof is complete.
In following the proof, we see that if we had assumed only that f e H'k in 9, we would have been able to conclude that u c- Hm, k. In fact we have proved the inequality Remark.
IIuIIn.j-k
< const. (Il f IIr + IIuilo)
This is one of a spectrum of "coerciveness" inequalities introduced
by Aronszajn [2] and studied by several authors (Browder [2],
L. BERS AND M. SCHECHTER
202
Nirenberg [4], Schechter [6], Agmon [3], Guseva [1], Koselev [1], Slobodeckii [1], and others). 4.8.
Inequalities in a Half-Cube
First consider a cube 2 with sides parallel to the coordinate axes. Employing the notation of Chapter 3, we state LEMMA 8.
(20)
Let m and t be any positive integers. Then
IIDnvlla < cont. (flD1v'2 n lf-1n+1 +
n-1 1JDwll t =1
for all periodic Cx functions v. Proof.
If v = E at eit'', then e . ()t Iati2
IIDnvII = const. IIDnvili -m+1 =
cont. Jen m(1 +
II Dtv II i = const.
f (1 +
. C)t-mr1 Ixt12 ?') t I xtl2
Hence (20) is equivalent to the inequality t ` n < cont. [ im ( 1 + '
)-"
-'+
n-1 a=1
which is true for any positive m. (We have tacitly assumed that the sides of 2 equal 2ir; this involves no loss of generality.)
Next, consider the intersection E of 2. with the half-space xn > 0. We recall the definitions of the classes V and V' given in the previous section. LEMMA 9.
If the norms are taken over ' E, then (20) holds for all
v E V'. Proof.
We employ a device due to Lions of extending
functions in V' to be in C. and have compact support in the whole of .2. They then can be considered periodic and inequality (20)
applies. We then show that the periodic norms in 2 of the extended functions can be estimated in terms of the corresponding norms over E.
HILBERT SPACE APPROACH, II
203
To carry out the procedure, if v e V', consider it (for the moment) to be zero outside Z in x > 0 and set X" > 0
v1(x',xn) = v(x',xn)
_
8-}-1
x,, < 0
.Z.v(x', k=1
... , xi_,) and s = max (m, t ± 1). The constants
where x' = (x,,
Ak are so chosen that v1 is in C. in 2, i.e., that RTL
(21)
k=1
j = 0, 1, ... , S
(-k)'2,, = 1
Thus v1 has compact support in .2, and ID'vlz dx
ID11v1I2 dx <
(22)
f.2
const.JP
Considering v, periodic in 2 we see that (20) holds for it. Hence, if all of the norms in (20) are non-negative (i.e., if t > m 1), it follows from (22) that (20) holds for v.
-
It remains to prove that for t < m - 1
,, j <_ const. 11Dn'vlli nn-1 (The norm on the left is the periodic norm for 2; the norm on the right is that of the type (19) for Z.) In order to do this we (23)
IID,,v, ll-
first note that for v e V' and wEV (24)
I (v,w) I -<
Il v ll t-m.+ 1 II W II
m- t_1
By completion, (24) holds for w e C,,, in E satisfying" (25)
onxn=0,
D;,w=0
j=0,1,...,m-t-1
Now let g be any periodic C. function in 2. Then (Dn v1,g) 0
= f D'; vlg dx
=
f D; vlg dx + f
Dnvlg dx
11 This may easily be seen by first observing that (24) holds for C,n_
functions which vanish near x,, = 0. Then if w satisfies (25),
W(-X', x - e) for x > e, = 0 for 0 <- x,, <. e, is such a function and
IIwt -wll,,,_t-1- 0 as
e-0.
L. BERS AND M. SCHECHTER
204
Converting the last integral, we obtain (Dn'v1,g) = JDV&' dx
where g'(X',Xn) =
k-1
k
By (21) we see that w - g' satisfies (25). Hence by (24), C III
I
vlll m+l llg'll
< const. II D;,'v ll
"1+1 Il; li
-l
1
Since this holds for all periodic g, it follows from Lemma 5 of Chapter 3 that (23) holds. This completes the proof. Remark. The extension of v employed in the proof of Lemma 9 shows that Lemmas 2 and 3 hold for functions in F. An immediate consequence of Lemma 9 is LEMMA 10.
Let L be a partial differential operator of order m with
coefficients continuous in Z. If the coefficient of D;;' in L does not vanish in Z, then
(26)
IIUII
_r
< const.
n-1
r
IID,vll
I
IlUllu)
,=1
for all v E P. Proof.
We know that L = aDn` + L1 + L2, where a
0,
L1 involves derivatives of the form D" with IpI = m, pn < m, and L2 is of order <m. Set ao = min lal > 0. Then (27)
IID','vlll
-1
ao
(IuLvll1-,n,1
-,
IIL1vlll_mY1 + IILZvll,-m+1)
If the norms are positive we see that (28)
IIL1vll -m+1 <
const."-I 1 IID,vIl 1-1
In addition, by the remark above, for any E > 0 (29)
IlL2vll l
-1 `- e
lip ;I
r. 1 - const. llvllo
Inequality (26) now follows immediately from (20, 27, 28, and 29).
HILBERT SPACE APPROACH, II
205
If the norms are negative, let w be any function in V and consider the expression (L1v,w). We can integrate by parts, throwing derivatives up to order m - t - 1 onto w in such a way that every derivative remaining on v involves some x i : n. Then
n-1 I (L1v,w) I <- const. Il w ll »,-t-1
II D,v II t i=1
Since this is true for all w e V, (28) holds. Also, in the expression
(L2v,w) we can integrate by parts, throwing derivatives up to order m - t - 1 on w. The derivatives remaining on v are of order
Hence II L2v II
t-, . 1
const. II v II t
and (29) follows from Lemma 2. The proof is thus complete.
We now observe that (26) will hold with the same constant in any subdomain consisting of the intersection of 2, with the half-
space x > h > 0. Letting a -o 0, we have Set s = max (m, t , 1) and suppose that v is in C, n HO in E and vanishes near all faces except x = 0. Assume that D,v E Ht in for i = 1, ... , n - 1, and that IILvll,,,-t+1 is finite. COROLLARY 3.
Then v E H`+1 in 2.' and (26) holds.
We use induction on I and show first that the assertion is true for t = m'. Let a' denote any difference quotient in the x, direction, i , n. Then Proof of Theorem 9.
L,,,r
L8;'v
where L,,, is the operator whose coefficients are the difference quotients of those of L. Thus IILb;'z'il
,n
IIa (Lz')'1<. IILz'11-
- I'L,,,z ll_ const. IIi'11,,,
and is finite. (Integration by parts was used in obtaining these estimates.) Hence there is a constant C such that (La'v,g) I < C iigll
206
L. BERS AND M. SCHECHTER
for all g E V and, a fortiori, for all test functions in 52,. Since v E Ho"", so is b,v for h sufficiently small. Hence by (6) I[bhv,g]l s C Ilgllm'
(30)
for all test functions g in S2,. By completion (30) holds for all g E Ho". We may therefore take g = bhv. Now by Garding's inequality [cf. (12) ]
Hence
[b"v,bhv] ? cl ci Il
61
'V112' - C2 IIb,vII
bhvllm < C Ilbhvll,,' + C2 Ilb"vIIO
showing that there is a constant C' such that Ilbhvllm' < C'
Now 116S - D'v II
0 in K2r_E for any e > 0 (cf. the end of
Appendix II to Chapter 1). Hence, by Lemma 7, D,v E Hm' in S2,_ and IID,vllm' s C'
Letting E -- 0, we see that D,v E H7" in Q,,. Once this is known, we
can apply Corollary 3. We therefore conclude that v e Hm'+i in QT.
Now assume that the theorem is true for t - 1 > m'. We have
LD,v = D,Lv + L'v where L' is an operator of order <m. Hence IILD,vII:-m <- IID,LVIlt-,n + IIL'vllt
s IILvIIt-m+i + const. IIvIIi
This is obvious if t >_ m. For t < m one must integrate by parts _,,, is finite. Moreover, if i 0 n, D,v E Ho m' and since v e H', D,v E Ht-'. Hence we may
in the usual fashion. In any event, II LD,v II
apply the induction hypothesis to D,v, obtaining D,v E H'. Another application of Corollary 3 gives v c- Ht+' and the proof is complete.
HILBERT SPACE APPROACH, II
207
Analyticity of Solutions
Appendix.
Hilbert space methods can also be used to give a proof of the analyticity theorem mentioned in Chapter 1. Assume that f and the coefficients of L are real analytic in qi and that u is a solution of Lu =f (we may even assume u is a weak solution or a distribution). Then by Theorem 1 of this chapter, u is in C.0 in q. We must show that for each point x0 in 9 there
is a small ball Ix - x01 < b in which u may be expanded in a power series. By a translation, we may assume that x0 is the origin. We shall find it convenient to use the following norms and seminorms Il w Il a,r
=
w18,r
=
DDwl2 dx
1P1--j xl
Let R > 0 be such that lxl (1)
I
Ir=a J xI
ID'wl2 dx
R is contained in 9. By hypothesis
1(IDsa91+ID8fI) <Mok!Bk
k=0,1,2,...
181=k
in IxI < R for some positive numbers M0, B. Let r < R be such
that R/2 < r < R, and h > 0 such that r + h < R. Let a(T) be a fixed Cc function of one variable which equals 1 for T < 0 and vanishes for T > 1. Set fi(x) = tr`IxI
- rl
h
Clearly = 1 in IxI < r and vanishes for IxI > r + h. It is easily checked that (2)
ID8l ( S cls1 h-1-q'
in IxI < R, where the constant c181 depends only on n, Ist and Q(T). If w is any C. function in IxI < R, set v = w. Then v is a test function in lxi < R. We may therefore consider it to be periodic in a cube containing IxI < R. We may alter the coefficients of L outside lxi S R so that they become periodic also. We now can apply Theorem 2 of Chapter 3 to obtain t = 1, 2, .. (3) Ilvlit,r+h < ct(IILvIIt-m,r+h + Ilvllt-m,r+h),
L. BERS AND M. SCHECHTER
208
where we have made use of the fact that v vanishes for Ixl > r + h. Now by Sobolev's inequality (cf. Lemma 3 of this chapter) there is
an integer N (>m) such that max IvI < x(n,N,R)
(4)
Izi _x
Employing the fact that Ivl g,1; and IIVII:,R are equivalent norms
for test functions (Lemma 5 of this chapter), we see that (3) implies (5)
C Const. (ILvi \.-m,r
IIVII.\',r+h
!h
+
Ivl.V-.,,+h)
Now if Isl = N - m, there are C,, functions bxo such that
DBLv = D3Lw + Hence (6)
.1 p'to
bD,D.Dv'4DD"w
N
II wll v,r-h <
C(ILwI.V-m.r+h
+
h-'
Ilwll,--;,r+h
j =1 where the constant C does not depend on w, r, or h. Now it is easily verified that IDBLD'ui
)
(I41)
ID-"TgfI + Kl IpI
lit Is'-s"I
Iq'4qI =IqI
14
I)
q' 7t o
x ID$ +q apI I De
-Fq'+puI
where K1 depends only on Isl. Hence by (1), lLD`'uI_\'-m,rlh <
Mo(N + IQI
-
m) !
Bn+IqI-m
X-m Iq
lyl){i +J)! B`' I -0 i=1 lq'I=IqI-i
+ K2
IIDq'ull-:.r+h
Inserting D"u for w in (6), we have (7)
1 II Dqull.\-,r+h < Cl WO(N + k - m) !
B`+k-m
IqI =k
\r
h-s IqI =k 1
I k
II D''u II N-,,rth
1
(k) (r + J) ! B'-'
S=1 ,=0181 =k-i J
where the constant Cl does not depend on u, k, r, or h.
I1Dqull.v-i,r+h}
HILBERT SPACE APPROACH, II
209
We now claim that it follows from (7) that there are constants
M and k such that for each r, R/2 < r < R and all integers
j>0
II D`u II:V,r-h, < :11j !
(8)
where h, = (R that
J=
R - r)
ICI =
- r)lj. Once (8)
0, 1, 2,
...
is proved, it follows from (4)
I max ID`uI < a(n,N,R)Mj! (R
A
...
j = 0, 1, 2,
r
ItI =j 1XI
which immediately implies analyticity of u. It therefore remains to prove (8).
We proceed by induction. Clearly, (8) can be made to be satisfied for] = 0. Thus let k be any integer > 1 and assume (8) is satisfied for all j such that 0 < j < k. We shall show that it holds for j = k. Let r be any number between R/2 and R and set
ho = R - r. Then h, = ho/j. Now by (7) and the induction hypothesis II
IQI =
k
<
Dv UII c,r-hx
C11 Mo(N
+ k - m) ! B` +k 'k
.IV
+ j=1 I hk'M(k -j) ! k J'-n: `k l
lx
+j)!Bi-,M(k -i -j)!
lk (N +k -m)! M k! (k-j)! o
Mk ((a ,
rn (k J. k j
-
/Z
Bx ''oh X11
- i -j) ! (i +j)! (B(ho - hk) (k -j)! j! -m ('2Bh0'-v
<Mk! (h) o k
+ C2
(h 0
/z
1
+
J
ho - hk,J
+
,>i
A'
A
hk)l'
2B(ho l
1
1
A
-k D
(
ho
J
ho
(k
- j) ! k!
hZ\)k )
I
kk
(k -
1)k-'
210
L. BF.RS AND M. SCHECHTER
where we have used the fact that
(N-m+k)! SC22kk! and (i +j)! SC22j!,
i SN- m
Now there are also constants C3 and C4 such that
(k -j)!
kk
k! k
(k-1)k-,SC3
1k
S C4
k
Hence
I IIDuIIv.T+n <Mk! I1.1' M' 2BN-m M
.
(2B/zo)k
t
+
N C3
+
N
C
2B(ho2 hk)
4 7
i1
If we choose M and A so that (9)
(10)
M>
3MOC2BN-m
2 z max (3NC3,6C2C4BR)
it follows that (8) holds for j = k. The proof given here is essentially that of Morrey-Nirenberg [1].
CHAPTER 5
Potential Theoretical Approach The Hilbert space approach described in the preceding two chapters is very elegant, but this elegance is paid for by neglecting certain finer aspects of the theory. Various questions in the theory of linear elliptic equations cannot be answered by the L2 theory, and this applies particularly to questions which are related to the more difficult theory of nonlinear equations. In order to investigate
these questions, one must use tools from potential theory. We devote the beginning of this chapter to a description of these tools. Fundamental Solutions. Parametrix
5.1.
The potential theory approach to elliptic equations is based on certain simple integro-differential identities and on three sets of
inequalities for convolutions of functions with powers of the distance from a fixed point. We shall discuss the identities first.
Let O(x) be an infinitely differentiable function with compact support [x = (x1, . . . , x,), n >_ 2]. Then Fundamental Identity.
(1)
O(x) =
(la)
O(x)
AO(y) dy
(n > 2)
IX -yln-2 O(3') dy
= c.Af
f
I
y
(n > 2)
2
fi(x) = c2 &/(y) log Ix -yl dy
(la') fi(x) = where Cn
_
c2+(y) log Ix -yl dy
(n = 2)
(n = 2)
-1
F(n/2)
2(n - 2)7rn12 - (n - 2) Q,,
_ C2
1
27T
(Q is the surface area of the n dimensional unit sphere). 211
L. BERS AND M. SCHECHTER
212
The proof is well known. We consider only the case n > 3. By Green's formula every sufficiently smooth function O(x) can be represented as the superposition of three potentials c"
'fi(x)
-y1)-2 d>'
J`M;)
fo, IX
-1
n-2
an ix ac(y)
1
an
Ix - yl,l-2
dS
dS
Here I is a domain with a sufficiently smooth boundary x is a point of 1, ajan represents differentiation in the direction of the outward normal, and dS is the area element on k For a function
of compact support, we may take 9 to be a sufficiently large sphere. In this case the boundary terms disappear and identity (1)
results. This identity can be rewritten, in terms of distribution theory, as
(n > 2) Oc2 log IXI = b(x) (n = 2) where b(x) is the symbolic Dirac function, i.e., a distribution defined by the relation Ac.,
Ix12-n
= b(x)
J(x)6(x) dx = 0(0) for every test function O(x). Now (la) follows immediately from (1), since O(x - z) dz = f ao(x - z) o r 0(y) dy = ASJ dz x
IX
IZI1--2
-yln-2
=f AM ') IX
yj 11
2
dy
J
IZIn-2
= c 'fi(x)
From (1) we obtain, integrating by parts, the identity (2)
O(x) = const.
y' a4 dy
f X` _ Ix J -yI" ay,
which represents a function with compact support in terms of its derivatives. For an elliptic operator L of order >2 with constant coefficients
and only highest order terms, one can construct explicitly a
POTENTIAL THEORETICAL APPROACH
213
function J(x) related to it in the same way as (xlz-n is related to the Laplacean A. More precisely, the function J (x) will have the following properties. (i) J (x) is a real analytic function for I x I 0. (ii) If n is odd or if n is even and n > m, then
(3)
J (X)
I(x)
where w(x) is positive homogeneous of degree 0, (w(lx) = w(x), t > 0). If n is even and n S m, then
J (x) = q(x) log
IxI +
I(x)
where q is a homogeneous polynomial of degree m - n. (iii) The function J (x) satisfies the equation
L0J (x) = b (x)
(4)
so that for every C., function with compact support (5)
f
f
0(x) = [Lo/(y)]J(x -y) dy = Lo s6(y)J(x -y) dy
The result is classical; we carry out the construction only for the case of an odd n > m and for m = 2 and all n. (As a matter of fact, we shall assume that n is odd and exceeds m whenever it is
convenient to do so during this chapter. This assumption, howdenote the characteristic form of ever, is not essential.) Let Lo, i.e., let a,,DP, I a,E" Lo = IPI =»6
IP1 =m
Consider the function
Ix = x1E1 + ... +
It is obvious that this function is positive homogeneous of degree m + 1 and real analytic for where x -
x 0 0. It is easy to see that Lol(x) is positive homogeneous of degree one and invariant with respect to any rotation about the origin. Hence LOI(x) = const. Ixt. Now set J(x) = cAn+112I(x), where c is a constant. This function has properties (i) and (ii).
L. BERS AND M. SCHECHTER
214
That it also has property (iii) for an appropriate choice of the constant c follows from the fact that L0J(x)
= Loci"+1/2I(x) = cA"+1I2LOI(x) = c0"-1/2 const. IxI = const. A(An-1I'2 IxI)
= const. A x12-" = b(x)
When m = 2 we can write down the fundamental solution just as easily. In fact if
=.I a,,$,s, let A denote the cofactor of a1, in the determinant la,,l. Then
J(x) = c(,J
A.1x,x,)2-,7/2
J (x) = c log (I A1,x,x,)
n>2 n=2
One readily verifies that these expressions satisfy the required stipulations.
The function J (x) just constructed is called a fundamental solution of the differential equation Loo = 0. More generally, for any homogeneous elliptic equation L¢ = 0, a function J (x,)r) depending on a parametery is called a fundamental solution if it satisfies the equation L.J (x,y) = b(x - y)
It should be remarked that bounds on the derivatives of w (x) in (3) depend only on n and the ellipticity constant of Lo, i.e., on
min IQ( ) IFI=1
Fundamental solutions for elliptic equations have been constructed by many authors, the most important results being due to John whose book [1] also contains an extensive bibliography. In
particular, fundamental solutions are known to exist for any equations with analytic coefficients (in which case the fundamental solution itself is analytic) and for any equation with C,o coefficients (in which case the fundamental solution is itself C.) as well as under much weaker assumptions. As a matter of fact, it
is probable that the fundamental solution can be constructed
POTENTIAL THEORETICAL APPROACH
215
for any equation with Holder continuous coefficients, though as far as we know, this has never been proved in the literature.
Fundamental solutions play an important part in the theory of elliptic equations. In particular, once a sufficiently nice fundamental solution has been constructed, one can easily prove analyticity and differentiability of solutions. An arbitrary solution
of an equation defined in a domain 9 with a sufficiently nice boundary ! can be expressed in terms of its Cauchy data at the boundary (values of the function and its derivatives up to the order m - 1) in terms of the fundamental solution of the adjoint equation. If a fundamental solution satisfies appropriate boundary conditions (in which case it is called Green's function), a solution can be expressed in terms of fewer boundary data. Fundamental solutions also play a part in the integral equation approach to boundary value problems. This method is classical for the potential equation and for second order equations in general;
an excellent account will be found in the book by Miranda [1]. The integral equations method has recently been extended to
equations of higher order (Agmon [4] and others). But our concern here will be only with the use of the simplest fundamental
solutions, namely the functions J (x) constructed above. These
applications are based on a simple device due to Korn and Lichtenstein.
To an elliptic operator of order m with variable coefficients a,(x) and to every point x0 we associate the "tangential operator"
=
v
lpl=yn
which is a homogeneous operator with constant coefficients. We shall denote by J.,°(x) the fundamental solution of the equation Lx°4 = 0 constructed above. JZ° is called a parametrix for the equation Lc = 0 with singularity at x0. We shall denote by Sx0
the operator which takes a function O(x) into the function ti = Sx00, where (7) V (x) =fix°(x -y)o(y) dy Finally, we define the operator Tx° by the equation (8)
Tx° = SX°(LZ° - L)
L. BERS AND M. SCHECHTER
216
Since S 0L, ,
= 1 (= identity) on functions with compact support
and similarly L 0ST0 = 1, by virtue of identity (5), we have the following If 0 has compact support, then
"LEMMA" A.
= Tx0 ± Sx0L¢
(9)
and if 0 = Txa¢ ± Sxu f
(10) then
is a solution of the equation Lc = f.
The quotation marks around the word lemma are in recognition
of the somewhat cavalier way in which the lemma has been stated. It goes without saying that in order to become a real lemma,
it must be supplemented by a statement on the assumptions involved. In applications to be considered, all this will be selfevident. The proof of A is clear. Equation (9) follows by noting that on
functions with compact support T,0 + S= L = 1. On the other hand, assuming that (10) holds we have (Tx0 ; Sx L)o = Tx0o -Sx0 f hence, S_L4 = S.,, ,f and Lq = L.,,, S,.0L0 = L,,Syo f =f. The applications of Lemma A are based on certain inequalities which are best stated in terms of certain norms. Some Function Spaces
5.2.
We consider first C X functions defined on the closure of a boun-
ded domain1i. For these functions we set
0,
(jIvdx)
1/r
(r > 1)
l.u.b. ICI
H..gr[o] = l.u.b. ,._0
Ix, - X "l
0 < x < 1
(diam V)' max IIDp0!Ic,(9), In!
k = 1, 2, .. .
POTENTIAL THEORETICAL APPROACH II
(diam V)k-" max H,,,,[Dp0],
IICktx() = It'IICk(l)
1PI =k
k=o,1,..., II0IIJi ()
Il
217
=
(diam
llNk(W) = I (diam q)Iv i -,nlr IPlsk
0<x<1
max II DI X11 La(s), IpI -k k = 1)
IIDD0IILr(F)'
2,...
k = 1, 2, .. .
All these norms satisfy the axioms of a normed space. By completing the space of C. functions in each of these norms, we obtain the Banach spaces
LrM, CkM, Ck+x(
),
BB(9),
Nk(9)
The space NC(I) consists of functions which have Lr derivatives up to order k. The space Ck(I) consists of functions which have derivatives up to order k. The space Ck+«( Y) consists of functions
which have continuous derivatives up to order k, the highest derivatives being uniformly Holder continuous with exponent a. The space Bx(Ii) consists of functions which have continuous derivatives up to the order k - 1 and Lr derivatives of order k. If the domain 1 is the ball of radius R (i.e., IxI < R), we shall write Lr(R), Ck(R), etc., instead of Lr(IxI < R), Ck(IxI < R), etc. We list the following properties of the norms which we shall need later (1 la)
II0IIrr(s)
(11b)
II4IIc,(f) s
(l lc)
IIOIIBk(f) <_ II
I!.\ (y )
for9c 9'. (12)
II4IIL,M < II0IIL,(s) II
UC,(S) < II0IIC,(#)
(14) (15a)
II4(Rx) IIc,(i)
=
(15b)
II#(Rx) JIB"(,)
=
(15c)
II0(Rx)
for s < t
(const. depending on s) I'!1C() IIWllc,(c4) Ilc,(R)
II0(x) II0(x) Il.Vk(ft)
218
L. BERS AND M. SCHECHTER
The proofs of (11), (13), (14), and (15) are obvious; (12) is the classical Holder inequality. We remark that it would be easy to modify the definition of the norm II II c,cy> so as to replace (14) by
the nicer inequality IlOPllc,(Ci)
In this case C3(c') would be a Banach algebra. We now discuss certain properties of the norms which are true only when the domain in question is sufficiently regular. For our purposes the following conditions will suffice. We shall say that 9 has the Cs extension property if there is a domain 9' containing
9 in its interior and a linear mapping 0 of C3(Si) into C.,(f') such that Ou = u
(16)
in 9
110ullc,c9'> < const.
It is easily seen that the ball Ixl < R has the C3 extension property for each s. In fact, we may take (17)
Ou = 1 A,u(R
- k IxI (lx) - R)x)
for R < Ixl < 2R, where k is any integer larger than s and the constants A, are chosen to satisfy k
=o
it
J-k
=1,
0
(This is just an adaptation of the method of Lions described in
the proof of Lemma 9 of Chapter 4.) We also note that by multiplying Ou by a test function in Or' which is identically one in 9, we can always achieve that Ou has compact support in 91'. LEMMA 1.
Assume that 9 has the Ct extension property, t > 0.
Then there is a constant kt depending only on t and the constant in (16) such that (18)
E
kte-Frc`-`
111011C0(Y)
,for0<s
POTENTIAL THEORETICAL APPROACH
219
By virtue of (13), Ct(9) may be considered as a subset of C3(W) for s < t. We show now that the injection of C1(F) into C8(T) is a completely continuous mapping, i.e., that LEMMA 2. Under the hypothesis of Lemma 1, for 0 < s < t every bounded sequence in Ct(V) contains a subsequence which converges in C.(W).
Let {u5} be a sequence in Ct(If) such that Ilu,llc,(,9) M. Since t > 0, the functions u, at least satisfy a uniform Holder condition on W and are thus equicontinuous. Thus by the AscoliProof.
Arzela theorem there is a subsequence (denoted also by {u,}) which converges uniformly on ej, i.e., such that 11 u, - u,llc.o(1) 0 as i, j --- oo. Let e > 0 be given and take N so large that \sl(t-e)/ e
111u, -
4M
-'
2kt
for i, j > N, where kt is the constant in (18). Then by Lemma 1, s/(t-8) Ilu,
-
4M
2M + kt . GMlll
for i, j > N. Hence II u: - u; ll c,(w) -'
Ilut - u,llco(s) < e
0 as i, j -* oo and the
lemma is proved.
We now show that Lemmas 1 and 2 have their counterparts
for the spaces Bk(1). We say that 1 has the Nf extension property if there is a domain NN(W) into NN(V') such that Ou = u
' 7)
const. Ilull.\x(-f)
(19)
The ball lxl < R has the Nk extension property for every k and r. In fact, the mapping (17) meets the requirements. LEMMA 3. Let V be a domain having the CJ1_1 and Nx extension properties. Then there is a constant C depending only on k, r, and the constants in (16) and (19) such that
(20)
e
for all j
CE-'/(k
L. BERS AND M. SCHECHTER
220
LEMMA 4.
