Multiple Integrals in the Calculus of Variations (Classics in Mathematics)

Classics in Mathematics Charles B. Morrey, Jr.

Multiple Integrals in the Calculus of Variations

Charles B. Morrey, Jr.

Multiple Integrals in the Calculus of Variations Reprint of the 1966 Edition

123

Originally published as Vol. 130 of the series Grundlehren der mathematischen Wissenschaften

ISBN 978-3-540-69915-6

e-ISBN 978-3-540-69952-1

DOI 10.1007/978-3-540-69952-1 Classics in Mathematics ISSN 1431-0821 Library of Congress Control Number: 2008932928 Mathematics Subject Classification (2000): 49-xx, 58Exx, 35N15, 46E35, 46E39 © 2008, 1966 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMXDesign GmbH, Heidelberg Printed on acid-free paper 987654321 springer.com

Die Grundlehren der mathematischen Wissenschaften in Ein2;eldarstellungen mit besonderer Beriicksichtigung der Anwendungsgebiete Band 130

Herausgegehen von

J. L. Doob • E. Heinz • F. Hir^iebruch • E. Hopf H. Hopf • W. Maak • S. MacLane W. Magnus • D. Mumford • F. K. Schmidt • K. Stein

Geschdjtsjuhrende Heramgeher

B. Eckmann und B. L. van der Waerden

Multiple Integrals in the Calculus of Variations

Charles B. Morrey, Jr. Professor of Mathematics University of California, Berkeley

Springer-Verlag Berlin Heidelberg New York 1966

Geschaftsfiihrende Herausgeber:

Prof. Dr. B. Eckmann Eidgenossische Technische Hochschule Zurich

Prof. Dr. B. L. van der Waerden Mathematisches Institut der Universitat Zurich

All right reserved, especially that of translation into foreign languagesi It is also forbidden to reproduce this book, either whole or in part, by photomechanical means (photostat, microfilm and/or microcard or any other means) without written permission from the Publishers (c) by Springer-Verlag, Berlin • Heidelberg 1966 Library of Congress Catalog Card Number 66-24 365 Printed in Germany

Title No. 5113

Preface The principal theme of this book is ''the existence and differentiability of the solutions of variational problems involving multiple integrals/' We shall discuss the corresponding questions for single integrals only very briefly since these have been discussed adequately in every other book on the calculus of variations. Moreover, applications to engineering, physics, etc., are not discussed at all; however, we do discuss mathematical applications to such subjects as the theory of harmonic integrals and the so-called ''^^Neumann" problem (see Chapters 7 and 8). Since the plan of the book is described in Section 1.2 below we shall merely make a few observations here. In order to study the questions mentioned above it is necessary to use some very elementary theorems about convex functions and operators on Banach and Hilbert spaces and some special function spaces, now known as ''SOBOLEV spaces". However, most of the facts which we use concerning these spaces were known before the war when a different terminology was used (see CALKIN and MORREY [5]); b u t we have included some powerful new results due to CALDERON in our exposition in Chapter 3. The definitions of these spaces and some of the proofs have been made simpler b y using the most elementary ideas of distribution theory; however, almost no other use has been made of that theory and no knowledge of t h a t theory is required in order to read this book. Of course we have found it necessary to develop the theory of linear elliptic systems at some length in order to present our desired differentiability results. We found it particularly essential to consider ''weak solutions'' of such systems in which we were often forced to allow discontinuous coefficients; in this connection, we include an exposition of the D E GIORGI—NASH—MOSER results. And we include in Chapter 6 a proof of the analyticity of the solutions (on the interior and at the boundary) of the most general non-linear analytic elliptic system with general regular (as in AGMON, DOUGLIS, and NIRENBERG) boundary con-

ditions. But we confine ourselves to functions which are analytic, of class C"^, of class C^ or C^ (see § 1.2), or in some Sobolev space H^ with m an integer > 0 (except in Chapter 9). These latter spaces have been

vi

Preface

defined for all real w in a domain (or manifold) or on its boundary and have been used by many authors in their studies of linear systems. We have not included a study of these spaces since (i) this book is already sufficiently long, (ii) we took no part in this development, and (iii) these spaces are adequately discussed in other hooks (see A. FRIEDMAN [2]» HoRMANDER [1], LiONS [2]) as wcU as in many papers (see § 1.8 and papers by LIONS and MAGENES). The research of the author which is reported on in this book has been partially supported for several years by the Office of Naval Research under contract Nonr 222(62) and was partially supported during the year 1961—62, while the author was in France, by the National Science Foundation under the grant G—19782. Berkeley, August I966 CHARLES B . MORREY, JR.

Contents Chapter 1 Introduction 1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7. 1.8. 1.9. 1.10. 1.11.