If
satisfies the hypotheses of Lemma 3, then for
1 < j < k every bounded sequence in BA (W) contains a subsequence ,
which converges in B(1).
The proof of Lemma 3 will be given in Appendix II to this chapter. The proof of Lemma 4 is similar to that of Lemma 2 and is omitted. Fundamental Inequalities
5.3.
We shall now state the three fundamental inequalities. It is convenient to write the integral
fix
0(y) dy
in the abbreviated form I yl' * 0(y) = r' * 0. The star represents convolution. If 0 is a Ca, function with compact support, the function r' * 0 is also in C, (for is > -n), though its support is not necessarily compact. Our first statement is that of SOBOLEV'S INEQUALITY [1].
If 0 is a C. function which vanishesfor
l xl > R and if q > 1, 0 < A < n, then we have
-+--1<0 A
1
(a) for
n
(const. depending on R, A, n, q)
Ilr-' *
(b) for
1 q
-1=0
-!n
and any
ql > 1
llr-' * 0II1.,,I(R) < (const. depending on R, A, n, q, q1) II0IILp(R) (c)
for Ilr-A
* 0111.,1(x)
1
1
q,
q
i n
1 >0
< (const. depending on ),, n, q) 11011La(R)
The proof consists of elementary but tricky applications of Holder's inequality and will be given in Appendix I to this chapter.
POTENTIAL THEORETICAL APPROACH
221
Here is an application of the Sobolev inequality. For functions 0 with compact support in I xl < R and having LQ derivatives of first order THEOREM OF SOBOLEV [ 1 ] AND KONDRASHOV [ 1 ].
(q > 1) II 0 II L_l (R'
q1 =
if q < n,
< const. max II D,0 II La(R)
if q > n
II0IILoo(R) < const. max IID,0IILa(R)
nq
n-q
and
0 < a < 1 - nn
if q > n,
const. max IID,CIIL2(R
q
It is part of the theorem that the left-hand sides are finite.
It is obviously enough to prove the inequalities asserted
Proof.
for the case of C., functions. The first two inequalities follow from the identity (2) and Sobolev's inequality [cases (a) and (c) ; 2 = n - 1]. In order to prove the last inequality, we observe that x -{- h Ix -- hits
(21)
x
< const. ihix
Ixi"
for every number a, 0 (x1,
... , x"), h = (h1, ... ,
1
ix
1
ixi"-i-x
hint-1+x
,-
a < 1 and any two vectors x h,,), as is easily verified. Now the
identity (2) shows that O(x)
-
x;
O(x + h) = const.
(IX
so that
I fi(x) - 0(x ; h) I < const. IhI°`
(
-
-yl" y'
h,
x' Ix
+ h -yI"
ay,
dy
Y IDt'(y) I dy
Ix+h -yl"-I4 fI I D,0(y) 14y J
IY
-y.l,t_i+=
and estimating the last two integrals by the Sobolev inequality [case (a), A = n - 1 ; x] we obtain the desired statement. COROLLARY OF THE SOBOLEV-KONDRASHOV THEOREM.
If (x) has
L,, derivatives of order k, (q > 1), and if s > 0 is the smallest integer such that q > n/s, then the derivatives of 0 of order k -- s are Holder continuous with exponent x for every x such that 0 < a < s -- (n/q)
L. BERS AND M. SCHECHTER
222
It is, of course, sufficient to prove the theorem assuming the function 0 to have compact support, since if it did not, one could consider instead the function $ = 0 where 4(x) is a C. function with compact support which is identically I in a given subdomain of For functions of compact support the assertion follows by repeated applications of the Sobolev-Kondrashov theorem. If s = 1, the corollary is just the third inequality of the theorem. Otherwise, we have the repeated inequalities II Dk-1c II Lg1(R)
< const. max II Dkc II Lq(R) const. max IID"`-l0 IIrg1(R)
IIDx-a}1IILgs_1(R) S const. max
IID-s+2
IILgs_2(R)
where Dl denotes the generic derivative of order t° and
ql -
n
nq
q2 =
g
ngl gi n
=
nq 2a n
nq
qs-1
= n - (s - 1)q > n
Moreover, H,,,R[Dk-d¢] < const. max II D'`-8+1011 LgJ(R)
since 0 < a < s - (n/q) = 1 - (n/qs_1). Combining the above inequalities, we have the desired result. We have already used a special case of the corollary in connection with Hilbert space methods (cf. Lemma 3 of Chapter 3 and Lemma 3 of Chapter 4). The next two inequalities refer to the case where the integral defining the convolution considered is conditionally convergent. We shall call a function K(x)
- ixl )
a singular kernel if cu (x) is positive homogeneous of order 0 (w(tx) _
w(x) for I > 0) and is a C. function for x (22)
w(x) dS = 0
J IxI =1
0, and if
POTENTIAL THEORETICAL APPROACH
223
where dS denotes the area element on the unit sphere. (The smoothness condition on w can be considerably relaxed while the condition (22) of vanishing mean value is essential.) It is easily seen that (22) is equivalent to (22')
K(x) dx = 0
I
R1
holding for all RI and R2 such that O
R2 co(E) dS6fAI r-11 . r'i-1 dr
r
K(x) d x
.-
J I`I=1
JR1
Moreover, if M(x) is any homogeneous function of order 1 - n, then M,(x) = (alax;) M(x) is a singular kernel. For (23)
M,(x) dx = I R1
f
R2
ICI=R,
=R1
M( ) _' dSS = 0 R1
by homogeneity. We also note that by taking the limit as R1 -* 0 and R2 - oo in (22') we see that for any singular kernel
dx = 0
(22 ")
in the Cauchy principal value sense. It is not difficult to verify that for a Holder continuous function O(x) with compact support the integral K * 0 =f0(y)K(x -y) dy
exists as a Cauchy principal value:
f 0(y)K(x -y) dy = lim CIO
Jr--&I>e
0(y)K(x -y) dy
The theorems to be stated below will show that the expression K * 0 can be defined for much wider classes of functions. HOLDER-KORN-LICHTENSTEIN-GIRAUD INEQUALITY. If O(X) is a C.',
function such that O(x) = 0 for I xI > R then for 0 < x < 1 11K * ¢ II C2(R)
<- (const. depending on (o, n, x)
II 0 II C,"(R)
L. BERS AND M. SCHECHTER
224
The proof is given in Appendix I; it is an extension of Holder's classical proof of the continuity of second derivatives of a Newtonian potential of a Holder continuous distribution of matter. CALDERON-ZVGMUND [1,2] INEQUALITY.
Under the conditions of
the previous theorem and for q > 1 11K * O II
L,(.r)
<_ (const. depending on (o, q)
II
II
L,,t x
This inequality has been proved relatively recently, but has already found many applications to the theory of partial differential equations. The proof is given in Appendix I. We should remark that Calderon and Zygmund also considered more general singular integral operators of the form P(_y)H(x, x -y) dy
These play an important role in further applications to partial differential equations (cf., e.g., Calderon-Zygmund [3]). It follows from the Calderon-Zygmund inequality that K * u can be defined for u E L?(oe) and that the inequality holds again. Also, we see that the Holder-Korn-Lichtenstein-Giraud inequality holds for any (b c Q R) which vanishes on Ixl = R.12
We show now that the operator T,;0 introduced above [cf. (7 and 8)] is a bounded operator in the spaces C,,,_x, NV? and B;,, and that its norm can be made arbitrarily small by considering functions with sufficiently small support. Assume that the function O(x) belongs to C,,12(R) and vanishes near lxl = R. Assume also that the coefficients of the elliptic LEMMA B.
operator L considered belong to the space
Then
IITOO11C,_z(R) < const. R"
where the constant depends only on the ellipticity constant of the equation and on bounds on the moduli and Holder constants for exponent x of the coeficients.
12 In fact, it follows from the remark before Lemma 1 that the Holder-Korn-
Lichtenstein-Giraud inequality holds for all 0 E C',,(Ri. We shall not make use of this fact here.
POTENTIAL THEORETICAL APPROACH Remark.
225
We stated the lemma for To. But it will be clear
from the proof that a similar statement, with the same constant applies to T?p for any x0, provided we have over-all bounds on the
quantities (ellipticity constant of the equation and moduli and Holder continuity of the coefficients). The same applies to Lemmas C and D below.
For the sake of simplicity
Proof of Lemma B.
assume that
n > 3 is odd and that R < 1. Write
iV=(Lo-L)0=wi+V2 v,(x)
=j _ IPi -in [a,(0) - a,(x)}DPO(x) 1pi=m
2(x) = - Y a, 1P1<.111
0 we have that max I
Noting that Hence
I < const. R".
Ix1 `R
IIbpIIC'a(R) < const. R"
Hence, by (14), IIbPDP0IICQ(/C) < const. R"
and by the definition of the norm IIV'iIIca(I{) < const. R"-m II'ilcm,a(R)
On the other hand, I IIDp4IICauc) < const. R1 p.-in
so that (24)
("a (R) < const. R" -m II0I!Cm.a(R)
Now set X = Too, i.e.,
f
/.(x) = Jo(x --y)W(y) dy For Ipi < m, it is not difficult to show that DPZ(x) = jDio(x - ),\V( y) dy
II/IICm+,(I(
L. BERS AND M. SCHECHTER
226
Thus II DPxII co(R)
const.
Ilrm-n-IPI
* ItVI IIC0(R)
const. Rm-IPI I10IC,(R) < const. R"-IPI
and hence RIP' IID9xII co(R) < const. R" II0IIC,.(R)
(25)
If IpI = m, we claim that
f
DPx(x) = DPJo(x -y)V(y) dy + const. ip(x)
(26)
We first prove (26) for W(x) in C1(R). If IpI = m - 2, then by partial integration D,DkDP f Jo(x -y)V(y) dy = D, f DkDPJo(x -y')V(y) dy
= f DiDPJo(x -y)DkV(y) dy f -y)V(y) dy = lim f Iz-vI>e DkD,DPJo(x D,D"Jo(x -Y)V(Y)Yk dy - lim f Iz-YI=e = f DkD,DPJO(x -y)V(y) dy + const. W(x) = D, DPJo(x -y)Dkv(y)
dy
IO
e, O
where yk is the k-th component of the unit normal to the sphere Ix - yl = e. Next we note that DP J (x) is a singular kernel for
IpI = m. This follows from (23). Hence by the Holder-KornLichtenstein-Giraud inequality and (26) we have (27)
IIDPxIIC2)R) < const. IItVIIC(R)
By completion it follows that (27) [and hence (26)] holds for y, c- C. (R).
Combining (24), (25), and (27) and recalling the definition of the norm we have IIXIICm+i(R) < const. R" IIOIICm+z(R)
and the lemma is proved.
POTENTIAL THEORETICAL APPROACH LEMMA c.
227
Let the function 0 belong to B7,(R) and vanish near
I xI = R. Assume that the coefficients of the elliptic operator L considered are continuous (these conditions can be weakened considerably) and that
q > n. Then IIT00IIB-(R) < a(R) 11011 mIm
where a(R) denotes a junction depending only on the ellipticity constant of
Lo and the bounds and moduli of continuity of the coefficients of L.
Moreover, a(R) -, 0 as R -- 0.
Using the same notation and conventions as before, we have that Proof.
Rm-n/q
11IP111L°(R) S max IlbDllco(R) -
Rm-n/q I
IID9911L°(R)
IpI =on
< o(l) II/IIBm(R)
and Rm-n/v
const. max Ilapllco (R) IID9
11tP211L (R) °
x
JjxI
IPI
IIc0 (R)
cont. R
dx
o(1)
J
so that (28)
R--"'q IIWIII.°(1(>
C 0(1)
Applying the Calderon-Zygmund inequality to (26) we have for IPI =m (29)
Rm-nI4 IID1%11L°(R) C o(1)
For 0 < IpI < m we estimate q > n. We have that
II Dny II C,, (R)
using the assumption
I D"J (x) I C const. lxI m-n-IDI
so that for Ixi < R ID"A
const.
Irm-n-IDI
* Iv'l
I
1 /q'
< const. 11+P11L°(R)
)( 1
Ix
l IvI
< const. Rm-Ivl-nlq
IIWIIL,(R)
-yl(m-n-IDI)q' dy)
L. BERS AND N1. SCHECHTER
228
where q' = q/(q - 1). Hence (30)
const. R""I" IIl'IILQ(R)
RI PI
o(1) II0IIJ (R)
by (28). Inequalities (29) and (30) give the lemma. LEMMA D.
If the coefficients of the elliptic operator L are continuous,
then for each q > 1 IIToOlI
(R) c a(R) II0II.\m(it)
for all S e NN,(R) which vanish near jxl = R. The function a(R) -> 0 as R -- 0 and depends only upon q, the ellipticity constant of Lo and bounds and moduli of continuity of the coefficients of L.
The proof of Lemma D is very similar to those of the preceding lemmas and is omitted. 5.4.
Local Existence Theorem
As the first application of the tools developed above we shall prove an existence theorem in the small. THEOREM 1.
Let L be an elliptic operator of order m with Holder
continuous coefficients and f a Holder continuous function. In a sufficiently
small neighborhood of a point, say of the origin x = 0, there exists a solution u of the equation Lu = f having the following properties: (i) u has Holder continuous derivatives of order m. (ii) u and its derivatives up to order m - 1 have at the origin prescribed values. (This solution is,
of course, not unique.) Proof.
We first note that
1 on C,,(R) functions which
vanish on Ix) = R. For if 0 is such a function, we can find a sequence (0v} of Cr functions which vanish for Ixl > R such that
-'IIC2(R) -' 0
asv-kcc.Nowby(5) and by (26)
LOSOO, = 0.
r
LoSoc,(x) =J LoJo(x - y)O,(y) dy ± const. ¢v(x)
POTENTIAL THEORETICAL APPROACH
229
Hence by the Holder-Korn-Lichtenstein-Giraud inequality in
LOS00, - LOSOO
C. (R)
giving LOSOO = 0
Let fi(x) be a Co, function which is identically one in IxI < 1/2,
vanishes for IxI > 1. Then by (15a) II
S(R-Ix)
11C'm+x(R) = II y(x) 11Cm+a(l)
Set
Mou(x) = Toq(x)u(x)
where n (x) = (R-'x) . Then by Lemma B, Mo is a bounded linear operator which maps C. ,(R) into itself. Moreover its norm 11Moll = l.u.b. {IIMou11Cm+x(R)/Ilullcm+«(R)}
less than const. R« II fi(x) Ilem+«ci) and hence can be made arbitrarily small by taking R small. If IIM011 = e < 1, then the "geometric series" 1 + M0 + Mo- + ... is dominated by 1 T e + e2 ; ... and is therefore easily seen to converge to a bounded operator is
x
(1 -Mo)-1=IMu j __o
which maps Cm+x(R) into itself, satisfies the relations
(1 - MO)-'(1 - M0) = (1 - Mo)(1 - Mo) -1 = 1 and has norm II(
1-M
0)-1II
OIIs
<1 +e+e2
j=0
1
1-e
Set
g(x) = I Y,,xp 1A --11,
where the yp are constants. Clearly Log = 0. If
u = (1 - K) -'g we claim that Lu = 0 for IxI < R/2. Indeed we have
u-Mou=u-S0(Lo-L)i7u=g and hence Lou - LOSO(L0 - L),qu = Lou - Loqu + L,1u = 0
L. BERS AND M. SCHECHTER
230
Since 77(x) = 1 for Ixl < R/2, u is a solution of Lu = 0 near x = 0. Also
Ilu -
IIM01I
s IIMoII 11(1 - Mo)-'1I IIgIIc,.+2(R)
<
IIgIIc.+,(R)
1
Recalling the definition of the norm II IIC,a+,,(R)' we see that this implies that
IDiu(0) - YPI s Ce' I
IPIsm-1
IyDI
for IpI < m - 1
where C is fixed constant and e' can be made arbitrarily small by making R sufficiently small. Now denote by u(PO) the solution just obtained for the following choice of the constants Y,,: yDO = 1, y,, = 0 for p : po. Every linear combination
u = I cDu(P) 1PI _-M- 1
is a solution of Lu = 0. We want to determine the constants c,, so as to satisfy the equations I cDDQu(P)(0) = YQ (IqI < m - 1) IPI<-m-1
i.e., as many equations as there are unknowns. We know already that by choosing c,, = yp we can satisfy these equations approximately (with an error not exceeding Ce'). Hence, by a well-known theorem in linear algebra, the equations are solvable.
The theorem is now proved for f = 0. In order to settle the general case it suffices to exhibit one solution of Lu =f which belongs to C,n+,,(R). (We assume that f c- C,,(R).) By our reasoning above it suffices to exhibit a solution u of the equation
u = Mou + S0f But if IIMoII < 1 such a solution is given by u = (1 - Mo) -'Sof 1. THEOREM 2.
Assume that the coefficients of the elliptic operator L of
order m are continuous and f is a function belonging to L q > 1. Then
POTENTIAL THEORETICAL APPROACH
231
there exists a solution u of the equation Lu = f which belongs to the space NN(R) for small R and satisfies condition (ii) of Theorem 1. If q > n, we can find such a u in B qm (R)
.
The proof is the same as the one given above, except that Lemmas C and D are to be used instead of Lemma B. The continuity assumption on the coefficients can be considerably weakened. An interesting corollary of Theorems 1 and 2 is the fact that by
introducing new dependent and independent variables an elliptic equation can be transformed, locally, into a form in which the (undiferentiated) unknown function and its first derivatives do not appear explicitly.
In particular, every homogeneous second order equation can be written, in the small, in the form
Iail
a2
=0
In order to prove this let 0o be a solution of Lq = 0 such that 00 = 1 at x = 0. In the neighborhood of the origin introduce y, _ (0/00) as a new unknown function. Then tp satisfies an elliptic equation Liy, = 0 and since y, = 1 is a solution, the coefficient
of y, in Li is zero. Next, let V,, j = 1, 2, ... , n, be a solution of LIV = 0 such that 1
any;
aXk
I
z=o
0
if j = k if j Ok
and y,, = 0 at x = 0. We may introduce ; = V, as new independent variables and obtain an elliptic equation L2W = 0. This equation is satisfied by every linear function. Hence the coefficients of y, and all first derivatives in L2 are zero.
5.5. Interior Schauder Type Estimates
Some of the most important applications of potential theoretical methods are a priori estimates of solutions of linear elliptic equations of the type given by Schauder for second order equations and then extended by other authors to higher order equations and systems.
L. BERS AND M. SCHECHTER
232
Schauder assumed the equations to have Holder continuous coefficients, but using the Calderon-Zygmund inequality the method can be extended to continuous coefficients and even to more general situations. In this section we prove two theorems on so-called "interior" estimates. Let L be an elliptic operator of order m with coefficients in C«(9). Let 9 be a bounded domain contained, together with its closure, in .9. Let u belong to C,,,+«(W) . Then, for every compact THEOREM 3.
subset To e (31)
Ilullc.,«(T0) S C(ItLullc«(,,) + Ilullco(f))
where the constant C depends only on m, W, WO, the ellipticity constant of L and bounds for the C«(Wl) norms of the coefficients of L.
If
and WO are given, we can cover WO by a finite number of open balls whose closures belong to W. Hence it suffices to prove
the theorem for the case where 9 and 90, are concentric balls of small radius. Without loss of generality we assume that the center is at the origin and prove Theorem 3 in the form (32)
C 1 - Rr)-' (IlLullc«(R) + Ilullco(R))
for 0 < r < R and R sufficiently small, where T The proof is based on the preliminary estimate n-«
(33)
IIUIIC..+«(R,) < Cl
1 - RZ
(IILuIIc«(R2) + Ilullc.-,T«(R2))
holding for all Rl and R2 such that 0 < Ri < R2 < R. Assuming (33) for the moment, set
A = sup
0
r'
R Il u 11 c
There exists an R1, 0 < Rl < R, such that A<2 1-
R-') IIUIIcm+«(R,)
POTENTIAL THEORETICAL APPROACH
and using (33) we have, for R1 < R2 < R -Rlna-a{IILuIIC(R,) <2(1 -RR-1)TCi(I
A
+ IIUIICm-,+a(R2) 2
<
-
2C1(1
-m-={IILuIICa(R)
1 - R2)
R11*I
+ lUll Rl
<
in-a
- W 1 - R2)
2C1(1
[by (lib)]
_l+acR2)}
{IILuIIC«(R)
(34) + E IIUIIC,n+a(R2)
<
-
2C1(1
Km+aE-(m-1+a)
-
R-Ri1T(1
(by Lemma 1)
II U II C0(R,)}
RR_1)-m-aE(1
-
RR21A
21
Rilr
Ri
m -a
IILuIIC«(R)
RI (1
R2) l-m_a 811(1-T2)
xr»+a
F-(m-l-a) IIUIIC0(R)
/
Set d = 1 - R1/R and choose R2 and E as follows : R2 1-R
b
2,
e=
2-2-m-2-*6m-)-XjC
1
Then
2<1-R
0
0<e<1
2
and
2C11-R
( so that by (34) A S 4C1(1
- R1
+ 4C1(I
'-Rl /
(1
-
R1 =
R) (
(1_Ti)e<
2
m-allLullCa(R) R-1)
R2 Y R1'
1
- m-z
R 2)
km+x
E-(m-1+s)
< 4C12(m+(x)(=+2) km+z (llLullca(R) + IIUIIC0(R))
IIUIIC0(R)
L. BERS AND M. SCHECHTER
234
It remains to prove (33). We choose R so small that (35)
IIToOllc.+«(R.) < k IIOIICm+,(R2)
for all R2 < R and 0 with support in Ixi < R2 (cf. Lemma B). Let cu(t), 0 < t < 1 be a fixed C. function such that w(t) = 1 for t < 3 and w(t) = 0 for t > I and set, for 0 < Rl < R2 for IxI < Rl
1
for R1 <x
R)
2
2
One verifies at once that there is a constant Co such that (36)
II IIC.+«(R2) < CO(I \
- Ri 2
4u. Then the support of 0 is in Ixl < R2 and by
Now set Lemma A
= Too ± SOLO
By (35) this gives II O II Cm+«(R2) < 2 it SOLO II c,.+0C(R2)
Now by (26) and the Holder-Korn-Lichtenstein-Giraud inequality IID'SOLOIIC(R2) <- const. IILOIIC«(R2)
for IpI = m while for IpI < m lrm-n-IPI
ID-'SoLOI S const.
* ILOI
< const. Rm-1111 JLcl
Hence (37)
IISoLMIICm+«(R2) < const. Rm IILOIIC«(RE)
But
Ld=CLu+N
where N is a linear combination of derivatives of u up to order m - 1 multiplied by derivatives of C up to order m, a derivative of u of order k being multiplied by a derivative of of order m - k. Hence Rm II J
it
C«(R,)
< const.
II
II
Cm+Z(R2) II u II Cm_,+«(R2)
POTENTIAL THEORETICAL APPROACH
235
and therefore R", IIL0IIc«(R2) < const.
(IILuIIc,,(R2) + IIUIIcm_1+a(R2))
Combining this with previous inequalities we obtain const. IIVCm+.(R2) (IILuIIcx(R2) + Ilullcm_,+z(R2))
Since 0 = u for IxI < Rl Ilullcn+«(R,) S
const. II IIc,p+a(R2) (IILukkc,,(R2) + IIUIIC,._j+,,(R2))
which is the desired result in view of (36). We consider next equations with continuous coefficients. In the same way as Lemma B implied (33), Lemma C leads to the preliminary estimate (for q > n) (38)
IIUIIR",(RI) <
Cl(i
-R
Yin
(IILuIILQ(R2) + IItIIRm-1(R2)) 2
From here we obtain, exactly as before, the inequality II1IIRm(t)
C(1
- R)
m2
(IILuIIL,,(R) + Ilulle"(R))
and thus we have Let L be an elliptic operator of order m with continuous coefficients defined on the closure of the bounded domain 9. Let 9o be a THEOREM 4.
compact subset of K If u belongs to Bn, (0), with q > n, then IluiIIim(fgo) s C(IILuIIL,(T) + IIUIIe0(1))
where the constant C depends only on m, !F, To, the ellipticity constant of L, and bounds and moduli of continuity of the-coefficients of L.
We do not state a corresponding theorem for the N--norm, since we shall not need it. 5.6.
Estimates up to the Boundary
There exist also estimates "up to the boundary." We state here the second order case for the Dirichlet problem which is due to Schauder [1,2]. The corresponding result for equations of higher
L. BERS AND M. SCHECHTER
236
order and more general boundary problems has been proved by Agmon-Douglis-Nirenberg [1]. An (n 1) dimensional hypersurface in n space is said to be of
-
class Cx +z if every sufficiently small piece of it can be mapped into
a piece of a hyperplane by a transformation of coordinates y(x) with positive Jacobians, the functions y = ( 31 , . . . , y,(xl,
... ,
being of class Ck
.
Let T be a bounded domain with a boundary hypersurface W of class C2;a. Let the coefficients of the second order elliptic operator L belong to C,,(V). Let u be a junction belonging to C2+,, (T) . THEOREM 5.
Then (39)
Ilull(.2+M(1) < C(IlLullca(1) + Ilullc2+Yc%4) + Ilullcoc9))
where the constant C depends on p, the ellipticity constant of L and bounds on the Ca(lf) norms of the coefficients of L.
The method of proof is similar to the one described above, except that one must also use explicit formulas for solving the Dirichlet problem for Lou =f in some standard domain. For the second order case, this is relatively simple and we sketch the procedure. We first note that may be covered by an interior subdomain and a number of small "boundary patches." By hypothesis, each
boundary patch can be mapped in a one-to-one way onto the semisphere Ixl < R, x,, > 0 with the boundary being mapped into
the hyperplane x = 0. Moreover, the mapping and its inverse have Holder continuous second derivatives. One then observes that by subtracting an appropriate function from u, it suffices to prove (39) for functions which vanish on 47. Thus it suffices to prove (40)
Ilullcp+a(s,) < C(1
r)-7 R
(IlLullca(ER) +
for r < R and functions u which vanish on x = 0, where 1. is the semisphere Ixl < r, x > 0. This is done, as in the previous case, by proving (41)
Ilullc+a(='R1) < C,1(1
R R2)
2-z
(IILuile2(r-RI) + Ilu11c,_=(ER:))
for all Rl and R2 such that R1 < R2 < R.
POTENTIAL THEORETICAL APPROACH
237
In order to prove (41), we first note that
f
W(x) = Lo W(y)Jo(x -y) dy
holds for W E Ca(R) vanishing on Ixi = R (cf. the proof of Theorem
1). Now if O(x) is a C,., function in >;R which vanishes on x = 0 and near Ixj = R, we can extend it to xn < 0 as an odd function of xn. Thus O(x) will be a C1(oo) function vanishing for Ixi > R and satisfy
0(x') = -0(x) where x' is the image of x under reflection with respect to the plane
xn = 0. We can therefore write
J (y)Jo(x -y) dy =U0(y)Jo(x -y) dy n>o
+f where
n
0(y)J(x -y) dy
Jo(x,y) = Jo(x -y) Set
=f R`o 0(y)Jo(x,y) dy
- Jo(x -y')
S0'0 (X) = f,vn o0(.y)Jo(x,y) dy
To = S' (Lo
- L)
Then we have, by partial integration,
0=
S0L
provided 0 vanishes on x = 0. The proof now proceeds as in the "interior" estimates. One shows that an inequality of the Holder-Korn-Lichtenstein-Giraud type holds for kernels of the form D"J (x) for Ipl = 2, i.e., s const. II0IIC%(
R)
IpI = 2
The proof then goes through with little change. 5.7. Applications to the Dirichiet Problem
In order to apply the Theorems 3 and 5 one must keep in mind that the constants in these estimates depend only on the domain
L. BERS AND M. SCHECHTER
238
considered, the ellipticity constants of L and the Ca norms of the coefficients of L but not on the particular operator L considered !
As an application of the Schauder estimates we will solve the Dirichlet problem for the second order equation 2
(42)
Lu =
a,,(x) ax ax i
+ I a,(x) ax
+ a(x)u =f(x)
in a domain T satisfying the conditions of Theorem 5. We assum that f and the coefficients of L belong to C2+x M and that a < 0,
so that the maximum principle is applicable. We consider first the boundary condition (43)
u=Oon1,
0eC2+(V)
We assume also that we know how to solve [in C2+ac(g)] the corresponding Dirichlet problem for the Laplace equation. (This can be done, for instance, by the method of integral equations or the methods of Chapter 4.) The proof (Schauder [1,2]) is by the continuity method. Assume
first that g - 0. For 0 < t < 1 set Lt = (1 - t) A + tL and denote by T the set of those values of t for which the Dirichlet
problem Ltu =f, u = 0 on 0 has a solution in C2.+2(T) for all f in Ca (T) . We already know that t = 0 belongs to T. We shall show that r is open and closed. This means that r is the closed interval 0 < t < 1, so that the homogeneous Dirichlet problem is solvable for t = 1, i.e., for L, = L. Proof That T Is Open.
Assume that to belongs to T and denote
the solution of Lt u =f, u = 0 on
by Mf. (This solution is
unique by the maximum principle.) Clearly M is a linear mapping of Ca (q) into C2+ (T) and by Theorem 5 we have that (44)
IIMf
C IIf II Cr(y)
The equation Lto+,u =f may be written in the form
Ltou =f + E(O - L)u so that a solution of the equation (45)
u = EM(L - L)u + Mf
POTENTIAL THEORETICAL APPROACH
239
would be the desired solution of the homogeneous Dirichlet problem for Lto+Eu =f.
Consider the linear operator N = M(i - L). Clearly 0 - L maps C2+«c91> into C
)
and
II (o - L) u ll c« (9) < const. II u II c:+«(V)
so that N maps C2+x (T) into itself, and by (44) IINII = l.u.b.
IINuIIra+s+«(g)
Ilullc
Equation (45) is equivalent to
(1 - sN)u = Mf and if IEI < K-1 it has a solution
u = (1 - sN) -1MF = I E'N' Mf 0
Hence to + e belongs to -r if lel is small enough. Proof That r Is Closed.
Assume that t j = 1, 2,
.
.
.
,
belongs
to -r and tf -* to. Let 0 be the solution of L,.u""> =f, u(') = 0 on #. By Theorem 5 IIUW IIc:+«(g) < C1
(a constant independent of j )
By Lemma 2 we may assume, selecting if need be a subsequence,
that u(' converges uniformly, together with its derivatives of orders 1 and 2 to a function u. Clearly u belongs to C21 «(g), vanishes on the boundary and satisfies the equation Ltou = f. Hence to belongs to -r. In order to solve the nonhomogeneous boundary value problem (43) we denote by h the harmonic function satisfying this boundary condition and solve, as we now know how, the equation
Lv=f,
f=f-Lh
under the boundary condition v = 0 on 0. Then u = v + h satisfies condition (43) and the equation Lu = f. Consider next the boundary condition (46)
u = 0 on (,
0 E Co(9)
L. BERS AND M. SCHECHTER
240
We construct a sequence of functions 0; defined on
j
II0;
such that
- OGc'o(4) - 0
(For instance, take the 0; as polynomials.) Let us be the function in C,,. (6) determined by the conditions Lu, =f, u,
on.
By the maximum principle u; converges uniformly to a function u. Thus I'Uj - uIlc-o(rs)
(47)
0
so that u c- Co(11) and u = cb on V. On the other hand, by Theorem 3 for a compact subset 10a e G°, II u; - u, II c$+a(vo) S const. 11u, - ut 1, ca(1o)
the constant being independent of i, J. It follows from (47) that u has Holder continuous second derivatives and satisfies the
equation Lu =f. In a similar way, using the estimates in Theorem 4 and approximating the coefficients of (42) by nice functions one can
show that if L has continuous coefficients, the equation Lu =f has solutions with continuous first derivatives and generalized LQ derivatives of second order in every subdomain. 5.8.
Smoothness of Strong Solutions
We now give a proof of the theorem on continuous differentiability of strong solutions (cf. Section 3 of Chapter 1). Assume that f and the coefficients of L satisfy Holder conditions with exponent x. Let u be a function having strong derivatives up to order m in
L. for some r > 1 and suppose that Lu =f almost everywhere. We shall prove that the m-th order derivatives of u satisfy a Holder condition with exponent x and hence u is a classical solution of Lu =f. Since the theorem is local in nature, we need only to prove it in the neighborhood of some point, say the origin. Let fi(x) be a Cx function which is identically one in Ixl < 1 and vanishes for lxl > 2. Setting r7(x) = (R-'x) we have by (15a) (48)
1117 IIc,(2m = II 0C,(2)
POTENTIAL THEORETICAL APPROACH
241
for any s. For v c- Nn(2R), r > 1, set Mov = T077v
Now by Lemma B, (14) and (48), we have for v e Cm+,,(2R) II T0nv I I Cm+:c (2R)
< const. R' II v II Cm+cx (2R)
Similarly, one sees by Lemma D, that for v c- NN(R) 0(1) IIvllA-m(2R)
as R - 0. Hence we may take R sufficiently small so that (49)
Il Mov ll Cm+a (2 R) `= z II V I1 Cm+%(2R)
and
(50)
11M0vll \m(2R) <
Ilvll
Now let Ri be any positive number less than R and let 0 be any
test function in lxl < R which is identically one in lxl < R1. One easily verifies by means of the Calderon-Lygmund inequality,
that Lemma A holds for functions in Nm(2R) having compact support (for a similar argument see the proof of Theorem 1). Thus cu = M0 u -1- S Lou since Ou = Oqu. Now by (50), 1 - M. has a bounded inverse in Nm(2R) (cf. the proof of Theorem 1). Hence ou = (1 - MO) -1S0LOu
Now Lou = Of 4- Liu, where L1 is of order less than m and has
Holder continuous coefficients. Now if r > n, we see by the Sobolev-Kondrashov theorem that L1u satisfies a Holder condition
with exponent fi < 1 - (n/r). If fi > x, we have Lou E C,,(2R). Hence, by the Holder-Korn-Lichtenstcin-Giraud inequality S0Lou is in Cm+ (2R) . But by (49), (1 - M,)-1 maps Cm+.x(2R) into itself and hence Ou is in Since 0 = I in Ixl < R1, we have u E C,,,+2(R1) and our assertion is proved. If j3 < x we know that Lou is in CC(2R) and hence SOL6u is in C,,,_g(2R). By taking
-
R sufficiently small we know that (1 M0) -1 maps C,.+,, (2R) into itself. Hence u r= Cml,3(R,). But this implies that Liu is in
L. BERS AND M. SCHECHTER
242
for any
C,,(R1). Hence by our reasoning above, u E
R2
-
Kondrashov theorem asserts that Llu E L,1(2R), where r1= nr(n r)-1. Hence LSu e Lrl(2R). Applying the Calderon-Zygmund in-
equality we see that S0Lciu e Nm(2R) into itself. Hence 6u E N», (2R) and u E A,, (R1) . If rl > n, the argument above then completes the proof. Otherwise, we note that L1u E Lr2(R1), where r2 = nr1(n - rl) -1 = nr(n - 2r). From this it follows that u e N,., (R2)
for R2 < R1. If k is the largest integer such that kr < n, we repeat the procedure k times to obtain u e NN(Rk), with rk = nr(n kr) > n. We then know how to proceed to the final result.
-
Appendix I. Proofs of the Fundamental Inequalities
We shall outline proofs of the three sets of inequalities stated in Section 3 of Chapter 5. 1. Sobolev's Inequalities.
The proof of (a) is trivial. In fact, by
Holder's inequality Ir--A
*
('
If Ix -y1-'o(y) dy
-yl-aq dy)
Ix
I
IyI
S const. R("17') -2
l f q'
110 II L0(f)
where
q'=
q
q-
1
n -,1=nI -1 -n) >0
and
q
q
We base the proof of (c) on a one-dimensional inequality of Hardy and Littlewood [1] (cf. also Zygmund [1]) which states that for f c- Lq,( - oo, oo) and g e L,, ( - oo, cc) x
-- 11-1f(s)g(t) ds dt fHs x
xx
< const.
'
If (s) I'1 ds) -
when 1
q>I'
p>I'
I +>I,
p
q
/F
14
l .j
Ig(t) 1-11 dl I
x
2-I
/
POTENTIAL THEORETICAL APPROACH
243
In order to prove (c) it suffices to show that I(r-i * 4, )I < const. II¢IILQcx)
(1)
II
IILQI(Go)
for all 0 and p with compact support, where qI =
ql f
(qi -I).
We make use of an observation due to Du Plessis [1] that
IxI-a < const. rl
(2)
Ix;1-a1fl
This follows from the fact that an arithmetic mean is greater than the corresponding geometric mean. Thus
fix; n j -i
?IIIxjI21"
j=1
which yields (2). Employing (2) we have I (r' * 4', V) I < const. I j I xj - yf l -'In 10(x) I I v (y) I dx dy. j=1
Setting p = qi and It = A f n, we apply the Hardy-Littlewood inequality to the integrations of x1 and y1. We then reapply it to each pair of integrations x5 and y; eventually obtaining (1). In order to prove (b) we note that if 0 is in Lq,(R), then it is in L,,(R) for all p < q. Let q2 be so large that q, < q2 and 1 + I < 1
(3)
Setting 1
_
P
we have q >p > 1 and
I
q2
llr_a * 4'IIr,Q,(R) < const.
=
1
q2
1
p
+
1
+q A
- 1. Hence by (c)
n
n((i141)-(1/q2)] 1jr-A * IIc.Q=cx)
< cont.
cont. R11171
IILQ(R)
Remark. Note that under assumption (3) we have proved Ilr-' * for case (b).
const.
R"QE
II4JL9(R)
L. BERS AND M. SCHECHTER
244
2. The Holder-Korn-Lichtenstein-Giraud Inequality.
We first note
that by (22") of Chapter 5,
f
K * 0 = [O(y)
(4)
- O(x)]K(x -y) dy
-
Let x' and x" be any two points in IxI < R and set 6 = Ix'x"I.
If ,=K*0,wehave w(x')
- '(x")
=f {[O(y) - O(x')]K(x' -y) =I1+I2
- [O(y) - sb(x")]K(x" -y)} dy
-
where Il is the integral over the sphere 01: ly x'I < 26 and I2 the integral over the domain 02: ly - x'I > 26. We note that if
y2 isIx"in-yl 02 and x' is such that Ix' - x'I S 6, then Ix' - yI < s 4 Ix' -yl. Now II11
<- H.,x[0] fa { IK(x' -y)I Ix' -yl' 1
(5)
cont. &H«,R[f]
+ I K(x" -y) I Ix" - yi"} dy
Also, (6)
- O(x')]K(x' -y) dy
I2 =fo [0(x")
+f !2
[O(y) - O(x")][K(x' -y) - K(x" -y)] dy
Now by (22') of Chapter 5
K(z) dz = 0 L>20 and hence the first integral on the right hand side of (6) vanishes.
Moreover, by the theorem of the mean, there is a point x' between x' and x" such that
IK(x' -y) - K(x" -y)I <
const. 6 1x
_yln+j
Thus ify is in 02 (7)
I K(x' -y) - K(x" -y) I <
cont. 6 Ix,
-y19+1
POTENTIAL THEORETICAL APPROACH
245
Employing (7) we have II21 S const. 6H«,R[9] (8)
f'_ ly -
x'l"-n-I dy
< const. 6"Hx,R[0]
The result now follows from (5) and (8).
Our proof is based on a
3. The Calder6n-Zygmund Inequality.
classical theorem of M. Riesz [2] (cf. also Zygmund [1]) which asserts that for q > 1 f- f(t) dt q dT S const. f 00 1 f(t)IQ dt J-oo j_ t-T -oo °°
I
Clearly, Riesz's theorem is just the Calderon-Zygmund inequality
for n=1. Assume first that the singular kernel K(x) in n space is odd:
K(-x) = -K(x). We carry out the integration in the integral defining K * 0 first along the radii and then along the surface of the sphere. Thus
K*0=
fo(y)K(x -y) dy =
fo(x - z)K(z) dz
= lim f _ico() dSS f O(x 00
rn-1 dr
eI
where r = IzI. In view of the oddness of K(z) [and hence of co(z)] the integrals along two oppositely directed radii can be combined into a single integral. Hence K*
=
and
I K* l a dx < const.
f
00
aj (E) dSf
f;i=1 dSJ J
O(x - r)r-1 dr 00
f(x - r) r'-1 dr
I
dx
Since r-1 is a one-dimensional singular kernel, we have, by Riesz's theorem, that for fixed
r
I (x + t - r )r-1 drl
J-
a
dt < const.
f I0(x + t )IQ dt
L. BERS AND M. SCHECHTER
246
Integrating along axes perpendicular to
l
gives
Q
dr
O(x -
fo.
dx < const.J I0(x) I,, dx
which, when combined with the preceding inequality proves the theorem for odd kernels.
If K(x) is a singular kernel, let us denote by K the operainto the function K * 0. We tor which takes the function consider the particular singular kernels (Riesz kernels) K,(x) = cxjxl-n-1, i = 1, 2, . . . , n and note that for an appropriate choice of the constant c, 1Q2 + ... + ten = 1 = identity. To see this we observe that K. (X) = c1
a
i = 1, ... , n
Ixl'-n' ax2
Thus by (4) ci
f 10 (Y) - 0 (x)] ax Ix -yi1-" dy
Now by partial integration we have for R1 sufficiently large,
f
i
[0(y)
- fi(x)] ax Ix -Y11- dy ,
a
JF<,v-zl
ay
I
x -'-nd yl y + ( x)
Ix
v-Z,=Rj
'
J- O(x)]
=d -Y1" X'
_ yly dy
The first boundary integral vanishes identically while the second tends to zero as r -- 0 by the continuity of 0. Hence
10 = ci
aa(3) y=
Ix
-_yl'-" dy = ci Izli-n
z)
ao (X
dz
,
Hence (9)
iV = clr'-" * OX =
C1
(ri-n
*)
for test functions,. Now by Sobolev's inequality and the usual limiting procedure, we see that (9) holds for a Cam, function having first derivatives in Lq(oo) for some q < n. But the same inequality
POTENTIAL THEORETICAL APPROACH
247
tells us that v = X,0 is such a function when 0 is a test function. Therefore, assuming for the moment that n > 2, we have
f
Ci
Izll-n
aR1 (x + z) dz ax,
J
c1J IZI1-n axt - J IUIi-n a0(x axz, + u) du dz
+ z + u) du dz _ C f Izll-n lull-n a2o(x ax2 Izli-my-zll-n a2o ax+ y) dv dz
= GIJ
_c
f
F(v)
a2¢(x
+ v)
ax!
where F(v)
=f
Izi1-n Iv - zIl-n dz
Clearly F(v) is positive homogeneous of degree 2 - n and invariant with respect to rotations about the origin. Hence F(v) = const.
IvI2-n
This means that X20 ,
Therefore
a2o (x
const.
+ v)
dv
J1v12-n
f
1X?¢ = const. IvI2-n 0O(x + v) dv which equals const. 0 by (1) of Chapter 5. When n = 2 the interchange in the order of integration above
is not justified by the behavior of
Izll-n Iv - zI1-n at infinity.
However, by being slightly more careful we can arrange things so that only finite integrals are interchanged. Assume that O(x) vanishes for IxI > Ro. Then by (9) we have (10)
a
Cl ai
(r-1 * K,O)
L. BERS AND M. SCHECHTER
248
But
r-1 * 1Q, = c1 lim fIzj R-. x
J'vl < Ro
a-0
= c1 lim
(11)
R-- x flyl < Ro ay:
-cl f
Ix
IzI-I
lim
f
+z
dy dz
a1',
IzI-1 Ix + z -yI-I dz dy
=I
zI
a
IzI-1 Ix + z - yI -1 dz dy
lyl
Now consider x, y, z as complex variables and set t = x -y, Z = z/t = pe'o. Then
f I
IzI-1 Ix + z
f2,,f,R/Itl Itl2 p dp dO
-y1-1 dz
ItZI Itz - tI
U
J'"Jo R/IrI
dp d
I pe'o - 1
o
Hence
f
IzI-IIx+z
ay, fzI
-yl-Idz=
f
2,r
U
dO
R e io
R
f
-1
l
2
Itl
t;
f21r
t2 211
de
eto_Itl R
and
Jim
Jfzl
ay,
z -yI-1 dz = -Tr t=2
Substituting back in (11) we have
f
r-1 *K,O = const. o(y)
xt -Y,
Ix -yl
dy
f y3 const. f a' log Ix - yI dy ya
const. 0(y) a log Ix - yI dy
Hence by (10) L20
= const.
0 log Ix -yI dy
fay,
t, ItI3
POTENTIAL THEORETICAL APPROACH
249
and R2)
f
= const. 0# log Ix -yl dy
2
= const. 0 by (1') of Chapter 5, and the assertion is proved for n = 2 as well. Now let K(x) be a given even singular kernel [K(-x) = K(x)]. We have that k = Z (IMI)1e=. If we can show that R i may be written as M where M=(x) is an odd singular kernel, then (L1 ,) and Xj will both be bounded in the L, norm by the result obtained above. This will then imply the Calderon-Zygmund inequality for K. Since every singular kernel can be written as the sum of an even and an odd kernel, the proof will be complete. It therefore remains only to show that kki = M=, where M,(x) is an odd singular kernel. Now R.0
= cifIx -Y1'-" aa(y)
dy
y=
and therefore
ciJ K(x -_)f aazz) 11.), - Z11- - Ix - yll-'=] dz dy = ci
r I
az
M(x
- z)
dz
where (12)
f
M(x) = K(x -y)
[Iyi'-n - Ixli-n] dy
Obviously, M(x) is positive homogeneous of degree 1 - n. Hence, for x 0 0, it will follow from (23) if we can show that it is in
of Chapter 5 that M=(x) = (a/ax;)M(x) is an odd singular kernel. In addition, for Ri sufficiently large, we shall have R=>lx-zl>a
M(x - z) aazz) dz =
M, (x - z) [0(z) - 0(x)] dz
Rl>Ix-zl>a
- z - (x) JIx-zh=Ri M(x - z) x' dz Ix -zI _
fix-ZI-8
M(x - z)[O(z)
- fi(x)] X,
z
Ix
zI
=
dz
L. BERS AND M. SCHECHTER
250
The first boundary integral vanishes since K(x) (and hence M(x)) is even. The second tends to zero as e -> 0. Thus it will follow that
9'0 = C,
[O(z) - O(x))M1(x - z) dz
and the proof will be complete. It therefore remains to show that M(x) is in C,,, for x 0 0. This is immediate if one writes (12) in the form M(x)
=fzj< (Ix -
IR,
Z11-n
- Ixll-n)K(z) dz
Iyil-n K(x -y) dy
z-vl>R,
where R1 is some fixed positive constant less than I xJf2. The function Ix - zIl-n is in C" for IzI < R1 and vanishes for z = 0. Hence we may differentiate under the integral sign. Since K(x -y) is in C., for Ix -yi > R1, the same is clearly true Ixll-n
for the second integral as well. This completes the proof. It is obvious from our proof how one may relax considerably the smoothness assumption on &)(x).
Appendix II.
Proofs of the Interpolation Lemmas
We now give the proofs of Lemmas 1 and 3 of Section 5.2..
Assume that 9' is contained in some cube with sides of length 2R. Since we may take Ou to have compact Proof of Lemma 1.
support, we may assume that 9' is the cube. Thus it clearly suffices to prove IIOIlc,((l) <_ E IIOIIc,(
') +
ktE-8/t-8
for functions 0 c- C,(9'). Setting M. = II O II
we first note that for 0 < r1 < r2 < r3
the inequalities M1.1
for all 0 <E < 1
< EM,, +
A-1,= < EAIra
+
-'2 MO
for all 0 <e < 1
POTENTIAL THEORETICAL APPROACH
251
imply
Mr, < PM,,, +
k12P-r,,r2 -r1Mo
k23a-ra/ra-rsM0)
P((,TMr3 +
+
k12P-rlrra-r,Afo
for 0 < p < 1, 0 < a < 1. Setting p = Era-r1/ra-r1 and a = Era-hlra-rl'
we have Mrl < EM,, + (k12 T k23) F -r,lra rtMO
Hence it suffices to prove that for every integer k z 0 and any a satisfying 0 < at < 1, there is a constant Ck+a depending only on k + a such that (1)
Mk+13 < EMk+x +
Ck+aE-k+Q1a-/3M0
for 0 < fi < a and 0 < e < 1. Moreover, we claim that (1) follows by induction if there are constants Cx+a such that (2)
Mk+p < EMk+2
i
Ck+aE
-fl/0-0
Mk
0 <#
Mk < EMk+x +
0 <E < 1
0<E<1
Ck+aE-11aMk_l
for k 0 and 0 < a < 1. For if (1) holds for all k < j and (2) and (3) hold for all k, then Mi PMt+a + Ci+aP-11aMi-1 C)v-ti-llMo)
PMs+a + Ci+aP-"(aM, +
(where we have employed (1) for k = j - 1, fi = 0, and a = 1). Taking p = e/2, a = E11a/2'1a+1C;+a, we obtain (1) for k
fi=0.By(2),if8>0 Mf+r3 < PM,+a
-«P -t1/a
-1_ C'
C
s
-rzla- QM
+C
v-i1%M
by the case of (1) just proved. Taking this time p = E/2 and a = e1( a-O)/21+#/(a we obtain (1) for k = j and fi > 0. Since (1) is equivalent to (2) for k = 0, the induction is complete. It therefore remains to prove (2) and (3). If x is any point of T, let x' be another point such that the segment xx' is along the x, axis in W, and set b = Ix - x'I. Then by the theorem of the mean there is a point . such that
IDik-10(x) - Dk-10(x')1 = Ix - x'I ID (. )I
L. BERS AND M. SCHECHTER
252
Moreover,
Ix - -41
D' 0(x)
Hence Dko(x) I < 8"Hx
I
I
< b"H".
2
IID; -'
IIco(s')
Since 2R is the diameter of T', we have by summing over i M,t < nR-" 6"Mk+" ± 4nR 6-1M.-1 Since e < 1, we may pick x' such that b"n = ER". This gives (3). In order to prove (2) we let x and x' be any two points of V'. Then by definition,
- D'O(x')I
D'O(x)
Ix -x'10
< Ix -
x'I"-iH",,i-[Dko]
where Dk is any derivative of order k. This implies '-P < e. Otherwise RO -" Ix - x' I ID ko (X) Rk+,6
(2) if
- Dko(x') I < 2M ak
Ix - x'I1
which gives (2) for this case as well. Thus the proof of Lemma 1 is complete. Proof of Lemma 3. By the nature of the hypothesis and inequality (20) of Chapter 5 we see that we may assume that the diameter of ig is one. Moreover, by Lemma 1, it suffices to prove 1 IID"0IIL,(9)
II0II 'toy) +
CE-;'k-j
II0IIL,(f)
IPI =i
We first consider a Ca, function f (t) of one real variable having compact support. By the theorem of the mean, we have for any
tand h f "(P) _ for some
f(t + h) -f(t) h
such that t < E < t + h. Hence fr IZ,
f'(t)I <
If-(T) I dT ±
h
(If(t + h) -f(t) 1)
POTENTIAL THEORETICAL APPROACH
253
and z+h
If"(t) Ir <
2r-1
(hr-if
f"() d+
I f(t + h) -f(t) I r
and consequently (4)
f
I f"(T)
-,O
if 1(t) r dt < 2r-lhr J
x
I r dT + 22-1h-rJ
If(t) Ir dt
Now returning to n-space, let W(x) be a C. function with compact support. We let t be one of the x;, set f (t) = DDW and integrate (4) over the remaining variables. This gives
I
I II DPW II i,(-) < 2r-1hr
IpI=i
II DPW IIi,(I)
1P1=j+1 2r-1
h-r 1P1=j-1
IID"Wlli,(u)
An easy induction argument (as in the proof of Lemma 1) then shows us that there are constants C1 and C2 such that +C2h-rj/k-j
I IIDD1IIL,(00)
IIWIIL,(-)
IpI =k
1A =J
If 0 is a Cc0 function in 9, then Ipl=i
IID"
U
.,(91)
<- I IIDne0IILr(c0) 1A =j
< C1hr I IID"e0IIL,(c0) +
C2h-J1k-i
IIe0IILr(c0)
1A =k
< const. hr
const.
Taking s = const. h given the required result.
h-rilk-i
Il4lli,(9>
CHAPTER 6
Function Theoretical Approach The intimate connection between elliptic partial differential equations and complex function theory became apparent after it was discovered that all solutions of elliptic partial differential
equations with analytic coefficients are themselves analytic functions. The possibility of continuing solutions of analytic elliptic equations into the complex domain gave rise to many important results (Bergman [2], Vekua [2]). But it turned out that, at least in the case of two independent variables, the deep analogy between analytic functions and solutions of elliptic equations does not depend on the analyticity of the equations considered. In particular, one can associate with a second order linear elliptic equation with nonanalytic coefficients a generalized
theory of analytic functions-the theory of so-called pseudoanalytic functions (Bers [1,8], Vekua [4])-which exhibits almost
all characteristic features of classical function theory. In the theory of pseudoanalytic functions, however, one still has to make some smoothness assumptions on the coefficients, though very mild
ones. It is of interest to note that some function-theoretical results can be obtained without assuming any smoothness or continuity conditions whatsoever. As a matter of fact, consequent application of function-theoretical methods to elliptic equations, and especially to nonlinear equations, leads one, almost against
one's will, to study linear equations with bounded measurable coefficients (Morrey [2], Bers and Nirenberg [1], Boyarskii [1], cf. also the supplement to Chapter IV in Courant-Hilbert [2] and the references given there).
In this chapter we report on some results obtained in this direction. Applications will be given in the following chapter. There also exist applications of the theory described in this chapter
to complex function theory itself, in particular, to the theory of Riemann surfaces and discontinuous groups. 254
FUNCTION THEORETICAL APPROACH
6.1.
255
Complex Notation
We shall be concerned with linear elliptic systems of two first order equations for the unknown functions u(x,y), v(x,y) : ux = a11vx + a12vY + b11u + b12v + C1 (1)
-uv = a21vx + a22vv + b21u + b22v + C2
and with second order linear elliptic equations for one unknown function O(x,y) : (2)
A110xx + 2A120xv + A22/v1, + A1cx + A20v + A0 6 = B
System (1) is called elliptic if a12 > 0 and (3)
4a12a21
- (all +
a22)2 > 0
uniformly elliptic if all coefficients are uniformly bounded and the left-hand side of (3) is bounded away from zero. If equation (2) is elliptic, we may assume, without loss of generality, that (4)
A11A22-A12-1,
All>0
In this case the equation is called uniformly elliptic if all coefficients are uniformly bounded. We shall always assume uniform ellipticity, but the coefficients a11(x,y), ... , B(x,y) of (1) and of (2) will be assumed only to be
measurable functions. In this case we have no right to expect solutions with continuous derivatives. By a solution (u,v) of (1)
we shall, rather, mean a pair of continuous functions which have L2 derivatives which satisfy the equations almost everywhere. In the case of equation (2) a function 0 will be called a solution if it has continuous derivatives 0x, 0v which in turn have L2 derivatives 0., 0xv, 0vv satisfying the equation almost everywhere. There exists an intimate connection between elliptic equations of the special form (5)
A11Oxx + 2A124xv + A220vv + A14x + A20v = B
and elliptic systems. To see this, associate with every real-valued function 0 its complex gradient w= u + iv = 0x - ic6v. It is clear
256
L. BERS AND M. SCHECHTER
that 0 will be a solution of (5) if and only if (u,v) is a solution of the elliptic system (6)
uz = (2A12/"A11)v + (`422/A11)v - (A1/A11)u + (A2/A11)v ± (B/A11),
-uY = vX
Furthermore, system (6) is uniformly elliptic if and only if equation (5) is. The more general equation (2) can be reduced to the special form (5) by introducing the new unknown function c = 0/00, 00 being some fixed positive solution of (2) (B = 0). Under the present weak hypotheses on the coefficients it is by no means obvious that such positive solutions exist. Their existence can, however, be established in every sufficiently small domain, and in every domain if A0 < 0. We shall not, however, discuss this here. We shall make use of the following Let there be given a sequence of functions u" > (x, y) defined in a domain 9' and assume that (i) the sequence u(') LEMMA ON CONVERGENCE.
conrtrges to a function u(x,y) uniformly on every compact subset of -9, (ii) the functions u(2) have L2 derivatives, and (iii) the Dirichlet integrals of the functions u(') over any compact subset of r', are uniformly bounded. Then (j) the function u has L2 derivatives, and (jj) there exists a subsequence u('^) such that u/' - u. and u/ --b u, weakly in 'r.
This is an immediate consequence of Lemma 7 of Chapter 4.
A convenient technical simplification results from using complex notations. We set x + iy = z, u + iv = w and denote the conjugate of a complex number by a bar. Functions of x and y (whether real or complex-valued) will be written as functions of z
without implying analytic dependence. The formal derivatives with respect to z and to 2 = x - iy are defined by the relations a __1 a a_1 a a 2(ax+Zay az a2 2(ax-ZTay) In most cases the z and 2-derivatives will, of course, be understood to be strong derivatives. The usual rules of calculus apply to these
operators. We note, in particular, that the equation w2 = 0 (7)
FUNCTION THEORETICAL APPROACH
257
equivalent to the Cauchy-Riemann system (ux - Uy = 0, uy + v = 0) and expresses the fact that w is an analytic function is
of the complex variable z.
An elementary though unpleasantly lengthy computation shows that every uniformly elliptic system (1) can be written in the complex form
Wj=µwz
(8)
where µ(z),
,
vtuZ+aw
;
fltF1+y
y(z) are complex-valued measurable functions satisfying inequalities of the form (9)
. . . ,
lµl + Ivl <_ k < 1,
Ixl + IFII <_ k',
Iyl < k"
Conversely, the complex equation (8) satisfying (9) is always equivalent to a real uniformly elliptic system (1). We note that equation (9) is homogeneous (y - 0) if and only if (1) is (ci - c2 = 0), that µ = v = 0 if and only if a12 = a21 =--- 1, ail = a22 = 0, and that a = = 0 if and only if b;; - 0. 6.2.
Beltrami Equation
We consider first a special equation of the form (8), the socalled Beltrami equation
wi = (z)w
(10)
where
IuI
(11)
In real terms equation (10) reads (l Oa)
gux = -g12v + gi1vy,
guy = -g22vx
+ 912VVI
g2 -911912 -g12
where the (real) coefficients g,, are defined by (12)
gil
j2
+ 2g12 dx dy + &22 dye = Idz + µ dz12
A Beltrami equation (with a smooth µ) has a simple geometric meaning. A topological mapping w = w(z) satisfying (10) is conformal with respect to the metric (12). This means that if two
258
L. BERS AND M. SCHECHTER
curves in the z plane intersect at an angle 0 (measured in the metric (12) ), then their images in the w plane also intersect at the
angle 0 (measured in the ordinary way). A function satisfying some Beltrami equation is called quasiconformal.
Beltrami equations play an important part in differential geometry, complex function theory, and the theory of differential equations. They occur, for instance, in the problem of reducing
an elliptic equation (5) to canonical form A0 + ... , by introducing new independent variables
computes at once that equation
= E(x,y), n = 77(x,y). One
_ + irk must satisfy the Beltrami
with
µ
A22 - All - 2iA12 A22 + All + 2
We list now some properties of Beltrami equations. They are proved in the Appendix. (a) If w is a solution of (10) and f (w) an analytic function, f (w (z) ) is a solution of (10).
In the "smooth case" (u of class Ca, w continuously differentiable) the proof is trivial :
af(w(z))/az =f'(w(z))w2 =f'(w)µ(z)wz = u(z) af(w(z))/az (p9) If w(z) is a solution of (l0) in a domain T, if w(z) is a topological mapping of 9, and if w (z) is another solution of (10) in 9, then there exists an analytic function f (w) such that W ^(z) = f (w (z)) .
Again the proof is trivial in the smooth case; one computes easily that aiu/aw = 0. (y) Let w(z) be a solution of (10) which maps the unit disc topologically onto itself leaving the origin fixed. Then
(13)
Iw(z1)
- w(z2)I < K 1z1 - z211
Iz1 - z21 < K Iw(zl) - w(z2)I' where a and K depend only on k.
This basic result is due to Morrey [2], Ahifors [I], and Lavrent'ev
[1]. The best exponent in (13) is a = (1 - k)/(l + k). Mori [1] proved that (13) holds with this exponent and K = 16.
FUNCTION THEORETICAL APPROACH
259
(8) For every measurable u satisfying (11) there exists a unique solution w of (10) which maps the closed unit disc conformally onto itself and satisfies the conditions w(0) = 0, w(1) = 1. (E) If a is a measurable complex-valuedfunction such that la(z) I S k' for I zi < 1, then the nonhomogeneous Beltrami equation
wt =µwa+a
(14)
possesses a unique solution in I z I < 1 which is continuous on I z I < 1, real
on the unit circle, and vanishes at z = 1. This solution satisfies a uniform Holder condition depending only on k and k'.
6.3. A Representation Theorem REPRESENTATION THEOREM (Bers and Nirenberg [1]; cff, also Boyarskii [11). Let w (z) be a solution of the uniformly elliptic equation (8) defined in the domain 9, a subdomain of the unit disc I z I < 1. Then w admits the representation
w(z) = e"ZJ [x(z)] + so(z) where s and so are continuous on I zJ < 1, vanish at z = 1, and are real on IzJ = 1, = x(z) is a homeomorphism (topological mapping) of the closed unit disc onto itself subject to the conditions x(0) = 0, x(1) = 1, and f (?) is analytic in the domain x (9), the image of T under X.
Furthermore, the functions s(z), so(z), x(z) and the inverse homeomorphism x-1(4) satisfy uniform Holder conditions depending only on the constants in (9).
If y = 0, the theorem holds with so = 0. If ,u = v - 0, the theorem
holds with x(z) = z. If a = # = 0, the theorem holds with s - 0. u = v - y - 0, the functions Also, in any open set in which a s, so, and x are analytic.
The significance of the theorem lies in the fact that it decomposes an arbitrary solution w of (8) into four functions, three of which are perfectly well behaved and can be estimated without
knowing anything about the particular solution, whereas the fourth is analytic. We remark that the unit disc may be replaced by another domain by means of a conformal transformation.
L. BERS AND M. SCHECHTER
260
We prove the theorem first for a uniformly elliptic equation of the form
w2=uwz±aw+y
(15)
By (e) there exists in (z(< 1 a solution s(z) of the equation
sf=usz±a which satisfies a Holder condition depending on k, k', and the boundary conditions
s = Oatz=1
Ims=0on(z( = 1, We note that
(y e-s( < kl
where k1 depends only on k, k', k" and set W = e-8w. A direct computation (which is legitimate, though we work with strong derivatives) shows that this function satisfies in 9 the equation Wf = uWz + ye-8
Again by (e) there exists in IzI < I a solution t(z) of the equation If = utz -,- ye-8 which satisfies a Holder condition depending only on k, k1 and the boundary conditions
t=0atz=1
Imt=Oon(z(=1, We note that the function W = W W2
- t satisfies in T the equation
= ,u W.
By (8) and (y) there exists in (z( < 1 a solution equation
= X(z) of the
XI = YX z 1, satisfies the which maps (z( < 1 topologically onto conditions X(0) = 0, X(1) = 1 and satisfies, together with its
inverse function X-1, a Holder condition depending only on k. such that By (f3) there exists an analytic function f T3'(z) =.f [XM] Setting s, = te, we obtain the desired representation.
FUNCTION THEORETICAL APPROACH
261
In order to pass from the special equation (15) to the general case we use a device, due to Morrey, which is applicable only because we admit discontinuous coefficients. Let w(z) be a fixed solution of equation (8) defined in 9. We set µ = u + v(wt/wz) at points of 9 at which wz # 0, and ,u = 0 at points of <9 at which W. = 0 as well as at all points not in 9. Similarly we set
a=a+fl(wfw) at points in at which w 0 and & = 0 at all other points. The functions ,fi(z), &(z) are measurable and
IaI < III + ICI On the other hand, the function w(z) satisfies the equation wf = Pwz + aw + y Ii I < IfuI + IvI,
Hence the validity of the representation theorem in the special case implies its validity in the general case. 6.4.
Consequences of the Representation Theorem
We note some qualitative consequences of the representation theorem. THEOREM ON ZEROS OF ELLIPTIC SYSTEMS.
Let w(z) be a solution of
the uniformly elliptic homogeneous equation (16)
wf=uw=+ytZ+MW +fizz
defined in a domain 9' and assume that w does not vanish identically. Then the zeros of w are isolated.
Without loss of generality we assume that 2' is a subdomain of the unit disc. The representation theorem takes Proof.
the form w (z) = e1cZf [X (Z) J
The analytic function f does not vanish identically. Hence its zeros are isolated and so are the zeros of w, since X is a homeomorphism.
L. BERS AND M. SCHECHTER
262
UNIQUE CONTINUATION THEOREM FOR ELLIPTIC SYSTEMS. Two
solutions of the uniformly elliptic equation (8) coincide everywhere if they
coincide on an infinite set of points which has an accumulation point within their common region of regularity. Proof.
The difference of two solutions of (8) satisfies the
homogeneous equation (16).
In view of connection between second order elliptic equations and elliptic systems noted above we also have the UNIQUE CONTINUATION THEOREM FOR UNIFORMLY ELLIPTIC EQUATIONS.
A solution of a second order uniformly elliptic equation is
uniquely determined by its values in any open set.
Another consequence of the representation theorem is the fact that the maximum principle holds for uniformly elliptic equations of the
form (5) with discontinuous coefficients (in this case the proof in Chapter 2 does not apply). For the sake of brevity we consider only the case of equation (5) with B = 0 and prove that a nonconstant solution has no interior maxima. Indeed, assume that such a
solution c attains its maximum at an interior point, say at z = 0.
Then w = z - ion, = 0 at z = 0 and by the theorem on zeros
of elliptic systems w 0 for 0 < Izj < E. Let the argument of w increase by 2nir as z goes once around the circle I zi = h(0 < h < e)
in the counterclockwise direction. Then n is called the index of w
at z = 0. By the representation theorem w(z) =f [x(z)], where x(z) is a homeomorphism and
f (x) = (x - x(o))m[l + a, (X - x(o)) + a2(x - x(o))2 + . ..], m > 0. This shows that n = m > 0. But it is easy to verify by a geometric argument that at a maximum point we must have n = -1. This contradiction proves our assertion. We conclude this section by a simple but useful EXTENSION OF THE MAXIMUM PRINCIPLE. Let 0 be a solution of the uniformly elliptic equation (5a)
Al1jxx + 2A120., + A2201v = 0
Then the derivatives 0Z, 0y are either constant or have no interior maxima.
(The corresponding statement for more than two dimensions is false.)
FUNCTION THEORETICAL APPROACH
263
The proof is trivial if the coefficients are of class C1 and the solution of class C3, since then 0x and 0y themselves satisfy elliptic equations. (To obtain the equation for 0x, for instance, divide (5a) by A22 and differentiate with respect to x.) In the general case the assertion follows from the representation theorem which implies,
in the case considered, that w = ox
- i/ = f(x(z)) where f is
analytic and X (z) a topological mapping.
6.5. Two Boundary Value Problems
We shall now apply the representation theorem to the Dirichlet and Neumann problems. For the sake of simplicity we consider an equation of the form (5) assumed to be uniformly elliptic and take as our domain the unit disc. The Dirichlet problem consists in finding a solution of (5) under the boundary condition
0 = -r on Izi = 1 (-r given function) Without loss of generality we assume that r = 0 at z = 1. We (17)
also assume that T has a Holder continuous derivative T' = dT f ds on the unit circumference. Hence we may write (17) in the form (17a)
ao
as
_
-T' onlzI=1,
0=0atz=1
The Neumann problem we formulate as follows : (18)
a-=c+vonlzl=l,
=0atz=1
Here v is a given function, assumed to be uniformly Holder continuous, c a constant to be determined. This formulation of the Neumann problem has the advantage that it leads to an unrestricted existence and uniqueness theorem. Solutions of both problems are required to belong to C1 (1) . Let 0 be solution of the Dirichlet problem (17) (or of the Neumann problem (18)) for equation (5). Set w(z) _ 0- io,,. A PRIORI ESTIMATE.
-
There exist positive constants a aid M depending only on the uniform ellipticity of (5) and on the Holder condition for T' (for v) such that (19)
IIWDc.cl> < M
L. BERS AND M. SCHECHTER
264
This estimate is stronger than the Schauder estimate of the preceding chapter since only the size but not the continuity properties of the coefficients are involved.
The proof of the estimate is based on the representation theorem and on a known theorem in function theory, Privalofj's theorem (proved in the Appendix), which states that iff (z), IzI < 1 is analytic and Ref (z) is continuous on I z I = 1 and satisfies there a Holder condition with exponent 6 < 1 and constant K, then f (z) satisfies on Izi < 1 a Holder condition with exponent 6 and constant CK, where C depends only on 6. It was noted above that the function w(z) satisfies equation (6),
i.e., an equation of the form (8) where the complex-valued coefficients u(z), . . . , y(z) are measurable functions satisfying the inequalities (9). The constants k, k', k" depend only on the uniform ellipticity of (5). Applying the representation theorem we have w(z) = e81zf[x(z)] + so(z)
(20)
Here C = x(z) is a liomcomorphism of the closed unit disc onto itself which leaves the points 0, 1 fixed; the functions s(z), so(z) are continuous in the closed unit disc, real on the unit circle, and vanish at z = 1, and the function f is regular analytic for Cl < 1 and is continuous on the unit circle. Also, we know bounds and Holder conditions for the functions s(z), so(z), x(z) as well as for the inverse homeomorphism z which depend only on the constants k, k', k". I
Since T'(e'0) is the tangential derivative of a single-valued
function it has mean value zero and hence vanishes somewhere on the unit circle, so that by hypothesis (21)
1 -r' l < 2K1,
Ki depending on the Holder continuity of T'
Also, from (17) we infer, by considering the variation of 0 on a curve of steepest descent, that there exists a point zl in the unit disc such that (22)
l ic(zi) I < K21
K. depending only on the Holder continuity of T'
FUNCTION THEORETICAI. APPROACH
265
The boundary condition (17a) may be written in the form
Re [izw(z)] = -r'(z),
(23)
IzI = 1
or, since s and so are real on Iz1 = 1, (24)
Rc {izf[x(z)]} = e-812>[T'(z) - so(z) Re (iz)],
We have that x-1 (ei°) = e;n(o)
IzI = 1
0<0<2ir
where the function SA(0) is monotonically increasing, satisfies a known Holder condition, and )(27r) = 27r
S2 (0) = 0,
By Privaloff's theorem there exists an analytic function H(t), I < 1, which is continuous on the closed unit disc and satisfies the conditions
-
0 < 0 < 2,r,
Re H(1) = 0 and for this function we know a bound and a Holder condition. Im H(ei") = SZ(0)
0,
Set
Re and
so[x-1( )] _ s[x-1(S)] = a(Q, Condition (24) may be rewritten in the form (25)
Re [i e" ehcc> e-of0
Re
c
[ix-1(4)])
Noting the Holder continuity of T' and (21) we conclude that the right-hand side of this equation satisfies a known Holder condition
and is bounded by a known constant. Applying again the Privaloff theorem we obtain a Holder condition for the regular function i exit f (i) on the closed unit disc. A bound on the real part of this function is known. On the other hand, inequality (22) at a point in together with formula (20) give a bound for If I
the unit disc I I < 1. Hence we also know a bound for the This implies that we know a bound and a function iC e"(; f Holder condition for the function f (4) on the unit circle 1 I = 1.
Applying once more Privaloff's theorem and the maximum
L. BERS AND M. SCHECHTER
266
modulus theorem for analytic functions we obtain a bound and a Holder condition for the function f on I I < 1. Returning to the representation formula (20) and noting that
by now we know Holder conditions and bounds for all terms occurring on the right-hand side, we obtain the desired inequality (19).
For the Neumann problem the argument would be the same if we could find a bound on I aolanl. If the equation is homogeneous (B = 0), this is easy, since the maximum principle implies that ac/an must vanish at some point on the boundary and we know
the Holder continuity of aO/an. Also, as before, a0/as must vanish at some point of the boundary, so that we again obtain inequality (22), this time for some point on the unit circumference. For B 0 the argument is more complicated and will be omitted. The Dirichlet and Neumann problems formulated above have unique solutions. EXISTENCE AND UNIQUENESS THEOREM.
Sketch of proof. We consider a sequence of equations (26) A(i)' Ali)ox _ A>i'o = B(i) _l._ 2Aci;J 11 xx
12 xv
= AP + 22
HY
y
with very smooth (say C,o) coefficients such that all equations
(26) are uniformly elliptic with the same constants and the coefficients of (26) converge almost everywhere to the corresponding coefficients of equation (5). Let 0(i) be the solution of (26) under the boundary condition (17) (or (18) ). The existence
of P) is classical; it can be established, for instance, by the methods of Chapter 4 or 5. By our estimate there is an a > 0 and an M such that (27)
II¢ici>Ilcl+a < M
and, selecting if need be a subsequence, we may assume that there is a (28)
satisfying the boundary conditions and such that 11 0 - P> II c,
0
It is not difficult to verify (either by elementary means or by the representation theorem) that the Dirichlet integrals of the derivatives of 00) are uniformly bounded over every compact subset in the unit disc. Using the lemma on convergence we conclude that satisfies (5).
FUNCTION THEORETICAL APPROACH
267
Uniqueness can be proved either by the maximum principle or by noting that our derivation of the a priori estimate implies that w= 0 if T' or v vanishes identically. Using the maximum principle and other estimates, which we shall not state here, the solvability of the Dirichlet problem can be proved also for arbitrary continuous boundary values. Appendix.
Properties of the Beltrami Equation. Privaloff's Theorem
This appendix contains a proof of Privaloff's theorem and the theory of the Beltrami equation to the extent needed to establish statements a-,- of Chapter 6. A solution of (10) of that chapter will be called a eu-conformal function, a homeomorphic solution, or
a ,u-conformal mapping. There are two main problems: to find a u-conformal mapping of the whole plane onto itself, and of a disc onto itself. In both cases a mapping exists, and with appropriate normalization, it is unique. For a Holder continuous ,u the existence of local eu-conformal mappings is classical (Korn [1], Lichtenstein [1]; modernized proofs have been given by Bcrs [7] and Chern [1]). The global mapping theorems follow from the local theorem by use of the general uniformization theorem. Vekua [3] and Ahlfors [2] gave direct proofs of the global theorems. For measurable u the mapping theorem is due to Morrey [2] ; earlier, the case of a continuous u had been treated by Lavrent'ev [2]. New proofs of Morrey's result were given by Bcrs-Nirenberg
[1] and Boyarskii [1]. One owes to Boyarskii the important observation that the generalized derivatives of the solution are of class L9 for some p > 2. The proofs given below are taken from Ahlfors-Bers [ 1 ].
For C.0 functions g(z), z = x define the operators P and T by (Pg)(z) =
-
IT
(Tg)(z) = - I
iy, witli compact support we
f gm ( 1 , - LI g(z) d$ dl)
Jf (4 - z)2
d$ d??
L. BERS AND M. SCHECHTER
268
where = E -;- in and integration is taken over the whole plane. We write II, instead of II i,(x for the sake of brevity. II
II
LEMMA 1.
(1) (2)
The following relations hold:
(Pg) Z = 7'g (Pg); = g, IPg(Z1) - Pg(z2)1 < c Ilgll, IZ1 -
for 2 < p < o0 for 1 < p < co 2211-2/n
(3)
11Te11, < C, IIg1I
(4)
1I T9112 = 119112
Relations (1) are easy to verify; (2) is an immediate consequence of the Holder inequality, whereas (3) is a special case of the Calderon-Zygmund inequality. The identity (4) follows from (1) and integration by parts. In fact, ffi Tg12 dx dy = ff (Pg) z(Pg)Z dx dy
- f f (Pg)zfPgdxdy = f f (Pg)Z(Pg)zdxdy f f lg12dxdy Let Cp denote the smallest constant for which (3) holds. We have LEMMA2.
C. - l asp, 2.
This is a consequence of the Riesz convexity theorem (M. Riesz [3]) and of (4). It follows from Lemma 1 that for g e L9, p > 1, Tg is defined, while Pg is defined if p > 2. Also Pg belongs to the Banach space B. consisting of complex valued functions w(z), Izl < ee, which
vanish at z = 0, satisfy a Holder condition with exponent 1 - (2/p), and have generalized derivatives in L,. The norm in Bp is (5)
II(1)IIBD = H1-2jp,x[w] ± IIwZII, + IIwOZII,
Clearly P is a bounded linear transformation from L,, to Bp, satisfying (1-3).
Throughout we shall assume that all functions denoted by µ, with or without subscripts, are bounded and measurable and
FUNCTION THEORETICAL APPROACH
269
subject to the inequality lp(z) I < k with fixed k < 1. The exponent p will be a fixed number satisfying
p > 2,
(6)
kCp < 1
By Lemma 2 there are such p, whatever the value of k. If a E Lp, the equation
THEOREM 1.
cot = µ w z
(7)
+a
has a unique solution w"° E B. Proof. To establish the uniqueness we show that a solution of the homogeneous equation
wz = uw
(8)
reduces to zero if w(0) = 0 and wz E L. It follows by (1) that
w = P(wz) + F where F is holomorphic. We obtain
(0 z = T(µwz) + F
(9)
and by (3) IIF'Ilp < (1 + kCp) Ilwzllp
But IIF'II is finite only if F is constant. Hence F = 0 and (9) yields 11(,)Zllp <
kCp
II(ozIID
a contradiction unless w2 = 0. Thus o) = 0. To prove the existence, we solve the equation
q = T(µq) + Ta
(10)
in L. by setting
q= Ta+TyTa+TuT,uTa+... This is possible because Ta E Lp and the series converges since the norm of the transformation Tju is < kCp < 1. We set (11)
w = P(1uq + a)
and verify that w; = q, w, _ pq + a. Hence co is a solution of (7).
L. BERS AND M. SCHECHTER
270
It follows from (10) that 11g11,
< kCn llgll,, + C, llaj,
and thus llg11, < c Ilall,
(12)
where we use the notation c for unspecified constants which may depend on k and p. This shows that w'',° E B and also establishes The mapping or , o,'`,° is a bounded linear transformation LEMMA 3. from L, to Bn with a bound that depends only on k and p. LEMMA 4. If Fen -+ Fe almost everywhere and Ilan - 611', -- 0, then co"^. co," in BD.
Proof. We set SP)
oil',' and
qn
q=
(011,(Y. From
fn)
pq + an - a = JunQ(Zn) + l n - lu) q + an - a
= lungn
with An = (Fen - µ)q + an - or. 0 since II (un - u) q II ,, - 0 by dominated convergence But II An II n 0 and Il an - a 1l D - * 0 by assumption. Hence, by Lemma 3, f2(n) we conclude that S2('b)
in BD.
In addition to the restriction 1 ,al < k < 1 we assume until the proof of Theorem 3 below that all functions denoted by µ vanish outside a fixed set. There exists a unique µ-conformal function f' which vanishes at the origin and satisfies f2" - 1 E L,. It is given by THEOREM 2.
f'(z) = z -f- cu1"''(z) It is evident that f" is a solution of
(13)
.fi = ,ufZ Since f'` is locally of class LD with p > 2, it is also locally of class L2. (14)
We remark that f' is independent of the choice of p. Indeed, since u has compact support,, f'' is analytic in the neighborhood of oo. Together with f," - 1 E L, this implies f' = 1 + o(1/Iz12), and hence fi' - 1 E Lo for 1 < q < p. LEMMA 5.
If u is a C. function, so is f -u. Moreover, f'' is a homeo-
morphism of the whole plane onto itself and the Jacobian is positive.
FUNCTION THEORETICAL APPROACH
271
Let us write the Beltrami equation (14) as a system of two real equations for u and v (f = u + iv) Proof.
v. = -a12ux + a22u. (15)
Uv = -allux + a12uv
(cf. (10a) of Chapter 6). Eliminating v by differentiation, we obtain for u the second order elliptic equation (16)
alluxx - 2a12uxv + a22uvv + blux + b2uv = 0
where bl = allx - a12v and b2 = a22, - a12x. By the local existence
theorem (Theorem 1 of Chapter 5) there exists, near every point zo, a C2 solution of (16) with prescribed first derivatives at zo. Finding v by means of (15) we see that in the neighborhood of each point zo there exists a C2 u-conformal mapping (z) with positive Jacobian. By Theorem I of Chapter 5, (z) is a C. function. A direct computation shows that f'`(z) is a holomorphic function of (z) near zo. On the other hand, f''(z) is holomorphic for large Izi and since I 1 - df'' (z) f dzI " is integrable, we see that
f(z) =z+ao+al/z+... near z = oo. The assertion follows from these statements. of C. functions with fixed It is possible to find a sequence f compact support such that y. -> u almost everywhere, and, as a 0. It follows by Lemma 4 that consequence, IIu,, - uII,,
f'n -P - 0 in BD. We employ this approximation to prove
LEMMA 6. f' is always a homeomorphism of the whole plane onto itself.
Let gn M be the inverse of fn (z) by the preceding lemma. One finds that Proof.
with
gn,r = 1'ngn,Z
vn - - (fn, zun) fn.i
Iynl = lun) ° gn
where h - g denotes h (g (Q) .
gn
known to exist
L. BERS AND M. SCHECHTER
272
By use of Holder's inequality we obtain
nl p (If2 - Ifa.zl2) dx d).
JfIvV ddi1
S f I 'U. I P-2 I J ,,ll2 dX d), d.Y).21p
C where
(fftitnID dx dyll
?1P (55 If,, 21
dx
_ + irk. But Ilf».fllp - II (Vi^,/lnllp
C I!'"nlly
and hence Ilvnlip C C Il,unllp
(17)
It is evident that gn = f"^, since gn, from Lemma 1 and Theorem 2 that Z
W'
Ign(t- l) -
1 as z -- oo. We conclude C llYnllr
2I1
2/p
or, equivalently, that IZ1 - Z21 C Ifn(Z1) -f.(Z2)I -1
C
Ifn(zl)
-fn(zn)I1_21"
On letting n tend to infinity, we see that f(zl) = f" (z2) implies that z1 = z2i and we have therefore proved that f" is one-to-one. The fact that f" -- I for z -* oo is sufficient to show that the mapping is onto. A homeomorphismf is said to be measurable if measurable sets
are mapped onto measurable sets. An equivalent condition is that null sets are mapped onto null sets, or that the set function mes f (e), defined for Borel sets e, is absolutely continuous. The mapping f" and its inverse are measurable. Moreover, 0 almost everywhere, and
LEMMA 7. fZ`
(18)
mesf"(e) =J.1 (If2I2 - If_
for any measurable e.
dxdy
FUNCTION THEORETICAL APPROACH Proof.
273
Let e be an open set. Let x denote the characteristic
function of f'(e) and x that off, (e). Then x < lim inf x and consequently mes f1`(e) < lim inf mes f (e)
We have mesfn(e) =fj.(Ifa.Z12 -- Ifn,Zl2) dx dy (19)
ff
<
I f,, , =12 dx dy
e
and by Holder's inequality 2IP
J'JIfn.z12 dx dy <
(JJILl2 dx dyl
e
(mes e) l-'-'1n
e
Now
(jjfiP dx
dy)u' Ilfn,Z
- 1119 +
(mcs e)11n
e
Cc
II Iu,, 11 P
+ (mes e) 11P
and we obtain -4- (mes e)1/1')2 (mes
(c
e)1-21n
these estimates show that mes f1'(e) tends to 0 with mes e. Since every null set is contained in an open set of
Because II µn II, - II U ll
arbitrarily small measure, we have proved that null sets are mapped on null sets, and thus f1` is measurable.
Precisely the same reasoning can be applied to the inverse function, since (f")-1 = lim f"n and Ilv II,, is uniformly bounded. If a is an open set and bounded, it follows from (19) and the preceding inequality that mes p(e)
(If''I2 - I f,112) dx dy
e
since f Z -fz' and f,j4
f2 in L2(e). For compact sets e the reverse inequality is true, either by the same argument or by taking complements with respect to a large disc. On approximating
an open set by compact subsets it is easily concluded that the equality (18) holds for all open sets. It will therefore hold for closed sets, for Borel sets, and finally for all measurable sets.
L. BERS AND M. SCHECHTER
274
Because the inverse function maps null sets on null sets, it is
I,ff I2 to vanish on a set of positive measure. It is important to formulate conditions under which a com-
impossible for I fi112
posite function h ° fµ = h(f1'(z)) has generalized derivatives. For convenience we simplify the notation p to f. LEMMA 8.
Suppose that hg and ht are locally of class LQ, q Z 2.
Then h of has generalized derivatives given by
(h °.f) t
=
(h x
°f ).f= + (ht °.f )JJ
(20)
(h °.f)2 = (h. °f).fi + (h2 °f)];
and
II(h °f)zIlr S M(IIhzII, + 11h2IIQ),
(21)
r=
pq
P+q-2
where the norms are over corresponding bounded regions c, f(S2o), and M is independent of h. Proof.
We show first that (20) implies (21). By Holder's
inequality ffwz
°.f)fIrdxdy
00 <_
(ffvzz° ,fI°
If'12dxdy) (ffv
dxdy)l-.1q
and by use of Lemma 7,
°fI°IfI2dxdy <1
1k2
ffJh of jq
(fI2
- 1h12) dx dy
fl, 1
k2
Ih214dxdy
A similar estimate applies to (h2 ° f) fZ and (21) follows. The formulas (20) hold if h and f are of class C. By well-known properties of generalized derivatives (cf. Appendix II to Chapter
FUNCTION THEORETICAL APPROACH
275
1) it is possible to find C. functions hm, fn in such a way that ff(Ihm I(no)
ff(if
- hI2 + Ihm,z - hz12 + Ihm,3 - hilt) dx dy --. 0 fi
fI2 + Ifn _j2) dxdy
+ ifn.z
0
no
Consider
I, = Jf1(hm.z °f)
- (hz -f )I IfsI dx dy
no
12 = f f I (hm.z °f) - (hri.z °fn) I IffI dx dy no
I. = f f Ihm,z
°fnI Ifz -fn,ZI dx dy
no
By the same estimate that was used to prove (21) it is seen that I1 can be made arbitrarily small by choosing m sufficiently large. When m is fixed it is evident that I2 and I$ tend to zero for n - oo. Consequently it is possible to choose m and n so large that JJI(hm.z °fn)fn,z
- (hz °f )ffI
dx dy
no
is arbitrarily small. The same'` is true of ffI(hm.2 °fn)Jn,z - (h2 °f)JJI dx dy no
and we conclude easily that the first relation (20) is valid. The second is proved in the same way.
The preceding lemma is used to prove LEMMA 9.
(22)
(fµ)-1 = f' with
v = - (L-2) ° (f14) -i 411
We remark first that v is measurable because f ` is a measurable mapping. Therefore f exists and f' is locally of Proof.
class L..
L. BERS AND M. SCHECHTER
276
We compute (f' o f")t by the second formula of (20) and find that it is zero almost everywhere. Hence ( =f" o f" is analytic, and since it is a homeomorphism, it must be a linear function.
But D(0) = 0 and V(z) -- 1 as z , oo. Hence (D(z) = z and f' is the inverse off'`. We collect in a single theorem a number of results which are easy consequences of Lemma 8. In this theorem we do not need to assume that y has compact support. Let f be any ,u-conformal homeomorphism, defined in a region S2o. The following are true : THEOREM 3.
(i) (ii)
ff is locally of class L.
ff
7-1 0 almost everywhere.
(iii) f-1 has generalized derivatives which are locally of class L. (iv) (f 1)s and (f-% are determined by classical formulas. (v) f and f-1 transform measurable sets into measurable sets. (vi) Integrals are transformed according to classical rules. (vii) If is any ,u-conformal function in 00, then , o f 1 is analytic, and vice versa.
Proof.
Since all assertions are of a local nature, it is no re-
striction to assume that SZo is a bounded region. We extend u to the whole plane by setting u = 0 outside S20. The results (i-vi) follow from the fact that f o (f') -1 is analytic, and (vii) follows because 0 o (f) -1 and f o (f'') -1 are both analytic.
Statements (x) and (9) of Chapter 6 can now be proved by direct computation, just as in the "smooth" case. We now let w" denote a µ-conformal homeomorphism of the
whole plane onto itself which is normalized by w"(0) = 0, w''(1) = 1, w''(oc) = oo. Similarly, W" is to be a µ-conformal homeomorphism of the closed unit disc onto itself which satisfies W"(0) = 0, W''(1) = 1. If they exist, it follows from Theorem 3 that w" and W" are uniquely determined. The existence is proved in several steps. LEMMA 10.
(23)
If u = 0 in a neighborhood of oe, then
w"(z) =f'(z)/f'(1)
If µ = 0 in a neighborhood of the origin, then (24)
w"(z) = 1/w'(1/z)
FUNCTION THEORETICAL APPROACH
277
where
A(z) = 1u(1/z)z2/22 LEMMA 11.
Set u = 91 + µ2 where µl and
1u2
vanish near 0 and
oo, respectively. Then w" = wA O w"=
with
A = (1 `1
(25)
1u21u Ji LEMMA 12.
o (w'')-
.f", )
If ,u (z) = µ (1 /2) z2/ z2, then W" is the restriction of
w" to the unit disc.
Lemmas 10 and 11 are proved by direct verification. Together they show that w" exists, for any arbitrary IA can be decomposed in the supposed manner. As for Lemma 12, one proves by means of the uniqueness that w"(z) = 1/V'(1/2). Hence Iw'`(z)I = 1 for
IzI = 1, and it follows that the restriction of w" maps the unit disc onto itself. It is evident that Lemma 12 proves the existence of W". There exist unique ,u-conformal homeomorphisms of the whole plane and the unit disc onto themselves with fixed points at 0, 1, o0 and 0, 1, respectively. THEOREM 4.
Theorem 4 contains statement (b) of Chapter 6. LEMMA 13.
If µ = 0 for IzI >_ 1, then c-1
Proof.
Isc
<
We have 11p II, < c, and therefore, by (13) and Lemma
3, IzI1-2/p I,f"(z)I < IzI + c1 By virtue of Lemma 9 and the inequality (17), the same reasoning can be applied to the inverse function and yields
IzI < If'A(z)I These estimates imply
(1 +
+ c1 I f"(z)I1-2/p
C1)-2p-2/p-2 <- Iw"(ej9)I
which proves the lemma.
(1 +
cl)-2p-2/p-2
L. BERS AND M. SCHECHTER
278
Let [zl,z2] denote spherical distance. The following spherical Holder condition holds for all µ. LEMMA 14.
(26)
[w"(zl), w"(z2)] S c[z1,z2]z,
a>0
We represent w" as in Lemma 11, choosing µ1(z) = 0 for Izl > 1, µ1(z) = µ(z) for IzI < 1. Then 121 < kl(k) < 1 and by Lemma 13 together with (25), 2 vanishes in the disc I zl < 1 /c. It is sufficient to prove (26) for w"' and w2. Also, because the spherical distance is invariant under inversion, we may replace wz by w'(z) = 1/wA(1/z), where v(z) = A(1/z)z2/Z2 (see Lemma 10). Thus we need to prove (26) only for the case that µ = 0 for IzI > R; c and a may depend on R. Proof.
We know that If(l) I > c-1 (see the proof of Lemma 13). It follows by (23, 13) and Lemma 3 that Iw"(z1) - w"(z2)I < c 1z1 -
2211-2/p
This implies (26) if, for instance, Iz11, Iz21 <_ 3R.
The function g(z) = l/w"(1/z) is holomorphic for IzI < 1/R and if R > 1, as we may suppose, Lemma 13 yields Ig(z) I < c for IzI < 1/R. Hence Ig'(z) I < 4cR for IzI S 1/2R. This implies (26) if I z11, I z21 ? 3R.
Finally, (26) is trivially fulfilled if Iz1I < 2R and Iz21 z 3R. COROLLARY.
I W"(Zl) - W"(Z2) I< C IZ1 - Z21"
This implies statement (y) of Chapter 6. We now prove statement (E). The uniqueness part follows at once from Theorem 3. To establish existence, we define u and a to be zero outside the unit disc. Then 11a II
w in the form
7r1l'k'. We represent
w=0+w"'"
We must determine 0 in such a way that Im
on IzI = 1 and
- ('01"') = 0
- co-"," = 0 at z = 1. Set /(z) = (D(W"(z)).
We must determine a holomorphic function V(z) in the unit disc such that Im tD = -r = Im co-"-" ° (W")-1 on IzI = 1 and D = w"0L
FUNC'T'ION THEORETICAL APPROACH
279
at z = 1. Note that we know Holder conditions for W'`, (W,,)-1, w'`.", and r depending only on k and k', as well as a bound on Iw'`-"(1) I. It follows from Privaloff's theorem (cf. below) that a 0 with the desired properties exists and satisfies a Holder condition depending only on k and k'. This proves (e). It remains to prove Privaloff's theorem. THEOREM 5 (Privaloff [1]). Let f (z) = u + iv be analytic in 1 and satisfies z I < 1. If u (z) is continuous in I z I u(ezas) I (27) Iu(eia,) < K Ie'e, e'0,I2
-
-
then f (z) is continuous in I z I < 1 and satisfies
If(zl) -f(z2)I
(28)
where the constant C depends only on x.
Proof.
It clearly suffices to prove that (28) holds for all zi and
z2 such that
Iz21 < 1. For r < 1, we have by Cauchy's
I z1$,
formula,
f(z) =
_rf(')
2-7r1 J
1
1
d
r2 Z
when IzI < r. First taking the minus sign, we get, after simplification, loan
1
(29)
f (rete) Ireio
=
.f (z)
I ZI12 d8
The plus sign yields 1
(30)
.f (z) = 2ir
j
2a
f(re") 1 + 2ir Im L
= rf (0) +
12rr
IreiO
-
ze_i
A
z$2
i8 ff(rei0) 2ir Im ze- d© Ireie - z12 J
by the mean value theorem. Taking real parts in (29) and imaginary parts in (30), we obtain 2n
u(z) =
2
Io
u(feie)
v(z) = rv(0) +
2
ire
ie-
1 2112 de
2 I
i
L. BERS AND M. SCHECHTER
280
or, combining into one formula,
r
f (z) = ira(0) ±
rei°
!
z dO
Since u(z) is continuous in Izl <_ 1, we may let r - 1 obtaining .f (z) == iv(0) + 2r
(31)
-z
u(es°) eye
j
dO
The proofis now similar to that of the Holder-Korn-Lichtenstein-
Giraud inequality (see Appendix I to Chapter 5). It clearly suffices to prove (28) for 1 z11, I Z21 > 1 /2 and 6 = I zl - z21 < 1/3.
Let Al be the (possibly empty) set of those 0 in [c1 - ir, i -;- 7r] satisfying le'0 - z,l < 26, and let 02 be its complement in that interval. For I z1 < 1 2- e'0 + z
1
2Tr
o
ei0
-z d0=1
Hence if zl = rle'-A' and zz -
then
f (zi) -f(z2) = u(e''') - u(e'k1)
f 1-
d1+n([u(e'o)
(32)
1
-
u(e'41)]
+ [u(e'0) - u(e'o')] e,o
-
e
Since I z, 1, I Z21 > 1/2, le"01
-
e'411
e - zl
Z2} dO z2
< ?t I Z2 - zl I
Hence by (27) l u(e'0=) - u(e'41) I < K2-0'6"
(33)
We express the integral in (32) as the sum of two integrals Il and
I2 with integration taken over Al and :,2, respectively. Since 6 < 5, there is a constant c such that le;°
-
et-b;l < 10
- oil < c Iei° - z,I,
j = 1, 2
FUNCTION THEORETICAL APPROACH
for ci - i < b <
; ir. Hence by (27) K
1111 <
281
77C- n
f
,1
(10
-
10
- 021-1)
dO
(34)
const. b"
Moreover,
12 =
(zl
-
[u(e'B)
z2)
-
to (a to
- z1) (e1 - 42)
Now for 0 E7r A21 le=e
- z1 < 2 Ie'e - z21
Hence 1121
10 7TK6 C
4 I I"-2 de
S const. ax
The result now follows immediately from (33), (34), and (35).
CHAPTER 7
Quasi-Linear Equations In discussing nonlinear elliptic equations we shall limit ourselves to second order quasi-linear equations of the form
+ 2B(... )0xv + C(... )#vy = D(...
(1)
A, B, C, D being continuous functions of their five variables which satisfy the ellipticity condition
AC-B2>0
(2)
in the domain considered. Typical examples are the equation AO
(3)
= D(x,y,0,0¢,Ov)
the equation of minimal surfaces (4)
(1 + O2%. - 20,OyO
v
+ (1 + Ox)Oyv = 0
and the equation of a potential gas flow (5)
(P&)x + (Pov)v = 0,
p=1-y
2
(02,
1
+
Oy)I'v-1
where y > 1 is a constant. The latter equation is elliptic only if the flow is subsonic, i.e., if Ol + Oy < 2/(y 1).
-
7.1.
Boundary Value Problems
In studying boundary value problems for equation (1) we shall assume, merely for the sake of simplicity, that the domain considered is the unit disc I z I < 1 (x + iy = z). We shall consider the Dirichlet problem
(6a)
= r (given function) on 1': 282
IzI = 1
QUASI-LINEAR EQUATIONS
283
and the Neumann problem which is convenient to formulate as follows :
ao = c + y (constant + given function) on I',
(6b)
0 = 0 at 1 (The constant c is not prescribed but is to be determined.) In order to establish uniqueness theorems for a boundary value
problem one usually makes use of the obvious but important observation that if the coefficients of (1) are continuously differen-
tiable, the difference co = 01 - 02 of two solutions satisfies a linear elliptic equation. In fact, we have that i = 1, 2
A1o:,xx + 2B,0t>xv + CA,vv + D, = 0,
where A; = A(xy,0t,0i,x,0t,,,), etc. Subtracting the two equations, we obtain that (7)
A2(k)x, + 2B2wxv + C2wvv + (A2
- A1) 01,
+ 2(B2 - B1)01,xv + (C2 - C1)01,vv + (D2 - D1) = 0. The differences (A2 - A1), ... , can be computed as follows. Set
F(t) = A[x,y, 02 + (1 - t)01, 02,x + (1 - t)41,x, 02,v + (1 - t)01,vl Then
dt + to,, f
A2 - Al = fol F' (t) dt = co f lA,6 dt + (1,S f
dt o1A,.
o
Thus (7) may be written in the form (8)
A2w= + 2B2wxv + C2wyy + awt + bw + cw = 0
where 1
a = 01, faAd= dt + 201, fo
1
1
dt + 01.vv oCo= di ±
1
o
D,= dt
and b and c are defined similarly. Uniqueness of the Dirichlet or Neumann problem will be established if we can assert that the
284
L. BERS AND M. SCHECHTER
linear equation (8) has only the trivial solution u.) = 0 under the homogeneous boundary conditions. This is so, for instance, if A, B, C do not depend on 0 and aD/ao S 0. In this case c < 0 and the maximum principle is applicable (cf. Chapter 2).
7.2.
Methods of Solution
Existence proofs for our two boundary value problems can be reduced to the search of a fixed point of a transformation in an appropriately chosen Banach space B, the elements of which are functions defined in the unit disc. Let 'D(x,y) be an element of B. We form the linear equation (9)
airy + 2box, ± co,, = d
where
a(x,y) = A(x,y,t,(x,(D), .. .
and solve for it the boundary value problem considered. (We assume, of course, that this is possible, and in a unique way.) Denote the solution by ¢ = TO and assume that it belongs to B. Then T is a (nonlinear) mapping of B into itself and a fixed point
of T, i.e., a 0 such that 0 = To is a solution of the nonlinear boundary value problem. We will now describe several methods for proving the existence of a fixed point of T or, what is the same, of a root of the equation (10)
K(D=(1-T)(D =0
(a) Successive Approximations.
One may try to find a fixed point
of T by choosing some elements 0 and forming successively 01 = T O O , D2 = T01i etc. If the method converges, that is if (D)
'V and if T is continuous, then TO = (D. The convergence of the method of successive approximations
is often assured by the so-called principle of contracting mappings :
If T maps a closed convex subsets of a Banach space B into itself, and if there exists a constant 0, 0 < 0 < I such that (11)
IIT'V1 - T'V211 <_ 0 110 1 -'V211
QUASI-LINEAR EQUATIONS
285
then T has a unique fixed point which can be found by the method of successive approximations. There is hardly any need to write down the perfectly obvious proof. We have already applied this method several times in Chapter
5. There we dealt with linear operators T and condition (11) took on the form IITII < 1. Application of the method of succes-
sive approximations to nonlinear equations of the type of (2) are classical and will be found in several texts. In this method one embeds equation (10) into a one-parameter family of equations (b) Continuity Method.
(12)
KjO=0,
0
in such a way that K depends continuously on t, Ki = K and the equation KO O = 0 is easily solvable. Denote the set of t for which equation (12) is solvable by -r; r contains 0 and one wants to show that T is both open and closed.
In order to show that r is closed one needs to find a priori estimates of the solutions of (12) which imply that all solutions lie in a compact subset Zo of B. Then if t, c- T and t, -- to one can conclude that to c- -r. In fact, there will exist elements , in lo with Kt,(D, = 0. Since Eo is compact we may assume, selecting
if need be a subsequence, that 0, , $o. Then K0O0 = 0, so that to E -r.
In order to show that r is open one usually tries to show that if a zero of K,0 is known, a zero of Klo +, can be found by the method of successive approximations for e sufficiently small.
We have given an example of the continuity method, for a linear problem, at the end of Chapter 5. (c) Leray-Schauder [1] Method. This is a more sophisticated version of the continuity method. We assume now that T has the
property of mapping every bounded set into a compact set (complete continuity). Under this hypotheses, if T is defined on the
closure r of a bounded domain and K
0 on the boundary
E of E, Leray and Schauder define an integer d = d(E,K) which measures, intuitively speaking, the algebraic number of zeros of K in Z. The "degree" d is 0 if K 0 0 in Z. If K maps Z topo-
logically onto a domain containing the origin, then d = -*-1. The most important property of the degree is its homotopy
286
L. BERS AND M. SCHECHTER
invariance: d does not change if K is changed continuously, provided that no zero of K appear on E during the transformation. Hence we can prove the existence of a solution if we succeed to deform T continuously into a mapping To in such a way that no zero of K appear on the boundary during the deformation and the degree of KO = 1 - To is 0. For quasi-linear equations it is often sufficient to use a somewhat simpler method described below. (d) The Birkhof-Kellogg [ 1 ]-Schauder [3] Method is based on the extension to infinitely dimensional spaces of the well-known fixed
point theorem of Brouwer. This extension (Schauder's fixed point theorem) reads: If T is a continuous mapping of a closed convex set Z in a Banach space B into a compact subset Eo E, then T has a fixed point in E.
Note that for the particular mapping T with which we are concerned we ought to have no difficulty in establishing complete
continuity. In fact, 0 = TO is a solution of a linear elliptic equation the coefficients of which depend on (D and we know that a solution is "smoother" than the coefficients. The main difficulty lies in finding sufficiently strong a priori estimates which permit
one to apply the Leray-Schauder method, or the Schauder fixed point theorem. In general, the Schauder fixed point theorem can be used if the a priori estimates come from the theory of linear equations; if the estimates are derived from properties of nonlinear equations, the Leray-Schauder method must be employed. 7.3. Examples
We shall now apply method (d) to the Dirichlet and Neumann problems. We consider first an equation of the form (13)
+ 2B( ... %1, + C( ... )0,, = 0
and assume that the equation is uniformly elliptic (that means that
the coefficients are bounded and AC - B2 z const. > 0). We also assume that the boundary functions T and v in (6) are of class Cl_, and C2, respectively. Finally, and for the sake of brevity,
we assume that A, B, C are Holder continuous in their five
QUASI-LINEAR EQUATIONS
287
variables, though mere continuity would suffice (cf. BersNirenberg [2]. For an application of the same idea to gas dynamics see Bers [6]).
For B we take the space Ci+a (1), fi to be determined later. If 0 e C,.,_,,(1) and we form the linear equation
A(x,y,O,(Dx,O.)0zz + 2B(... ) 0., + C( ... )0y, = 0 then, according to the results of Chapter 6, our Dirichlet problem (or our Neumann problem) will have a unique solution 0 = TO. If we choose y sufficiently small we will have (14)
(15)
II0 II c,+Y
=
11TO ll el+V
<M
where M depends only on the uniform ellipticity of (13), the boundary function, and y but not on 0. Since the coefficients of (14) are Holder continuous, 0 will have Holder continuous second derivatives.
We choose
j9
so that 0 < fI < y.
Since
114IIc1,R <_ const. Ilollc,+,,
we see that T maps the closed convex set E :
II
Il
c,+ < Ml
of C,+,, (1) into itself and more precisely into the convex subset Ea
characterized by (15). But this subset is compact in Cl_, by Lemma 2 of Chapter 5. Next, T is continuous. For suppose that we have a sequence 0, in E and that 110, Oil c,+ ft , 0. Then there is a subsequence oo, ;n = Tl;y {.;n} such that for n in the Cl+. sense. We note that 0 satisfies the required boundary conditions. Also, the coefficients of the (linear) equation satisfied by 0;. converge to the coefficients of (14) and using the estimate in Theorem 3 of
-
Chapter 5 we can conclude that 0 satisfies (14). Thus 0 = TO and since TO is uniquely determined the selection of a subsequence
was unnecessary and we have that TED, - TO. Thus all hypotheses of the Schauder theorem are satisfied and we conclude that the Dirichlet and Neumann problems have solutions.
Note that we established existence under weaker conditions than those needed for a uniqueness proof. The argument would be almost the same if we were to assume
that the coefficients of (13) are only continuous. In this case
288
L. BERS AND M. SCHECHTER
solutions of (14) would have generalized L2 derivatives of second
order and the lemma on convergence from Chapter 6 would have to be used in establishing the continuity of T.
The same method works for equations of the form (1) if we require, for instance, that the right-hand side D( ... ) should satisfy the inequality K(1 + 10.1 +
(16)
For in this case every solution of (9) may be considered as a solution of a uniformly elliptic linear equation of the form (17)
acx,
- 2boxIl + co., + dcz + ecY = f
and we may again apply the estimates of Chapter 5.
We note also that in the case of the Dirichlet problem the assumption that the boundary function be Holder continuously
differentiable can be removed by approximating the given continuous boundary data by functions of class C1i
The assumption of uniform ellipticity, which is not satisfied
by such important equations as the equation (5) of minimal surfaces, can be removed in the case of the Dirichlet problem if we restrict ourselves (1) to equations of the form (13), (2) to boundary data of class C.2, and (3) to strictly convex domains bounded by a curve having continuous curvature. (In the previous results we worked with the unit disc for convenience only. In the case of the Dirichlet problem our argument could be carried out even for multiply connected domains.) We consider again the Dirichlet problem (6a) for equation (13) without assuming uniform ellipticity. Suppose we knew that for a fixed -r of class C2 the function 0 = TO (the solution of (14)
satisfying (6a)) satisfies the inequality (18)
II0IIc, <_ A1o
A10 being independent of 1. It is easy to construct a uniformly elliptic equation `1(X3',0,0s,Oti)Orz i 2B( ... )dry + C( ... )OYY = 0 such that (19) is identical with (13) for 11011r, < .110. This new equation has, by our previous result, a solution 0 satisfying (6a). This function is, however, also a solution of (13). (19)
QUASI-LINEAR EQUATIONS
289
Thus everything hinges on establishing the following A PRIORI ESTIMATE. Let 0 be of class C, on x2 -;- y2 < 1, of class C2 on x2 y2 < 1, and let it satisfy an elliptic (not necessarily uniformly elliptic) equation (20)
aox= + 2b¢xv + cOv,
=0
Furthermore, assume that 0= Ton x2 + y2 = 1, T being of class C2. Then (18) holds, M0 depending only on T.
This is often proved by using the so-called Rado-v. Neumann three-point-theorem. A simpler argument, going back probably to S. Bernstein, is as follows. First of all, 141 cmaxITI by the maximum principle. Since T is twice continuously differentiable we can find for every point (xo,yo) on the unit circle six numbers xi, fl1, YD 7-2, F'2, y2 such that (21)
gq
(22)
7-i2
1
,
2
2
2
#i, 7-? T 7-2
Kg
(K0 a constant independent of x0i (23)
alxo + P1yo + yl = 7-2x0 + P2yo + Y2 = T (xo)y0)
and (24)
7-,x + I'ly -F Y1 < T (X,_)') < a2x + 192J + Y2 for x2
-ry = 1 2
[Geometrically speaking: the planes 0 = 7-,x + /3,y --i- y, have uniformly bounded slope, pass through the point (x0,yo) and "enclose" the boundary curve of the surface 0 = O(x,)-).] Since every linear function satisfies (20), the maximum principle and (24) imply that (25)
7-1x + fl1y + Y1 < 0(x,y) < x2x + /2}' -J- 72
so that we obtain from (23) and (22) the inequality I
anI SK0
290
L. BERS AND M. SCHECHTER
for the normal derivative. The tangential derivative of 0 is bounded by that of T. Hence there is a constant Kl such that
on r
10.1 + I0yI < Ki
Since 0z and 0, satisfy the maximum principle (cf. Chapter 6) (26)
10.1 + Ioyl s Ki
everywhere, and from (21), (24), we obtain (18). In the present circumstances (i.e., without uniform ellipticity)
the condition that r be smooth cannot be removed, as can be shown by an example (Finn [1,2]). We have given only a few examples of nonlinear existence
theorems. Many more will be found in the literature-but it must be kept in mind that all such proofs (including those obtained by variational methods which we had no occasion to discuss) are
based on a priori estimates. It was the absence of methods for obtaining sufficiently strong a priori estimates in more than two dimensions which up to now restricted the theory of nonlinear elliptic equations to the plane. Quite recently, however, something
of a breakthrough has been achieved. On the one hand, one possesses now a priori estimates and existence proofs for quasi-linear
second order equations which have coefficients deviating not too
much from constants (Cordes [2], Nirenberg, and, for the gas dynamical equation, also Finn and Gilbarg [2]). On the other hand, De Giorgi [1] and Nash [1] obtained independently an interior a priori estimate for the Holder continuity of the solution of a uniformly elliptic equation written in divergence form
=0 the estimate depending only on max 101 (or the L2 norm of 0) and the ellipticity of the equation (cf. also Morrey [3], Stampacchia [2]). This work has been simplified and extended by Moser [1]. We have, therefore, good reasons to believe that significant progress in the general theory of quasi-linear elliptic equations is due in the near future.
291
Bibliography Agmon, Shmuel
1. Maximum theorems for solutions of higher order elliptic equations, Bull. Amer. Math. Soc., 66, 77-80 (1960).
2. The L. approach to the Dirichlet problem, Ann. Scuola Norm. Sup. Pisa (3) 13, 49-92 (1959). 3. The coerciveness problem for integro-differential forms, J. Analyse Math., 6, 183-223 (1958).
4. Multiple layer potentials and the Dirichlet problem for higher order elliptic equations in the plane I, Comm. Pure Appl. Math., 10, 179-239 (1957).
Agmon, Shmuel, Douglis, Avron, and Nirenberg, Louis 1. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, Comm. Pure Appl. Math., 12, 623-727 (1959). Ahlfors, Lars 1. On quasi-conformal mappings, J. Analyse Math., 4, 1-58 (1954). 2. Conformality with respect to Riemannian matrices, Ann. Acad. Sci. Fenn Ser. Al, 206, 1-22 (1955). Ahlfors, Lars and Bers, Lipman
1. Riemann's mapping theorem for variable metrics, Ann. Math., 72, 385 -404 (1960).
Aronszajn, Nachman 1. A unique continuation theorem for solutions of elliptic partial differential
equations or inequalities of second order, J. Math. Pures Appl. (9) 36, 235-249 (1957). 2. On coercive integro-differential quadratic forms, Conference on Partial Differential Equations, Technical Report No. 14, University of Kansas, 1954, pp. 94-106. Berg, P. W. 1. On the existence of Helmholtz flows of a compressible fluid, Comm. Pure Appl. Math., 15, 289-347 (1962). Bergman, Stefan
1. Jber die Entwicklung der harmonischen Functionen der Ebene and des Raumes nach Orthogonalfunctionen. Math. Ann., 86, 238-271 (1922).
2. Linear operators in the theory of partial differential equations, Trans. Amer. Alath. Soc., 53, 130-155 (1943).
Bergman, Stefan and Schiffer, M. M. 1. Kernel functions and elliptic differential equations in mathematical physics, Academic Press, New York, 1953. Berstein, Serge
1. Sur la nature analytique des solutions des equations aux derivees partielles due second ordre, Afath. Ann., 59, 20-76 (1904).
L. BERS AND M. SCHECHTER
292
2. Demonstration du theoremc de M. Hilbert sur la nature analytique des solutions des equations du type elliptique sans l'emploi des series normales. Math. Z., 29, 330-348 (1928). Bers. Lipman 1. Theory of Pseudoanalytic Functions. Lecture Notes, New York, 1953.
2. On mildly nonlinear partial differential equations of elliptic type, J. Res. Nat. Bur. Standards. 51, 229-236 (1953).
3. Univalent solutions of linear equations, Comm. Pure Appl. Afath., 6, 513-526 (1953). 4. Function theoretic properties of solutions of partial differential equations of elliptic type, Ann. Math. Studies, No. 33, Princeton, 69 -94 (1954). 5. Local behavior of solutions of general elliptic equations, Comm. Pure App!. Math., 8, 473 496 (1955). 6. Existence and uniqueness of a subsonic flow past a given profile, ibid., 7, 441-504 (1954). 7. Riemann Surfaces (mimeographed lecture notes), New York University (1957-38). 8. An outline of the theory of pscudoanalytic functions, Bull. Amer. Math. Soc., 62, 291 331 (1956). Bers, Lipman and Nirenberg, Louis 1. On a representation theorem for linear elliptic systems with discontinuous coefficients and its applications, Atti del Conregno internazionale sulle Equazionz alle derirate parzialli, Trieste, 1954, pp. I1 1 140.
2. On linear and non-linear elliptic boundary value problems in the plane, ibid., pp. 141-167. Bitsadzr, A. V. 1. On the uniqueness of solutions of the Dirichlet problem for elliptic partial differential equations, Uspehi Alai. Nauk. 3, 211 212 (1948) (Russian). Birkhoff, G. D. and Kellogg, O. D. 1. Invariant points in function space, Trans. Amer. Math. Soc.. 23, 96-115 (1922).
Bochner, Salomon
1. L'ber orthogonale Systcme analytischcn Functionen, Math. Z., 14, 180 207 (19221. Boyarskii. B. V.
1. Generalized solutions of systems of differential equations of first order
and elliptic type with discontinuous coefficients, Afat. Sb.. 43 (85), 451 -503 (1957) (Russian). Browder, F. E. 1. Strongly elliptic systems of differential equations, .4nn. Afath. Studies, No. 33. Princeton, 15-51 (19541.
2. On regularity properties of solutions of elliptic differential equations, Comm. Pure Appl..11ath. , 9, 351-361 (1956).
3. A priori estimates for solutions of elliptic boundary value problems, Nederl. Akad. lVetenrch. Indag..11ath.. 22, 145-169 (1960).
4. Estimates and existence theorems for elliptic boundary value problems, Proc. Vat..4cad. Sci., U.S.A., 45, 365- 372 (1959).
BIBLIOGRAPHY
293
Calderon, A. P.
1. Uniqueness of the Cauchy problem for partial differential equations, Amer. J. Math., 80, 16-36 (1958). Calderon, A. P. and Zygmund, Antoni 1. On the existence of certain singular integrals, Acta Math., 88, 85-139 (1952).
2. On singular integrals, Amer. J. Math., 78, 289-309 (1956). 3. Singular integral operators and differential equations, ibid., 79, 901-921 (1957).
Carleman, Torsten 1. Sur les systemcs lincaires aux d6riv6es partielles du premier ordre a deux variables, C. R. Acad. Sci. Paris, 197, 471-474 (1933).
2. Sur un probkme d'unicit6 pour les systc'mes d'equations aux &rivees partielles a deux variables independentes, Ark. Alai., 26B, 1-9 (19391). Chern, S. S. 1. An elementary proof of the existence of isothermal parameters on a surface, Proc. Amer. Math. Soc., 6, 781-782 (1955).
Cohen, P. J. 1. The non-uniqueness of the Cauchy problem, Technical Report No. 93, Applied Mathematics and Statistics Laboratory, Stanford University, 1960.
Cordes, H. O. 1. Uber die eindeutige Bestimmtheit der Losungen Elliptischer Differentialgleichungen durch Anfangsvorgaben, Nachr. Akad. 11ss. Gottingen Math.-Phys., Kl. IIa, 239-258 (1956).
2. Uber die erste Randwertuafgabc bei quasilinearen Differentialgleichungen zweiter Ordnung in mehr als zwei Variablen, Math. Ann., 131, 278 312 (1956). Courant, Richard 1. Dirichlet's Principle, Conformal Mapping, and Minimal Surfaces, Interscience,
New York, 1950.
Courant, Richard, Friedrichs, K. 0., and Lewy, Hans 1. Ober die partiellen Differenzgleichungen der mathematischen Physik, Math. Ann., 100, 32-74 (1928).
Courant, Richard and Hilbert, David 1. Methoden der :Matematischen Physzk, Springer, Berlin, 1937.
English
translation of Vol. I published by Interscience, New York, 1953. 2. Methods of Mathematical Physics, II, Interscience, New York, 1962. De Giorgi, Ennio 1. Sulla differenziabilita e l'analiticita delle estremali degli intcgrali multipli regolari, Mem. Accad. Sci. Torino, (3a) 3, Parte I. 25-43 (1957). Douglis, Avron 1. Uniqueness in Cauchy problems for elliptic systems of equations, Comm. Pure Appl. Math., 6, 291-298 (1953). 13, 593-607 (1960). Douglis, Avron and Nirenberg, Louis
1. Interior estimates for elliptic systems of partial differential equations, Comm. Pure Appl. Math., 8, 503 -538 (1955).
L. BERS AND M. SCHECHTER
294
Fichera, Gaetano
1. Alcuni recenti sviluppi della teoria dei problemi al contorno per le equazioni alle derivate parzialle lineari, Atti del Convego internazionale sulle Equazioni alle derivate parzialli, Trieste, 1954, pp. 174-227.
Finn, Robert
1. On equations of minimal surface type, Ann. Math., 60, 397-416 (1954).
2. Growth properties of solutions of non-linear elliptic equations, Comm. Pure Appl. Math., 9, 415-423 (1956).
Finn, Robert and Gilbarg, David 1. Asymptotic behavior and uniqueness of plain subsonic flows, Comm. Pure Appl. Math., 10, 23-63 (1957). 2. Three dimensional subsonic flows, and asymptotic estimates for elliptic partial differential equations, Acta Math., 98, 265-296 (1957).
Frank, Philipp and v. Mises, Richard 1. Die Differential- and Integralgleichungen der Mechanik and Physik, Vol. I, Mary S. Rosenberg, New York, 1943.
Friedrichs, K. O. 1. The identity of weak and strong extensions of differential operators, Trans. Amer. Math. Soc., 55, 132-151 (1944). 2. On the differentiability of solutions of linear elliptic differential equations, Comm. Pure Appl. Math., 6, 299-326 (1953).
Garding, Lars
1. Dirichlet's problem for linear elliptic partial differential Equations, Math. Scand., 1, 55-72 (1953). Giraud, Georges 1. ProblZmes mixtes et problPmes sur de varietes closes relativement aux equations lineaires du type elliptique, Ann. Soc. Polon. Math., 12, 35-54 (1934).
2. Generalizations des problcmes sur les operations du type elliptique, Bull. Math. Soc. France, 56, 248-272, 281-312, 316-352 (1932).
3. Problemes de valeurs a la frontiere relatifs a certaines donnces discontinues, ibid., 61, 1-54 (1933). Guseva, O. V. 1. On boundary problems for strongly elliptic systems, Dokl. Akad. Nauk SSSR, 102, 1069-1072 (1955) (Russian).
Hardy, G. H. and Littlewood, J. E. 1. Some properties of fractional integrals, Math. Z., 27, 565-606 (1928). Hartman, Phillip and Wintner, Aurel 1. On the local behavior of solutions of non-parabolic partial differential equations, Amer. J. Math., 77, 453-483 (1955). Heinz, Erhard 1. fiber die Eindentigkeit beim Cauchyschen Anfangswertproblem eines elliptischen Differentialgleichung zweiter Ordnung, Nachr. Akad. Wiss. Gottingen, Math.-Phys., Kl. Ha, 1-12 (1955).
Hopf, Eberhard 1. Ober den funktionalen, insbesondere den analytischen Charakter, der
BIBLIOGRAPHY
295
Losungen elliptischer differentialgleichungen zweiter ordnung, Math. Z., 34, 194-233 (1931).
2. Elementare Bemerkungen Tuber die Losungen partiellen Differentialgleichungen zweiter Ordnung vom Elliptischen Typus, Preus, Akad. Wiss. Sitzungsber., 19, 147-152 (1927).
Hormander, L. V. 1. On the uniqueness of the Cauchy problem, Math. Scand., 6, 213-225 (1958) ; 7, 177-190 (1959).
John, Fritz 1. Plane Waves and Spherical Means Applied to Partial Differential Equations, Interscience, New York, 1955.
2. Derivatives of continuous weak solutions of linear elliptic equations, Comm. Pure Appl. Math., 6, 327-335 (1953).
Kellogg, O. D. 1. Foundations of Potential Theory, Springer, Berlin, 1929; Dover, New York, 1953.
Korn, Arthur 1. Zwei Anwendungen der Methode der sukzessiven Annaherungen, Schwarz Festschrift, Berlin, 215-229 (1919).
Kondrashov, V. I. 1. Some estimates for families of functions satisfying integral inequalities, Dokl. Akad. Nauk SSSR, 18, 235-240 (1938) (Russian). Koselev, A. I. 1. A priori L. estimates and generalized solutions of elliptic equations and systems, U.spehi Mat. Nauk, 13, 29-89 (1958) (Russian). Ladyzenskaya, O. A. 1. On the closure of an elliptic operator, Dokl. Akad. Nauk SSSR, 79, 723-725 (1951) (Russian). Landis, E. M. 1. On some properties of solutions of elliptic equations, Dokl. Akad. Nauk SSSR, 107, 640-643 (1956) (Russian). Lavrent'ev, M. A.
1. A general problem of the theory of quasi-conformal representation of plane regions, Mat. Sb., 21, 285-320 (1947) (Russian). 2. Sur une classe des representations continues, Mat. Sb., 42, 407-434 (1935).
Lax, P. D. 1. On Cauchy's problem for hyperbolic equations and the differentiability of solutions of elliptic equations, Comm. Pure Appl. Math., 8,615-633 (1955).
2. A stability theorem for solutions of abstract differential equations and its application to the study of local behavior of solutions of elliptic equations, ibid., 9, 747-766 (1956). Lax, P. D. and Milgram, A. N.
1. Parabolic equations, Ann. Math. Studies, No. 33, Princteon, 167-190 (1954).
Leray, Jean and Schauder, Julius 1. Topologie et equations functionelles, Ann. Sci. Ecole Norm. Sup., 51, 45-78 (1934).
L. BERS AND M. SCHECHTER
296
Lewy, Hans
1. 1\euer Beweis des analytischen Charakters der Losungen elliptischer Differentialgleichungen, Math. Ann., 101, 609-619 (1929). Lichtenstein, Leon
1. Zur Theorie der Konformen Abbildungen. Konforme Abbildungen nicht-analytischer singularitatenfreirer Flachenstucke auf ebene Gebiete, Bull. Acad. Sci. Cracoi,ie, 192-217 (1916).
Lions, J. L.
1. Problemes aux limites en theorie des distributions, Acta Math., 94, 1-153 (1955). 2. Lectures on elliptic differential equations, Tata Institute of Fundamental Research, Bombay, 1957.
Lopatinskii, Ya. B. 1. On a method of reducing boundary problems for a system of differential equations of elliptic type to regular equations, Ukrain. Mat. Z., 5, 123-151 (1953). Magenes, Enrico
1. Sul teorema dell'alternativa nei problemi misti per le equazioni linear ellittiche del secondo ordine, Ann. Scuola Norm. Sup. Pisa (3) 9, 161-200 (1955). Malgrange, Bernard
1. Existence et approximation des solutions des equations aux derivees partielles et des equations de convolution, Ann, Inst. Fourier, Grenoble, 6, 271-355 (1955-6).
Miranda, Carlo 1. Equazioni Delle Derivate Parziali di Tipo Ellitico, Springer, Berlin, 1955.
2. Teorema del massimo modulo e teorema di esistenza e di unicita per it problema di Dirichlet relativo alle equazoni ellittiche in due variabili, Ann. Mat. Pura Appl., (4) 46, 265-312 (1958).
3. Sui problemi misti per le equazoni lineari ellittiche, Ann. Mat. Pura Appl., (4) 39, 279-303 (1955). Mizohata, Sigeru
1. Unicite du prolongement des solutions des equations elliptiques du quatrieme ordre, Proc. Japan Acad., 34, 687-692 (1958). Mori, Akira 1. On an absolute constant in the theory of quasi-conformal mappings, J. Math. Soc. Japan, 8, 156-166 (1956). Morrey, Jr., C. B. 1. On the analyticity of the solutions of analytic non-linear elliptic systems of partial differential equations, Amer. J. Math., 80, 198-237 (1958). 2. On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc., 43, 126-166 (1938). 3. Second order elliptic equations in several variables and Holder continuity, Math. Z., 72, 146-164 (1959). Morrey, Jr., C. B. and Nirenberg. Louis
1. On the analyticity of the solutions of linear elliptic systems of partial differential equations, Comm. Pure Appl. Math., 10, 271-290 (1957).
BIBLIOGRAPHY
297
Moser, Jurgen 1. A new proof of de Giorgi's theorem concerning the regularity problem for elliptic differential equations. Comm. Pure App!. Alath., 13, 457-468 (1960).
Motzkin, T. S. and Wasow, Wolfgang 1. On the approximation of linear elliptic differential equations by difference equations with positive coefficients, J. Math. Phys., 31, 253-259 (1953). :Miller, Claus 1. On the behavior of the solutions of the differential equation Au = F(x,u) in the neighborhood of a point, Comm. Pure Appl. Math., 7, 505-551 (1954).
Nash, John 1. Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., 80, 931-954 (1958). von Neumann, John
1. Jber einen Hilfssatz der Variationsrechnung, Abh. Math. Sem. Univ. Hamburg, 8, 28-31 (1931) .
Nikodym. O. M. 1. Sur un theoreme de M. Zaremba concernant les fonctions harmoniques, J. Math. Pures App!., 12, 95-108 (1933). Nirenberg, Louis 1. Remarks on strongly elliptic partial differential equations, Comm. Pure App!. Math.. 8, 648-674 (1955). 2. Uniqueness of Cauchy problems for differential equations with constant leading coefficients. Comm. Pure App!. Math., 10, 89-105 (1957).
3. On a generalization of quasi-conformal mappings and its application to elliptic partial differential equations, Ann. Math. Studies, No. 33, Princeton, 95-100 (1954).
4. Estimates and existence of solutions of elliptic equations, Comm. Pure Appl. Math., 9, 509-530 (1956). Pederson, R. N.
1. On the unique continuation theorem for certain second and fourth order equations, Comm. Pure App!. Math., 11, 67-80 (1958).
Peetrc, Jaak 1. `I'heoremes de regularite pour quelques classes d'operateurs differentiels, Thesis, Lund, 1959.
2. Mixed problems for higher order elliptic equations in two variables, I, 11, Ann. Scuola Norm. Sup. Pisa, (3) 15, 337-353 (1961) ; (3) 17, 1-12 (1963). Petrowsky, I. G.
1. Sur l'analyticite des solutions des systemes d'equations differentielles, Afat. Sb. N.S., 5 (47). 3-70 (1939). du Plessis, Nicolaas 1. Some theorems about the Riesz fractional integral, Trans. Amer. Math. Soc., 8, 124-134 (1955). Plis, Andrew
1. Non-uniqueness in Cauchy's problem for differential equations of elliptic type, J. Math. Mech., 9, 557-562 (1960). 2. A smooth linear elliptic differential equation without any solution in a sphere, Comm. Pure App!. Math., 14, 599-617 (196 1) .
L. BERS AND M. SCHECHTER
298
Privaloff, J. 1. Sur les fonctions conjugees, Bull. Soc. Math. France, 44, 100-103 (1916).
Protter, M. H. 1. Unique continuation for elliptic equations, Trans. Amer. Math. Soc., 95, 81-91 (1960). Rado, Tibor 1. Geometrische Betrachtungen Tuber zweidimensionale regulare Variationsprobleme, Acta Literarum ac Scientiarum Regiae Universitatis Hungaricae Franciso-Josephine, Sectio Scientiarum Mathematicarum, Szeged, 228-253', 1924-1926. Riesz, Marcel
1. L'integrale de Riemann-Liouville et le probleme de Cauchy, Acta Math., 81, 1-222 (1950). 2. Sur les fonctions conjugees, Math. Z., 27, 218-244 (1927). 3. Sur les maxima des formes bilineares et sur les fonctionelles line aires, Acta Math., 29, 465-497 (1926). Schauder, Julius 1. t ber lineare elliptische Differentialgleichungen zweiter Ordnung, Math. Z., 38, 257-282 (1934). 2. Numerische Abschatzung in elliptischen linearen Differentialgleichungen, Studia Math., 5, 34-42 (1934). 3. Der Fixpunktsatz in Funktionalraumen, Studia Math., 2, 171-180 (1930). Schechter, Martin 1. Solution of the Dirichlet problem for equations not necessarily strongly elliptic, Bull. Amer. Math. Soc., 64, 371-372 (1958). 2. Remarks on elliptic boundary value problems, Comm. Pure Appl. Math., 12, 561-578 (1959).
3. Various types of boundary conditions for elliptic equations, ibid., 13, 407-425 (1960).
4. Mixed boundary problems for general elliptic equations, ibid., 13, 183-201 (1960). 5. General boundary value problems for elliptic partial differential equations, ibid., 12, 457-486 (1959).
6. On estimating elliptic partial differential operators in the L2 norm, Amer. J. Math., 79, 431-443 (1957). Schwartz, Laurent 1.
The'orie des Distributions, I, Hermann, Paris, 1950.
Shamir, E. 1. Mixed boundary value problems for elliptic equations in the plane. The LP theory. Ann. Scuola Norm. Sup. Pisa, (3) 17, 117-139 (1963). Slobodeckii, L. N. 1. Estimates in L,, of solutions of elliptic systems, Dokl. Akad. Nauk SSSR, 123, 616-619 (1958) (Russian). Sobolev, S. L. 1. On a theorem of functional analysis, Mat. Sb. N.S., 4, 471-497 (1938) (Russian).
BIBLIOGRAPHY
299
2. On a boundary value problem for polyharmonic equations, Mat. Sb. (44) 2, 465-499 (1937) (Russian). 3. Some Applications of Functional Analysis to Mathematical Physics, Leningrad, 1950 (Russian).
Stampacchia, Guido 1. Contributi alla regolarizzatione delle soluzioni dei problemi al contorno per equazioni del secondo ordine ellittiche, Ann. Scuola Norm. Sup. Pisa, (3) 12, 223-245 (1958).
2. Problemi al Contorno ellittici con dati discontinui dotati di soluzioni holderiane, Edizioni Universitarie `f," Genova, 1960.
3. Problemi al contorno misti per equazioni del Calcolo della variazioni, Ann. Mat. Pura Appl., (4) 40, 177-193 (1955).
Vekua, I. N. 1. New Methods for Solving Elliptic Equations, Moskow-Leningrad, 1948 (Russian). 2. Systems of partial differential equations of first order of elliptic type and
boundary value problems with applications to the theory of shells, Mat. Sb., 31, 217-314 (1952) (Russian). 3. The problem of reducing differential forms of elliptic type to canonical form and the generalized Cauchy-Riemann system, Dokl. Akad. Nauk SSSR, 100, 197-200 (1955) (Russian). 4. Generalized analytic functions, Moscow, 1959 (Russian). English translation published by Pergamon Press, London, 1962. Visik, M. I. 1. On strongly elliptic systems of differential equations, Mat. Sb. (71) 29, 615-676 (1951) (Russian). Weyl, Hermann 1. Die Idee der Riemannschen Flache, Teubner, Leipzig-Berlin, 1923; Chelsea, New York, 1947.
Zaremba, Stanislas 1. Sur le principe du minimum, Bull. Intern. Acad. Sci. Cracow., 197-264 (1909).
2. Sur un probleme toujours possible comprenant, a titre de cas particuliers, le probleme de Dirichlet et celui de Neumann, J. Math. Pures Appl., 6, 127-163 (1927). Zygmund, Antoni 1. Trigonometrical Series, Monografje Mtematyczne, Warsaw, 1935; Chelsea, New York, 1952; Cambridge, 1959.
SUPPLEMENT I
Eigenfunction Expansions By Lars Garding
Eigenfunction Expansions I. Introduction
The spectral theorem for self-adjoint linear operators is one of the most successful abstractions of mathematics. It has scores of applications to analysis and to classical physics, and it is one of the mainstays of quantum mechanics. Since the theorem was first
proved by Hilbert [10] fifty years ago, it has passed through various stages. The full-dressed version for unbounded operators is due to von Neumann [ 18.1 ], the theory of normed rings has
contributed a very elegant proof [14] and the theorem itself has branched out into a general spectral theory (see, e.g., [6, 9, 12, 20, and 22.2]) and into the theory of noncommutative rings of operators [ 16, 18.2, and 18.3]. We shall be concerned with one simple aspect of the theorem which has been somewhat neglected until a few years ago, namely, its application to singular eigenfunction expansions. There is a historical reason for this neglect. When Hilbert had proved the spectral theorem for completely continuous operators, which have a complete set of strict eigenfunctions, and wanted to
generalize it to continuous, i.e., bounded, operators, he had to find a substitute for the strict eigenfunctions since they no longer
formed a complete set and even could be entirely missing. In modern notation, this substitute was a set {Ex} of commuting projections labeled by a real parameter a and nondecreasing with I. In terms of these projections, the spectral theorem is symbolized by the celebrated formula (1)
A=fAdEA
where A is the self-adjoint operator. It has a sense also for nonbounded operators. Although fairly satisfactory from a theoretical point of view, the spectral theorem in this form is not easy to
apply to concrete cases and provides no immediate way of dealing with the generalized eigenfunctions which appear, e.g., in the theory of the Fourier integral and in quantum mechanics. 303
304
L. GARDING
Since the singular eigenfunctions do not belong to the Hilbert space and the spectral theorem is concerned exclusively with that space, it is clear that some work is required in such an application,
but it can be reduced considerably if we replace (1) by an alternative statement which is implicit in the work of Hahn and Hellinger (sec Stone [22.1]). It is formulated explicitly by von Neumann [18.3] and says that any self-adjoint operator can be diagonalized by a unitary transformation. This is also the usual formulation in modern spectral theory. We shall illustrate it in the finite-dimensional case.
The simplest way of stating the spectral theorem for a selfadjoint operator A on a finite-dimensional Hilbert space H is to say that A has a complete set of orthonormal eigenvectors. We shall restate it in another form. First let m be the dimension of H and let run through a set Q of m elements, e.g., real numbers and eigenvalues and label the eigenvectors 0 =
2_
A with these numbers. Secondly, let L be the Hilbert space of functions from the set Q to the complex numbers with the scalar product
Then 5EQ
(2)
defines a unitary mapping from H to L such that the operator UAU*
(3)
which transforms L into L, is diagonal in the sense that it is multiplication by a function, in this case A
This is nothing
but a restatement of the fact that
If12=I If,0(s)I2 and
(U.4f)() _ (4f(5)) _ (f4()) _ %()(f,()) _ %()(Uf) so that
(U.4U*F)(E) _ %($)F() if we put F = Uf. The explicit form of U* = U-1 is (4) U*F = F()O(E) Hence, via the formula
f = U*(Lrf)
EIGENFUNCTION EXPANSIONS
305
U gives the eigenfunction expansion of any element f of H. We shall call Uf a Fourier transform off. It is important to note that the diagonalization (3) of A is not unique. In fact, a relabeling of the eigenfunctions will lead to a different function A. Additional requirements may be imposed to give unique canonical diagonalizations. They will be discussed below. In the general case of the self-adjoint operator A on a separable Hilbert space H, the spectral theorem is true in the form we have
just stated it, provided we take Q to be all real numbers and let L
be all functions which are square integrable with respect to a suitable measure. The orthonormal set of eigenfunctions of A disappears from the scene, but the unitary mapping U does not, and A is still diagonalized in the sense that UAU* is multiplication by a function. We may say that Uf f gives the Fourier coefficients of F without using a basis, and we shall refer to Uf as a Fourier transform off. The main object of this lecture is to restore the formula (2) by defining suitable generalized eigenfunctions outside the Hilbert space. How this is done will depend on how A and H are defined. We shall see that if H is, e.g., all square integrable functions on
an open subset of real n-space, then (2) and (4) still hold independently of how A is defined, with the eigenfunctions replaced by distributions. When A is an elliptic differential operator, these eigendistributions are functions which are not square integrable in general. Also in the abstract case it is possible to get a complete set of eigenfunctions by making them linear functionals on some suitably topologized linear dense subspace of the Hilbert space H. The choice of this subspace is, however, arbitrary, and the present state of the theory is fragmentary and unsettled.'
Apart from a few remarks, the contents of the lecture are not new; references to the literature will be given as we go along. Before continuing to our main subject, we shall state the spectral theorem in a more precise form without giving complete measure-
theoretical details. For simplicity, we limit ourselves to eigenfunction expansions connected with self-adjoint operators. Most ' Added in proof. Since this was written, the introduction of rigged Hilbert spaces has clarified the theory. See Gelbaud and Vileukin, Generalised Functions, Vol. 4, Chapter 1, Moscow, 1961.
L. GARDING
306
of the results carry over to symmetric operators (see [1] for the abstract theory and [5.3 and 17] for the applications). U. Direct Integrals. The Spectral Theorem
Let R" be real n-space and C the set of complex continuous
functions with compact supports in R". We shall deal with Stieltjes integrals
t = (ti,
51(1) dy(t),
. ,
t")
where f is a function in C, or, more generally a Borel function, and u is a measure. We assume that it is a Radon measure, which means that the integral is non-negative when f is non-negative and that it is bounded by the maximum of If (t) I times a constant which depends only on the support off. Conversely, by a classical
theorem by F. Riesz, any linear functional on C with these properties is the integral with respect to some Radon measure u. We get a measure ,u of this sort by putting, e.g., du (t) - h (t) dt,
dt = dill
... ,
dt,d
where h > 0 is locally integrable, but the measure can also be concentrated to a countable subset Q of R", in which case the integral is a sum
if (t) P (t) I
teQ
where u(t) > 0 is the measure of the point t. In the latter case we say that It is discrete. We shall denote by LZ(R",y)
the Hilbert space of all Borel functions f from R" to the complex numbers with finite square integral (1)
1112
=f If (1)21 du(t)
and the scalar product (2)
(f,g) = ff(t) g(t) du (t)
EIGENFUNCTION EXPANSIONS
307
Strictly speaking, we get a Hilbert space only if we identify two functions f and g when
If --gI =0 or, which is the same thing, when f (t) = g(t) except in a set of ,u-measure zero. More generally, let v be a dimension, i.e., a Borel function from
R" to the integers 0, 1, 2,
... and ac and consider vector-valued
Borel functions
k = 1, ... ) v(t) where the number of components is v(t) and varies from point to point. If we put
f (t) = (fx(t)),
f(t)
g(t) _ jJJ(t)gx(t) If(t)12
(3)
and
= f(t) '.f(t)
the integrals (1) and (2) have a sense for our new functions and we denote the corresponding Hilbert space by (4)
L2(R",#,v)
In the sense of von Neumann [18.3], it is a direct integral with
respect to y of a collection of Hilbert sequence spaces H, of dimensions v(t). The formula (3) means that we have referred each Ht to an orthonormal basis. This is done only for simplicity. In the last section we shall make use of other bases. We can now state the spectral theorem for a finite number of self-adjoint, bounded, and commuting operators
A,,...,Am on a separable Hilbert space H. It says that they can be simultaneously diagonalized by a suitable unitary' mapping U from H to a suitable
H, = L2(R",,,v) There are many such diagonalizing transformations U and we
can, if we want to, even prescribe that n = 1 and v = 1 (see [18.4]). It is, however, customary to choose a canonical diagonalization by requiring that m = n and that UAkU* is multiplication by the kth coordinate
L. GARDING
308
This means, in particular, that
now gives us the multiplicity
of the composite eigenvalue rr
rr
of the operators A. For canonical diagonalizations there is a uniqueness theorem (see [18.3]) : the measure y and the dimension function v are determined up to equivalence in the sense that this term is used in measure theory.
The spectral theorem extends to nonbounded self-adjoint operators A if we take commutativity in the sense that the bounded operators (A1 ± 1)-r, ... , (A. ± i)-1
commute with each other. Let us now consider some examples. First of all, we assume that the measure ,u is discrete. Then we have complete analogy with the finite-dimensional case: the diagonalizing transformation U can be expressed in terms of a complete set of mutually orthogonal eigenfunctions of the operators A. In fact, let y be concentrated to a countable set Q and construct for every in Q a set of functions *)5
k = 1, .
. .
, v( )
which vanish except at the point E where
Then Fk(E) = (F,G($,k))
for any F in H1 so that (f,g($,k))
for any f in H, if g($,k) = U*G(E,k) is the image of G(e,k) in H. It is clear that the functions G and g form complete orthogonal sets in H, and H, respectively, and are eigenfunctions of the operators UAU* and A, respectively. Conversely, if the operators A have a complete orthogonal set of eigenfunctions, we can, as we did in the introduction, construct a diagonalizing transformation U from H to a suitable space of the form (4) with a discrete measure
ju; if we want to, we can take n = 1 and v = 1. Well-known examples are the completely continuous operators, e.g., the various classical inverses of Laplace's operator in a bounded region.
EIGENFUNCTION EXPANSIONS
309
We shall now give an example of a diagonalization with a nondiscrete measure. We choose the`Fourier integral
i:rf (x) da
(W) ( ) =J e-
(5)
where
da = dxl ... dx It is well known that U is a unitary mapping from H = L2(R",a)
x = 1x1 +
. . .
,
to
H1 =
du = (27T)-n dal
... dEn
and provides us with a canonical diagonalization of the differential operators (6)
k = 1, ... , n
Dk = a/iaxk,
They are self-adjoint if we take the domain of definition of D. to be all f such that Dj is square integrable, or, equivalently
JI(ii2d,U < ac The transformation U-1 = U* is given by (U*F) (x)
=f
dy
Observe that we can write the right side of (5) as (f,o) where 0 = eiz6 is an eigenfunction of the operators (6) which is not square integrable. We shall meet more examples of this kind of degeneracy of (0.2) in this and the next section. By way of the formula
f (x) = LT*U.f(x) =
f
dy
we get an expansion of an arbitrary element f of H in terms of the eigenfunctions 0. If p(s) = p(EU .
. .
, Sn)
is a polynomial, U diagonalizes the differential operator (7)
B = p(D,i
. . .
, D,a)
L. GARDING
310
If p is real, B is self-adjoint if restricted to functions such that
dµ < 00
We have in this case a noncanonical diagonalization. It is instructive to make it canonical. This can be done by a transformation of variables as follows. Setting aside the trivial case when p is a constant, let t be a real parameter and let St be the
-
nonsingular (n 1)-dimensional part of the level surface p($) = t. Let N be the set of numbers t, for which St is not empty and define for t c N a measure cut in St by putting dtu = dt
dmot
Consider the Hilbert space Ht = L2(St,wt) of functions on St, square integrable with respect to wt, and let k = 1, ... , v(s) be a complete orthonormal set in Ht. If n = 1, is the number of zeros of t and if n > 1, we have to put x. Put H2 = L2(R,a,v) where da = dt on N and 0 otherwise. Finally, define a mapping V from Hl to H2 by putting (8)
(VF)k(t) =f
;F(E)hk(t,E)
when t e N and VF(t) = 0 otherwise. The reader will have no trouble verifying that V is unitary and that W = VU, which is a unitary transformation from H to H2i diagonalizes B canonically. An easy example is furnished by or, in quantum mechanical terms, the nonrelativistic Hamiltonian of n free particles. Observe that the spectrum of any B of the form (7) has uniform infinite multiplicity when n > 1. From (8) we obtain an analogue of (0.2) by expressing F = Uf in terms off. Formally the result is where (9)
( 4Tf)k(t) = (f,4k(t))
Ok(t,x) =Jet
`hk(t,)
- dwt()
EIGENFUNCTION EXPANSIONS
311
If Se is compact, which it is if B is elliptic in the sense that the polynomial p has definite principal part, then S6k is an eigenfunction
of B, but it is not square integrable. When St is not compact, but the functions h have compact supports, we still get functions
0, but for arbitrary orthonormal sets this is no longer true. To get an example of this, we can, e.g., put n = 2 and p = i8E2 with a larger integer s. Parameterizing St with dcul =
we get
28 d$j
so that any function vanishing at infinity and behaving like for small values of $, is square integrable on St. For such a function, however, the right side of (9) is not defined, but it is easy to see that it makes sense as a distribution. In fact, we shall see later that the eigenfunctions of differential operators always exist as distributions. III. Generalized Eggenfunctions
We are going to study situations when the formula (0.2) degenerates without losing its sense altogether. We begin with an interesting half-way abstract case treated by Gelfand and Kostyucenko [8]. First, a few notations. We let S = S(V) be the set of all complex infinitely differentiable functions vanishing outside compact subsets of an open subset V of real n-space. The results are true
also for functions with values in a finite-dimensional complex space, but we stick to complex functions for simplicity. Derivatives
will be denoted by Dk = a/iaxk and we put
D. = Dal, ... , D«,n and denote its order by Ial = m. To these symbols we add Da = 1, of order 101 = 0 and the norms max IDJ(x)I,
1«I < m
and
IDmfI = sup IDmf(x)I,
xeV
A subset W of V is called precompact if its closure W is compact and contained in V. We let S' be the space of distributions in V
(see [21]). It consists of all linear functionals L on S with the
L. GARDING
312
following continuity property. To every precompact open subset V' of V there exists an integer m and a constant c such that
IL(f)I
f
(f19) = fg dx,
f,g e s
any locally integrable function g gives rise to a distribution LD (f)
= (f g)
and any distribution is in fact a weak limit of such distributions with g infinitely differentiable. Following L. Schwartz, we identify
L. and g in this case and write
L(f) = (f,L) also in the general case. Now let (f,g) = P (.f g)
be a scalar product in S which is a distribution in V x V. This means that to any given precompact subset V' of V there exists an integer m and a constant c such that I (fg) I< c I D'f I IDmgI
when f and g belong to S(V'). The least integer m for which the inequality holds with some constant c is called the order of the scalar product P. We can, e.g., put
(f,g) = fj b«#DJDflg dt where the coefficients b are locally integrable and all except a finite number vanish outside any given compact subset of V. The simplest choice of P is (.f g) = (f, g) Let H be the Hilbert space obtained by completing S in the norm
(2)
IfI =-N/(ff)
EIGENFUNCTION EXPANSIONS
313
let A be a self-adjoint operator on H whose domain of definition includes S and let
Hqf
Uf E H1 =
be a diagonalization of A. Then we have ([8]). For almost all the components of the Fourier transform of a function f in S are linearly independent distributions. Their
THEOREM 1 :
orders have bounds which depend only on the order of P and the dimension n.
We shall sketch a simple proof which uses an idea of Berizanskii
[3]. The measure-theoretical details we leave to the reader. Consider the function
C(x) = (27r)-" f e_iz;(l Ex)-1 dE,
Ixl
p>n
It is p - n - 1 times es continuously differentiable and it is a fundamental solution of the differentiable operator
L = L (D) = Y
DX,
17.1
p
in the sense that
fC(x -y)Lf(y) dy =f(x) for any f in S(R). Let V' and V" be open precompact subsets of V which telescope so that V' c V" c: 17" c V, let h c- S be 1 on V' and put
b(x,y) = h(x)C(x -y)h(y) Then, if f E S(V') we have (3)
f
f (x) = b(x,y)Lf(y) dy
It is clear that if we choose p large enough, then b(-,_y) E H
for every}', and an easy approximation argument will show that operator L' commutes with the integration in (3) so that (4)
fB(E,y)Lf(y)
dy
L. GARDING
314
for almost all , where (Ub(-,y))($)
This is true in fact for any bounded operator from H to Hl. It is not difficult to prove that the function B can be chosen to be a Borel function in both arguments. Since U is unitary, Ib(-,y)12 =
f
da(d)
so that
j'Ib(.,.y)I2d y=f
da(d) dy <
00
This shows that
5IB(,y)I2dY < 00
for almost all $ and hence the components of the right side of (4) are indeed distributions. It is plain that the orders of these distributions in V' do not exceed n + I plus the order of the scalar product P in V". Varying the set V', we find that for almost such that all there exist distributions (5)
(Uf)k() _ (f, ()),
k = 1, 2,
... ,
for almost all E and all f in S. It is not difficult to prove that they
are linearly independent (see [2]). The spectral synthesis of a function f from its Fourier transform F = Uf is given by the formula
Mf =J
da(d)
where M is a mapping of H to S' defined by
(Mfg) = (f g) It is clear from (5) that the distributions 0x play the role of
(6)
eigcnfunctions of A although they do not belong to the Hilbert space H. There are special situations where this statement can be made more precise, e.g., when A maps S into a set of functions
EIGENFUNCTION EXPANSIONS
315
with compact supports which are so smooth that almost all eigendistributions are linear functionals in the set. Then (7)
(Af ,o(ff)) _
feS
which may also be written as
where A' is the adjoint of A with respect to the scalar product (1). An interesting example of this situation occurs when (2) holds
and A is a differential operator with sufficiently differentiable coefficients. If A happens to be elliptic ([4.1, 7.1, and 19]) or hypoelliptic ([11, 4.2, and 4.3]), then (7) implies that 0($) is a smooth function and we get an eigenfunction expansion similar to the classical Fourier transform. It turns out, (sec [8]), that the growth of the eigenfunctions at the boundary of S is limited by the growth of the coefficients of A. We obtain another interesting
application of theorem 1 if we let the operator M in (6) be an elliptic differential operator and let A be given (on S) by another differential operator L so that (Af,g) = (Lff g),
fgeS
(see [8 and 4.3]). Both M and L are supposed to have sufficiently smooth coefficients and to be symmetric in the sense that
( f,Mg) = (Mfg),
(Lf g) _ (f,Lg) In that case there exist vectors V(E) whose components are distributions S such that
(Uf)(E) _
If M' and L' are the adjoints of M and L with respect to the scalar product (1), this means that
L is elliptic or hypoelliptic, tp(E) is a function.
Theorem I was devised to apply mainly to differential operators.
There is another simple result (see [7.2]), due essentially to
L. GARDING
316
Mautner [15] and generalized in [2], which applies to integral operators. Let H = L2(W,a) be the space of square integrable functions on a measure space W with the measure a. We assume that W is a countable union of sets of finite measure and that His separable. We say that an operator B on H has the Carleman property if its domain of definition D(B) is dense and if (Bf) (x)
is a linear functional off e D(B) for almost all x in W. This means that
(Bf)(x) = Ib(y,x)f(,) dot(y)
(8)
where
r
c2 (x) = I I b (y,x) I2 da (y) < oo
so that B has a Carleman kernel b. It is possible to choose b to be measurable on W x W. We let C(B) be the set of functions f in H for which r (9)
I
I f(x) I c(x) da(x) < oo
It is dense in H. .1 Now let A be a self-adjoint operator on H and let
H3f-* LfeHI =L2(R,a,v) be a canonical diagonalization of A. We say that a complex measurable function h defined on R is essentially different from zero if the set where it vanishes is the sum of a a-nullset and a (necessarily countable) set contained in the point spectrum of A. THEOREM 2:
If B = h(A) has the Carleman property for some
function h which is essentially d ferent from zero, then for almost all A,
.(10)
((-f)(A) = (f,o(A)),
f c- C(B)
where the components of O(A) are linearly independent functions on W. When A is in the point spectrum of A, then
5I(A,x)I2d(x) = fI9k()I2 da(x) < 00
EIGENFUNCTION EXPANSIONS
317
Otherwise
ft(x) I0(A,x)12d% (x) < oo
(11)
for almost all
,
and every function t > 0 such that ft(x)c2(x) dot (x) < oo
Stripped of measure-theoretical detail, the proof of this theorem is very simple. From (8) we conclude that
f
(B*f) (y) = b(y,x)f(x) do, (x)
where f c C(B). It is not difficult to see that the operator U commutes with the integration and this gives (UB*f)(A)
=f B(1.,x)f(x) d«(x)
where B(A,x) = (Ub(x, -))(A)
(12)
Dividing by h we get (10) with
4(A,x) = h-'().)B(2,x)
(13)
It is easy to prove that the components of O(2) are linearly independent (see [2]). To get (11), we combine (12) and Parseval's formula, getting
f
c2(x) = I b(x,y) I2 dx(y) = f IB(2,x) I2 dd(.)
so that multiplying by t(x) and integrating
f
ff IB(2,x)12 t(x) dx(x) da(A) = t(x)c2(x) - dx(x) < oo
By virtue of (13), this implies (11) for almost all A.
Clearly, Theorem 2 applies where A itself has the Carleman property and this makes it possible to apply it to any self-adjoint operator A on a separable Hilbert space H. In fact, if we refer
L. GARDING
318
H to a complete orthonormal set we can think of it as a space of functions L2(W,x) where W consists of all integers of arbitrary sign and y assigns the measure one to every integer. If we choose all elements g1, x e W, of the orthonormal set in D(A), then where
f c D(A)
(Af) (x) = I a(x,y)f (y), a(x,y) =
is the matrix of A. Hence, A has the Carleman property, (9) reduces to
I If(x) I IAg.I < oo we have
and
A (A,x) = (Ua(x, ))(A)
(14)
I0().,x)I2
t(x)
< Co
where now t(x) IAgsI2 <
°C'
It is clear that difference operators furnish interesting examples of this situation. The estimate (11) shows that almost all eigenfunctions ck give linear functionals (15)
(ff ck(A))
on suitable subspaces of C = C(B) which are still dense in H. One small such subspace can be obtained as follows: we choose the elements gZ of the orthonormal set in the intersection I of all D(Ap), p = 1, 2, ... , and let Co be the subspace of H characterized by
I If (x) I
I Ag=I °
I APgxl 1(x) < x
for all integers p and q > 0. It is left to the reader to verify that Co is dense in H and contained in 1 and that A maps Co continuously
into CO. The linear forms (15) are clearly continuous on Co and we get (P(A)f ,0k(A)) = p(1) (f k(A)) for every polynomial P, everyf in Co, and almost all A.
EIGENFUNCTION EXPANSIONS
IV.
319
Ordinary Differential Operators
The best understood examples of singular eigenfunction expansions are those connected with ordinary differential operators of the second order (Sturm-Liouville operators) or, more generally,
ordinary differential operators of any order, and in fact, the first
general theory (Weyl [24]) preceded the abstract spectral theorem by twenty years. The literature on the subject is very large and there are expositions in, e.g., [5.1 and 17]. The fact that we now have a basis for the eigenfunctions makes it possible to obtain very explict results. In particular, if the basis is analytic in the eigenvalue parameter, the diagonalization can be obtained from Green's function via a formula due to Titchmarsh [23] and Kodaira [13]. The majority of authors in this field including, of course, Weyl, rely on complex integration or limiting processes as substitutes for the spectral theorem. For this reason, an abstract approach to the subject still has some freshness. Following [7.3],
we shall give a brief deduction of the main results from the general theory of the preceding section.
Let I be an open interval of the real line, bounded or not, let ..22 be the space of locally square integrable functions on I and Y02 the space of the functions in .72 which vanish outside compact subsets of I. Let
L = amDm + ... + a0i
D = d/idx
be a linear differential operator in I of order m > 0 and
+. + a L* = Dma. m
0
its adjoint with respect to the scalar product (1)
(f,g) = f fg dx
We require that L be formally self-adjoint, i.e., that L* = L. The coefficients of L cannot be arbitrary. We require that at has locally square integrable derivatives of order
320
L. GARDING
We make L into a closed operator from Y2 to 22 by letting
its domain of definition D(L) be all functions in 22 whose derivatives of orders m - 1 and m are absolutely continuous and locally square integrable, respectively.
Now let H = L2(I) be the space of all square integrable functions in I with the scalar product (1). If both its domain and its range are restricted to H, L becomes a closed densely defined operator B with the property that
B*cB i.e., B* is symmetric. If the deficiency indices of B* are equal, which we assume, then B has self-adjoint restrictions obtained by imposing linear boundary conditions on the functions of D(B). They have the form
W(.fg) = (L.fg) - (f ,,-g) = 0 where g runs through a maximal set M of solutions of B, = ig with the property that W(M,M) = 0. Let A be any such selfadjoint restriction of B and
H3f-, Eye H, = L2(R,or,v) a canonical diagonalization of A. By the general theory, there exist for almost all A linearly independent eigenfunctions 0k(A,x), (k = 1, ... , v(A)), solutions of (2)
14k = AOk
such that
(U )k(A) = (f,c k())
when f E 2. In particular, the dimension function cannot exceed the order m of the differential operator L. To prepare the ground for the Titchmarsh-Kodaira formula, we are going to put this result into another form by choosing new nonorthogonal bases for the eigenfunctions. Let (3)
Si(A,X),
. . .
, S.(A'X)
be a basis for the solutions of Ls' = As' whose elements are Borel functions of the pair A, x. If we express the functions k(A,x)
EIGENFUNCTION EXPANSIONS
321
in terms of this basis, Parseval's formula takes the form
(J,f)
-J
(fsj(A))(JiSk(A))C,k() da(A)
where (c,k(A)) is a positive semidefinite Hermitian matrix of
rank r(A). More precisely, let H2 be the Hilbert space of vectorvalued Borel functions F(a) = [F1(A), . . . , F,.(1)] with the scalar product (F,, G)
= f .1 F,Gk dP,k(2)
where dp,k(A) = c,k(A) da(%) is the differential of nondecreasing positive semidefinite Hermitian matrix p, called a spectral matrix. Then (4)
H3 f - Vf = ((fs,(1)), ...
)
is a unitary mapping from H to H., such that V.-4 V*
is multiplication by a. Clearly, the spectral matrix depends on A and the choice of the basis (3). As we shall see, it can be expressed explicitly in terms of Green's function. In this connection, we have to define Green's function as the kernel of the resolvent
R=(A-CE)-1,
Iml00
where E is the identity operator and t a complex number. Quite independently of the spectral theorem it is easy to show that
f
(R1f) (x) = g(e x,y)f(y) dy
where the kernel g is a continuous function of the triple f., x, y
except for a jump at x =y when m = 1. It is analytic in / and square integrable in one of the variables x and y when the other is fixed. Further g(I', y,x) = g(t,x, )')
(5)
and (6)
(Ly
- C)g
= (L. - ()g = 0
L. GARDING
322
when x y. For x =y, Green's function has the classical singularity: for fixed x and (, the function Dm-'g is continuous except
at the point y = x where it has an appropriate jump. It also satisfies the boundary conditions that determine A in the sense that hg e D(A),
(7)
(/and y fixed)
if the function h e D(L) vanishes in a neighborhood ofy and equals
1 in a neighborhood of the endpoints of the interval I. Most of these properties hold also when A is a self-adjoint restriction of an elliptic partial differential operator L with suitably differentiable coefficients defined in an open subset of the real n-dimensional space. The resolvent is still given by a kernel g(e,x,y), continuous
in the triple C, x, y when x y and Im t 0 and analytic in The formulas (5), (6), and (7) are still true, but the singularity has to be described otherwise. In a neighborhood of the diagonal x =y, Green's function behaves like rm-", where r is the distance between x and y, or when m - n >_ 0 and n is an even integer like r"-" log r-1. It is a Carleman kernel if, and only if, 2m - n > 0.
Returning to ordinary differential operators, we can now connect the spectral matrix with Green's function. First of all we choose the basis (3) in such a way that all its elements are entire analytic functions of A. This means that they are defined when A = /,is complex and
Ls =fs for every I. We get such a basis, e.g., by solving an appropriate
Cauchy problem. Since the functions s are now analytic, it follows easily from (5) and (6) that Green's function has the form
g((,x,y) _ I Mk(f)S!(l,x)sk(fy),
yzx
g(fsx,y) = I Mi
vSx
(f)si(f,x)sk(/ y),
where sk(C,y) = sk (l,y) and the functions A11(1)
are analytic off the real axis. Let M(f) be the matrix whose elements are
M,k(P) = Al (f) I Mix (10)
EIGENFUNCTION EXPANSIONS
323
Then we have the Titchmarsh-Kodaira formula M(1)
(8)
_
dP(A)
J
A-f
which should be taken in the sense that rN
M(f) -
dp(A) ,
N>0
is analytic across the real axis in the interval -N, N. More explicitly, this means that p(A) = 7r-1 lira lim
ra+a I
e--. +0 d
(M(,u + ie) - M(,u - ie)) dµ/2i
if we normalize p so that it is continuous from the right and p(O) = 0. That (8) holds, is a fairly easy consequence of the formula
ffg(t,x,y)f(x)f(y) dx dy = (R efj )
= f (A -
8)-i Y- (fss(A))(fsrk(;)) dAJk(A)
which in turn follows from the spectral theorem.
Bibliography 1.
N. I. Aehieser and I. M. Glazman, Theorie der linearen Operatoren im Hilbertraum, Berlin, 1954.
2. 3.
W. G. Bade and J. T. Schwartz, On Mautner's eigenfunction expansion, Proc. Nat. Acad. Sci. 42, 519-525 (1956). Yu. M. Berezanskii, Eigenfunction expansions of partial differential and difference equations, Proc. Third Congress of Soviet Math., 36-37 (1956).
4.1. F. E. Browder, The eigenfunction expansion theorem for the general self-adjoint singular elliptic partial differential operator I, II, Proc. Nat. Acad. Sci., 40, 454-463 (1954).
4.2. F. E. Browder, Regularity theorems for solutions of partial differential equations with variable coefficients, Proc. Nat. Acad. Sci., 43, 234-236 (1957).
4.3. F. E. Browder, Eigenfunction expansions for formally self-adjoint partial differential operators I. II, Proc. Nat. Acad. Sci., 42, 769-771, 870-872 (1956).
L. GARDING
324
5.1. E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955. 5.2. E. A. Coddington, On self-adjoint ordinary differential operators, Math. Scand., 4, 9-21 (1956). 5.3. E. A. Coddington, On maximal symmetric ordinary differential operators, loc. cit., 22-28. 6.
J. M. G. Fell and J. L. Kelley, An algebra of unbounded operators,
Proc. Nat. Acad. Sci., U.S. 38, 592-598 (1952). 7.1. L. Girding, Eigenfunction expansions connected with elliptic differential operators, The Twelfth Congress of Scand. Math., Lund, 1953, pp. 44-55.
7.2. L. Girding, Applications of the theory of direct integrals of Hilbert spaces to some integral and differential operators, Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, 1954.
7.3. L. Girding, Kvantmekanikens matematiska bakgrund, mimeographed lectures (in Swedish), Lund, 1956. 8.
9. 10. 11.
I. M. Gel'fand and A. G. Kostyucenko, Expansion in eigenfunctions of differential and other operators, Dokl. Akad. Nauk SSSR (N.S.), 103, 349-352 (1955). P. R. Halmos, Introduction to Hilbert Space and the Theory of Spectral Multiplicity, Chelsea, New York, 1951. D. Hilbert, Grundzuge einer Allgemeiner Theorie der linearen Integralgleichungen, B. G. Teubner, Leipzig, 1912.
L. Hbrmander, On the theory of general partial differential operators, Acta Math., 94, 161-178 (1955).
12.
J. L. Kelley, Commutative operator algebras, Proc. Nat. Acad. Sci.,
13.
K. Kodaira, On ordinary differential equations of any even order and the corresponding eigenfunction expansions, Am. J. Math., 72, 504-544
U.S., 38, 598-605 k 1952).
14.
(1950). L. H. Loomis, An Introduction to Abstract Harmonic Analysis, Van Nostrand, New York, 1953.
15.
F. Mautner, On eigenfunction expansions, Proc. Nat. Acad. Sci., 39, 49-53 (1953).
16.
F. J. Murray and J. von Neumann, On rings of operators I, II, Ann.
17.
Math. (2), 37, 116-229 (1936) ; 41, 208-248 (1937). M. A. Naimark, Linear Differential Operators, Moscow, 1954.
18.1. John von Neumann, Allgemeine Eigenwerttheorie hermitscher Funktionaloperatoren, Math. Ann., 102, 49-131 (1929). 18.2. John von Neumann, On rings of operators, III, Ann. ,Bath. (2), 41, 94-173 (1940). 18.3. John von Neumann, On rings of operators. Reduction theory, Ann.Math. (2), 50, 401-485 (1949). 18.4. John von Neumann, Ueber Funktionen von Funktionaloperatoren, Ann. Math. (2), 32, 191-226 (1931). 19. A. Ya. Povsner, On the expansion of arbitrary functions in characteristic functions of the operator -Au + cu, Mat. Sb. N.S., 32, 109-156 (1953).
BIBLIOGRAPHY
325
I. E. Segal, Decompositions of operator algebras, Afem. Am. Math. Soc. No. 9, 1-66 (1951). 21. L. Schwartz, Theorie dec Distributions, I, 11, Paris, 1951. 22.1. M. H. Stone. Linear transformations in Hilbert space, Am. Math. Soc. Coll. Publ. (1932). 22.2. M. H. Stone, Boundedness properties in function-lattices, Can. J. Math., 1, 176-186 (1949). 23. F. C. Titchmarsh, Eigenfunction Expansions Associated with Second Order 20.
Differential Equations, Oxford, 1946. 24.
H. Weyl, Ueber gewohnliche Diferentialgleichungen mit Singularitaten and die zugehorige Entwicklungen willkurlicher Funktionen, Math. Ann., 68, 220-269 (1909).
SUPPLEMENT II
Parabolic Equations By A. N. Milgram
Parabolic Equations In this lecture we shall discuss in a general way some of the methods leading to proofs of existence and uniqueness of solutions
of boundary value problems for parabolic partial differential equations. At first we shall restrict our attention to equations of the second order. General methods applicable to equations of order higher than the second can be exhibited just as well for second order equations. Moreover certain relations (e.g., the maximum principle) hold for equations of the second order which do not hold for equations of higher order. The first three sections will describe three different approaches to solutions of the heat equation Au - au/at =f. The first will be a direct Hilbert space approach, the second the method of potentials or integral equations, and the third will be Perron's method.
We shall limit our attentiion only to Dirichlet type boundary
problems. The methods can be adapted to equations with coefficients which are functions of space and time. In the fourth section we shall discuss the interrelation between these methods and also some problems whose investigation might be fruitful.
The fifth and concluding section consists of some remarks concerning parabolic equations of order 2m, m > 1. We call attention here to an obvious gap in the existent general theory. Reference in the sequal were chosen more or less at random to illustrate subject, without considerations of priority or even best results. I.
Direct Hilbert Space Approach to Solutions of the Heat Equation
We describe in this section a proof (based on a method of Vishik') of the existence of solutions of (I-1)
-Au +
at =f(X,t)
1 M. I. Vishik, Mixed boundary problems, Dokl. Akad. A'auk SSSR, 97, 193-6 (1954). 329
A. N. MILGRAM
330
where
0=
a2
axe
+
...
+
a2
and
x = (x1, ... , x,,)
a n,
0
The solution u(x,t) is sought in a domain D, where D = 0 x [0 < t < T] and 0 is a domain in Cartesian n-space En. Denoting by a0 the boundary of 0, we require further that u(x,0) = uo(x) and 0 for e aO, where uo(x), the initial value, is a given function. Let C* denote those functions of class C°° in D which are zero in a neighborhood of the upper boundary of D, i.e., in a neighbor-
hood of the lateral surface aO x [0 < t < T] and the of "top," 0 x [t - T]. For simplicity all functions are assumed to be real. for each 4 = O(x,t) E C* we denote ¢(X,0) by 0o. Multiplying I-1 by 0 and integrating over D, we obtain (1-2)
LO(U)
=[
-
aO
u] + [0,u] i = (0o,uo) + [O,,f ]
where the symbols in (1-2) are defined as follows: (go)uo)
=LOO(x)uo(x) dxi
[/,u] = (I-3)
u Y= (0o)o)
... dxii T
D
oO(x,t)u(x,t) dx2 ...dxn dt
- u dx dt = o
auo
n Y1=1
axi axi
fol, (0,u)1 dt
dx 1
...
dxn
foT J(.-.--) dxi ... dx dt i
1100112 We denote the associated norms by 1100112 = = 110112 11011 In addition we define = o = [010], and (0o) 0o) 1, for 0, ,u e C* an inner product
(O,t,)* := (0o' 'o) + [O"v]i
and complete C* to a Hilbert space H* by means of this inner product and norm 110 11 * =
(,)
-
PARABOLIC EQUATIONS
331
Recalling the well-known inequalities II0II0 < A 11011 (1-5)
II0II0 < A II0II1
where A is a constant depending only on 0, we apply Schwartz's inequality to (I-2) and obtain ILO(V')I s K(/) (II+'IIo + IIVII1) <_ K(0)[1 + A] IIjpQI*
L,4(V) is considered here to be a linear functional in y, E C* for
fixed, and K(0) is a constant depending on 0. By the FrechetRiesz theorem there exists To E H* such that (I-6)
L4(ip) = (V, TO) *
T¢ is clearly a linear operator with domain C*. Taking y, _ (I-6), we have
in
(,T)*
= L4(o) = (0o,0o) + [0,0J = 11011 *2 And this combined with Schwartz's inequality (O,TT)* s II0II* II TTII* gives II Toll* >_ II0II*
Thus T is a 1-1 mapping and has a bounded inverse T-1 of norm
<1, which may be regarded as extended to the closure of the range of T. We now consider the right hand side of (I-2). Let (I-7)
B,,O,f (0) = (c6o)uo) + [0,f J
Evidently,
f
IBuo, (O) l <_ K1 [1100110 + II0 II0]
-
K2 11011*
where Kl = II UoIIo + llf IIo, and K2 = 2 - K1 - max (1,A). Not
only is B01() a bounded linear functional for all 0 in H*, but also Buo, f (T-ltp) is a bounded linear functional on the closed subspace M of H*, and hence (I-8)
BUo.f (T 'V) = (', U) *
for some U E M, and arbitrary V E M.
A. N. MILGRAM
332
Replacing T-'y, by 0 in I-8, TO, U)
BUa.f
and by (1-6)
(TT,u) * = L (u) = B,,.f (0) = (00,uo) which establishes t% as a weak solution of (I-1). It is worthwhile observing that by replacing y, by u in (1-8), we obtain 1jul1 *2=B«a.f(T-1u)
and hence II UII * c 2 max (1,A)
(1-9)
-
(Il U0110 + Iif Ilo)
If in equation (I-1) u is replaced by e+"v, the equation assumes the form (I-10)
Acv
- iv - at = e-'(f (x,t) = g(x,t))
For equations of this form it is evident that II4II* should be and then the preceding discussion will apply without change. If now we consider a function
defined by the relation 11 'I *2 = 110'12
f =f
(txu,
I
u =f (1,
i 110112 + 1!011 ,
au
`
au
ate ,
,
.
,
which satisfies a uniform Lipschitz condition
f
all
t,X,u, ax)
-f
(t,x,v,
aL'
ax)
AO I U - vI -
A,
au
av
ax,
ax,
it is readily seen that (I-11)
e-xf (t,x,e I Atu,e,-u au
ax)
- e-; f
( . [III,
--Al
ar
-
where the constant is independent of A. In the equation (1-12)
-.1u
av
u, =f (t.x,u, au)
V112
2 110
it U -
ti112 2]
PARABOLIC EQUATIONS
333
replace u by e-' A - v, obtaining
-Av - Av + vit = e-xf I t,x,zveAt,eA1
(I-13)
ax1
It is readily seen that by successive substitutions the functions defined by vo(x,t) = e-uuo(x)
a,.-,
-Ov1 -}- Av1 -' ati = e-"f
will converge in *-norm (for sufficiently large A) to a solution of I-13 and hence after substitution of v = e - Atu, we obtain a solution of 1-12 with prescribed initial data and 0-boundary values. The requirement of a Lipschitz condition in u and au/ax imposed on f (t,x,u, auf ax) can be weakened somewhat. H.
Potential Theory for Parabolic Equations
The method of potentials was applied to parabolic equations many years ago.2 The fundamental solution of the heat equation Z(x,t,y,-r) = a(t - T) where
1x12 = xi -
X2
-n/2e-Ir-sI2/4(t
- ... + x
7)
and a =
2-nir-,,12
plays a role in this approach analogous to the role of Ix -A -"-2 in the Laplace equation. For parabolic equations of second order with coefficients depending on space and time the fundamental solution was investigated by F. Dressel (Duke Journal, 1940 and 1946).
Observing the relations
J0
Jf van
y,T) dy dT =
-.fo
f
aT
Z(x,t,y,T) dy dr
d6; dT = - f Z(x,t,y, t - e) dy +J Z(x,t,y,0) dy
2 Cf. Gevrey, Journal de Math., 1913.
A. N. MILGRAM
334
and passing to the limit as a --k 0, we obtain
where
Z(x,t,,T) dodT = -E(x) + f Z(xt0) dy
£JO
o
E(x) = 1 if x c- O
IifxE6and
is smoothatx
These are the basic jump relations. Continuing the analogy with classical potential theory, set
U(x,t) = rt
f6
dd, dt
an
U(x,t) is a solution of the heat equation with initial value 0 at be a given function defined on all interior points of 0. Let
the boundary O of 0. The requirement lim U(x,t) = z- ' o
where x approaches through interior points, after an application of the jump relations, gives rise to the following integral equations, for (µ00,t) :
A(Eo,t) = [E(go) - 1] µ00,t) +f00 J
U(E,T)
!n Z($o,t,$,T) dog; dt
Si nce this integral equation is of Volterra type, a discussion of existence and regularity of its solutions can be achieved by known methods, providing the boundary is sufficiently regular. Naturally
the question of what is meant by sufficiently regular can be an interesting and delicate problem about which we have yet to hear the last word. Also the formulation of minimal assumptions on the coefficients of the second order operator is still a subject of investigation.
111.
The Maximum Principle and Perron's Method
Let E x J be the Cartesian space of n --I,- 1 dimensions. Denote points of E x J by (x,t) = (x1) ... , xn, t) and let D be a bounded
domain in E. x J. If v(x,t) is a solution of Au = aul at, it is not
PARABOLIC EQUATIONS
335
difficult to establish the relation
(III-1)
u(xo,to) S sup u(x,t),
(xo,to) e D
xjeaD
where aD denotes the boundary of D. If D is a cylindrical domain,
D = 0 x [0 < t < T], the assertion (III-1) remains valid for Xo e 0, 0 < to < T if (x,t) in sup U(x,t) ranges over the lower boundary of D, i.e., the combined lateral surface aO x [0 < t < T]
and bottom 0 x [t = 0]. In the light of this maximum principle, it is natural to ask whether Perron's method, used first to obtain solutions of the first boundary problem for the Laplace equation, is also applicable to the heat equation.
After defining subparabolic and superparabolic functions, Sternberg,3 Petrowsky,4 and others showed that for each continuous function f defined on aD there exists a solution u of the
heat equation such that U(P) =f (P) at each boundary point P e aD around which there exists a barrier. The problem of determining nontrivial sufficient conditions for the existence of a barrier is fundamental in such investigations.
A function 0 defined in a domain D of E x J may be called superparabolic if (1) 0 is continuous in D except at the points of a
finite number of hyperplanes t = constant, and (2) if C is any cylinder with axis parallel to the t axis and contained in D and if u is any solution of Au - Ut = 0 in C such that U < on the lower boundary of C then u satisfies the same inequality, u interior to C.5 Reversing inequalities yields the definition of subparabolic functions.
A barrier at a point P0 of D is a superparabolic function u,(P) defined for some r > oo in D n S(Po,r), where S(Po,r) is the sphere with center P0 and radius in E,, x J, satisfying the conditions (1)
lim co(P) = 0 and (2) P-'Pt
inf
w(P) = CE > 0 for each E > 0.
fP -Pol ? e
3 Sternberg, Math. Ann., 101, (1929).
4 I. Petrowsky, Zur ersten randwertaufgabe der warmeleitungsgleichung, Compositio Math., 1 (1935). b This definition is contained in an unpublished paper of Professor W. Fulks, Department of Applied Math., University of Minnesota. See also, J. L. Doob,
A probablity approach to the heat equation, Trans. Amer. Soc., 80 (1955).
A. N. MILGRAM
336
The solution to the boundary value problem is sought as the infimuin of supcrparabolic functions with boundary values > a given function]. In his treatment of the above problem, Petrowsky restricted himself to the case of one space dimension: a2u/ax2 = au/at, and considered a domain bounded by two curves
x=0i(t)
X = 02(t),
0
He proved that there exists a solution with boundary value a preassigned continuous function f given on the base and sides of
D if the boundary curves have the property that for each t, 0 < t < T, there exists a function p(h) (= p(t,h)) such that: (1)
p(h) 10
(2)
01(t -{- h)
- 41(t) > - -%Yh log p(n) 02(t -- h) - 02(t) < Vih log p(h) E
(3)
as hf 0-
p(h)'Vi
J
g p(h) I
A - - cc
as e --) -0 and C < 0
If (1) and (2) hold but (3) fails, counterexamples can be constructed. These results represent an improvement on Gevrey's condition of Holder continuity on the functions 01, 02 with an
exponent a > - . Such refined conditions have not been obtained for the higher dimensional case. A crude sufficient condition for the existence of a barrier at a point PO E D5 is that PO be the vertex of a cone in the complement of D along the axis of which the t coordinate of a point decreases as the point recedes from PO (see figure above).
The problems discussed in the preceding sections have one feature in common in that all seek solutions of Au = U= which assume prescribed boundary values on D the boundary of a general domain in space-time. The solution sought cannot in general achieve all prescribed boundary values. For example, if IV.
PARABOLIC EQUATIONS
337
D is a cylinder with axis parallel to the t axis, the boundary values assigned at the top of the cylinder will not influence the solution
if the solution conforms to preassigned values on the lower boundary. Thus the boundary value problems considered must be regarded as a search for solutions of Au = u, which in some sense minimizes the difference between a prescribed function f and the values assumed by u on D. The word "minimize" as used in Section III means equal on a boundary set of "greatest measure." But if minimize were to mean the solution u(p) = u(x,t) is sought
so as to minimize the integral f If(p) - u(p)12 dS taken over the boundary, a new class of problems would arise. In such a problem prescribed boundary values on the top of a cylindrical domain would have material influence on the solution. A question related to the above which might be asked is the
following: let R be a subdomain of D. Let f be a prescribed function on R the boundary of R. What boundary values should be assigned on aD so that the solution u(x,t) best approximates f
on aR ? We leave open the sense in which the words "best approximates" are used. Of the methods used in Sections I and II we can say that the first is modern, the second classical. In the direct Hilbert space approach a solution of the boundary value problem is obtained by first solving Au - au/at = f with 0 boundary values. Then using f = - Af0 - afol at, where fo is a function defined and belonging to C2 in D, the function v = u + f0 provides a solution of Av aav/at with prescribed boundary values. The listener will observe at once that this seems to limit the boundary values to those which admit an extension into the interior of D of class C2(D). This would indeed be a very serious defect in the Hilbert space approach. Fortunately, in the case of second order equations the maximum principle comes to the rescue by guaranteeing that a sequence of solutions with uniformly convergent boundary values is uniformly convergent in the interior and the limit is readily shown to be a
solution. Hence the fact that the Hilbert space method is not directly applicable to all preassigned boundary values is not of vital importance. For equations of order higher than 2 the situation is different.
A. N. MILGRAM
338
V.
Equations of Higher Order
Let D be a domain in E" x J, and a = Jti, a2, ... , xK a sequence of integers where 1 < a, < n for each v. Set lal = K $12, ... , 41K. The and D" = aK/(axai 4,2 . . . 4,K), equation
a xt D"u =
V-1 jaj:z_ 21n
"
au at
will be called parabolic if Z a,D" is a strongly elliptic operator, i.e., if there exists a constant A > 0 such that
Rt[,ja.(x,t)e] z (-1),,1+1A
for2, ... ," real and
I
IEI2"
I2 = ($1)2 + ...
($1)2
0,
and (x,t) E D. Systems of equations can also be treated; definitions will be found in the other lectures. Various methods for treating
(V-1) have been given. The most direct and comprehensive is probably the method described in Section I, suitably modified. Of course, the solution obtained is at first only known to be a weak solution, this difficulty is easily overcome. Using the fundamental
solution whose existence and regularity was established (for systems) by Eidelman6 it is easy to verify by known methods that the weak solution is regular. Other methods including Friedrich's mollifiers can be used to verify regularity of the weak solutions in the interior. However the requirement that the boundary values can be extended into the interior, i.e., assumed by a function fo
which has finite m-norm, cannot this time be eliminated by a simple limiting process. For higher order equations no maximum principle is known. Thus even for elliptic equations the general existence theory based on Hilbert space methods is inadequate
to prove, say, the well-known theorem that the biharmonic equation L2u = 0 admits a solution u in the interior of a sphere so
that u = g, au/ an =f for arbitrary continuous functions f and g on the boundary. It is likely that at least for sufficiently smooth boundaries, the method analogous to that discussed in Section II will enable us to 6 S. D. Eidelman, On the fundamental solution of parabolic systems, Mat. Sb. N.S., 38, 80 (1956).
PARABOLIC EQUATIONS
339
obtain more complete results. Recently R. Juberg (Thesis, University of Minnesota) observing that for the fundamental solution (Eidelman) similar jump relations continue to hold, derived integral equations analogous to those described in Section II. The resulting system of equations, however, are complicated,
and his results are definitive in certain special cases. It is clear that an investigation of the integral equations both for parabolic and elliptic equations in an effort to obtain a Poisson integral type representation of the solution must lead to new and deeper insight into the boundary value problem for parabolic and elliptic equations.
Index
A priori estimates, 231, 263, 286, 289 Adjoint operator, 138 Analyticity theorem, 136, 139, 207
Direct integrals, 306 Dirichiet and Neumann problems, 263, 286
Dirichiet principle, 164 Dirichiet problem. 141, 152, 153, 154, 155, 159, 190, 196, 197, 237, 263, 282, 329 for the difference equation, 157 Discontinuities, propagation of, 53 Distribution, 138 Duhamel's integral, 17, 63
Banach-Saks theorem, 181 Beltrami equation, 257, 267 Boundary conditions, 141 Bicharacteristics, 58 Bounded linear functional, 179
Calderon-Zygmund inequality, 224, 245
Cauchy and Kowalewski theorem, 46 Cauchy data, 44, 62 Cauchy problem, 18, 45 standard, 63 uniqueness of, 47 Cauchy-Riemann system, 257 Characteristic form, 43, 70, 135 Characteristic lines, 7 Characteristic matrix, 43 Characteristic surface, 21, 43 Classical solution, 137 Closed linear subspace, 178
Elastic waves, equation of, 74 Elliptic, 143 Elliptic equations, 134, 135 Elliptic operators, 143 Elliptic systems, 255 Energy integrals, 24
Coerciveness, 201
Compact support, 7 Complete, 178 Completely continuous, 183,199 Complex notations, 256 Continuity method, 238, 285
Finite differences, solution of parabolic equations, 109 stability of difference schemes, 115 Fourier transformation, 65 Fredholm pair, 142, 175 Fredholm-Riesz-Schauder theory, 183
Descent, method of, 16
Free surfaces, 43 Function spaces, 216
Differentiability theorem, 138,
Estimates "up to the boundary," 235 Finite differences, 155 Finite differences, method of, 108
see also Symmetric hyperbolic systems
Fundamental identity, 211 Fundamental solution, 214, 215, 333
139,
174 341
INDEX
342 GArding's inequality, 170, 198
Liebmann's method, 159
General boundary value problems,
Linear equations with wave operator as principal part, 31
142
Generalized derivatives, 138 Generalized eigenfunctions, 305, 311 Gradient vector, 38 Green's function, 215 Green's identity, 48, 104
Heat
conduction,
boundary initial
value problem for rectangle, 104 equation of, 94 initial value problem, 98 maximum principle for, 96, 97 smoothness of solutions, 101 uniqueness of solutions, 96, 97 Heat equation, 329 Hilbert space, 177 Hilbert space approach, 329 Hilbert space methods, 165 Holder condition, 136 Holder continuity of derivatives, 136 Holder continuous, 138 Holder- Korn-L ich tenstein-Giraud inequality, 223, 244 Holmgren, uniqueness theorem of, 47 Huygens' principle in strong form, 15 Hyperbolic equations, linear, 69 solution by plane waves, 72 standard Cauchy problem for, 63 wave equation, 1, 2
Hyperbolic systems, Green's identity for, 48 with constant coefficients, 70
see also Symmetric hyperbolic systems
Integral equation approach, 215 Integral equations, 329 Interior estimates, 232 Interior regularity, 190 Interpolation lemmas, 218, 219, 250 Laplace equation, 134, 136 Lax-Milgram lemma, 180, 199 Leray-Schauder method, 285
Lions, 202
Maximum principle, 150, 160, 262, 334
Maximum principle for heat equation, 96
Maxwell's equations, 87 Minimal surfaces, 282 Mixed boundary problem, 143 Mollifiers, 145
Morrey-Nirenberg, 210 µ-conformal mapping, 267 Multi-index, 135 Negative norms, 200 v. Neumann condition, 118 Neumann problem, 154, 283 Norm, 135 Normal boundary conditions, 142 Normal velocity, 23,61
Oblique derivative problem, 154 Ordinary differential operators, 319 Parabolic equations, 2, 94, 329
see also Finite differences and Heat conduction equation Parseval's identity, 65 Periodic distribution, 168, 169, 174 Perron's method, 329, 334 Plane waves, decomposition into, 72 Poisson equation, 134, 137 Potential gas flow, 282 Potentials, 333 Potential theory, 211
Principle of contracting mappings, 283
Privaloff's theorem, 264, 267, 279 Projection theorem, 164, 177, 179 Propagation of discontinuities, 53 Properly elliptic, 144
Quasiconformal, 258
343
INDEX Regularity at the boundary, 200
Representation theorem, 164,
179,
259
Riesz-Schauder theory, 164 Runge property, 140
Scalar product, 178 Schauder estimates, 231, 238 Schauder's fixed point theorem, 286 Schwarz inequality, 166 Second-order equations, 150 Self-adjoint operators, 303, 305 Single hyperbolic equations, 62 Singular kernel, 222 Smoothness of strong solutions, 240 Sobolev and Kondrashov, 221 Sobolev's inequality, 220, 242 Sobolev's lemma, 30, 77, 167 Spaces Ht, 165, 167 Spacelike surface, 23
Spectral theorem, 303, 304, 305, 306, 307, 308 Stability of difference schemes, 116 Strictly hyperbolic, 72 Strong derivative, 137 Strong solution, 138 Strongly elliptic, 143, 144 Sturm-Liouville operators, 319 Subspace, 178
Support of a function, 7 Symmetric hyperbolic systems, 86, 87 difference methods for, 117
Tangential operator, 215 Test functions, 137 Timelike surfaces, 23 Titchmarsh-Kodaira formula, 320
Uniformly elliptic, 150, 255 Unique continuation, 162 Unique continuation property, 140 Unique continuation theorem, 262
Uniqueness, Holmgren's theorem of, 45
Uniqueness for heat equation, 97 Uniqueness theorems, 283
Wave equation, d'Alembert's solution for, 5 characteristics, 21 characteristic cone, 21 characteristic lines, 7 characteristic surface, 21
domain of dependence, 7 Huygens' principle in strong form, 15 inhomogeneous, 17, 18, 19 initial value problem for, 6, 10 mixed problems for, 35
mixed problem for finite x interval, 9
mixed problem for semi-infinite x interval, 7 one-dimensional, 4 spacelike surfaces, 21 three-dimensional, 10 timelike surfaces, 23 two-dimensional, see Descent, method of
Waves, see Plane waves and Elastic waves
Weak derivative, 137 Weak equals strong, 144 Weak solution, 138, 198 Weak vs. strong, 137 Well-posed problems, 2
Zeros of elliptic systems, 261
NBSI E-6h00-9T29-0
6
Z64008u1Z808L
SyjV uo ayi gaA&
L'£/Wb1
9JO-SUQE'AAAMAi