Introductory remarks T h e p l a n of t h e b o o k : n o t a t i o n V e r y brief h i s t o r i c a l r e m a r k s T h e EuLER e q u a t i o n s O t h e r classical n e c e s s a r y c o n d i t i o n s Classical sufficient c o n d i t i o n s The direct methods Lower semicontinuity Existence T h e differentiability t h e o r y . I n t r o d u c t i o n Differentiability; reduction t o linear equations

1 2 5 7 10 12 15 19 23 26 34

Chapter 2 Semi-classical 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 2.8.

results

Introduction E l e m e n t a r y p r o p e r t i e s of h a r m o n i c f u n c t i o n s WEYL'S lemma P O I S S O N ' S i n t e g r a l f o r m u l a ; e l e m e n t a r y f u n c t i o n s ; G R E E N ' S functions Potentials Generalized potential t h e o r y ; singular integrals The CALDERON-ZYGMUND inequalities T h e m a x i m u m p r i n c i p l e for a l i n e a r elliptic e q u a t i o n of t h e s e c o n d o r d e r

39 40 41 43 47 48 55 6I

Chapter 3 T h e s p a c e s H"^ a n d H'^Q 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7.

D e f i n i t i o n s a n d first t h e o r e m s G e n e r a l b o u n d a r y v a l u e s ; t h e s p a c e s ii/^Q (G); w e a k c o n v e r g e n c e . T h e DiRiCHLET p r o b l e m B o u n d a r y values E x a m p l e s ; c o n t i n u i t y ; s o m e SoBOLEV l e m m a s Miscellaneous a d d i t i o n a l r e s u l t s Potentials and quasi-potentials; generalizations

. .

62 68 70 72 78 81 86

viii

Contents Chapter 4

4.1. 4.2. 4.3. 4.4.

Existence theorems The lower-semicontinuity theorems of SERRIN Variational problems w i t h / = f{p); the equations (1.10.13) with N = \, Bi = 0, A^" = A 2 . . Extensions of the DE GIORGI-NASH-MOSER results; v > 2 The case I' = 2

126 128 134 143

5.5.

Lp and SCHAUDER estimates

149

5.6. 5.7. 5.8. 5.9. 5.10.

The equation a-V^w + ^ *Vw + c w — Aw = / Analyticity of the solutions of analytic linear equations Analyticity of the solutions of analytic, non-linear, elliptic equations Properties of the extremals; regular cases The extremals in the case 1 < ^ < 2

157 i64 170 186 191

5.11.

The theory of LADYZENSKAYA and URAL'TSEVA

5.12. A class of non-linear equations

194

203

Chapter 6 R e g u l a r i t y t h e o r e m s for t h e s o l u t i o n s of g e n e r a l elliptic systems and boundary value problems 6.1. Introduction 209 6.2. Interior estimates for general elliptic systems 215 6.3. Estimates near the boundary; coerciveness 225 6.4. Weak solutions 242 6.5. The existence theory for the DIRICHLET problem for strongly elliptic systems 251 6.6. The analyticity of the solutions of analytic systems of linear elliptic equations 258 6.7. The analyticity of the solutions of analytic nonlinear elliptic systems 266 6.8. The differentiability of the solutions of non-linear elliptic systems; weak solutions; a perturbation theorem 277 Chapter 7 7A. 7.2. 7.3. 7.4. 7.5. 7.6. y.y. 7.S.

A v a r i a t i o n a l m e t h o d in t h e t h e o r y of h a r m o n i c i n t e g r a l s Introduction Fundamentals; the GAFFNEY-GARDING inequality The variational method The decomposition theorem. Final results for compact manifolds without boundary Manifolds with boundary Differentiability at the boundary Potentials, the decomposition theorem Boundary value problems

286 288 293 295 300 305 309 314

Contents

ix

Chapter 8 T h e ^ - N E U M A N N p r o b l e m on s t r o n g l y p s e u d o - c o n v e x m a n i f o l d s 8.1. Introduction 316 8.2. Results. Examples. The analytic embedding theorem 320 8.3. Some important formulas 328 8.4. The HiLBERT space results 333 8.5. The local analysis 337 8.6. The smoothness results 341 Chapter 9 I n t r o d u c t i o n to p a r a m e t r i c I n t e g r a l s ; two dimensional problems 9.1. Introduction. Parametric integrals 349 9.2. A lower semi-continuity theorem 354 9.3. Two dimensional problems; introduction; the conformal mapping of surfaces 362 9.4.

The problem of PLATEAU

374

9-5-

The general two-dimensional parametric problem

390

Chapter 10 10.1. 10.2. 10.310.4. 10.510.6. 10.7. 10.8.

T h e h i g h e r d i m e n s i o n a l PLATEAU p r o b l e m s Introduction V surfaces, their boundaries, and their HAUSDORFF measures . . . . The topological results of ADAMS The minimizing sequence; the minimizing set The local topological disc property The REIFENBERG cone inequality The local differentiability Additional results of FEDERER concerning LEBESGUE v-area

400 407 414 421 439 459 474 48O

Bibliography

494

Index

504

Multiple Integrals in the Calculus of Variations

Chapter 1

Introduction 1.1. Introductory remarks The principal theme of these lectures is ''the existence and differentiability of the solutions of variational problems involving multiple integrals/' I shall discuss the corresponding questions for single integrals only very briefly since these have been adequately discussed in every book on the calculus of variations (see, for instance, AKHIEZER [1], B L I S S [1], BOLZA [1], CARATHEODORY [2], F U N K [1], P A R S [1]. Moreover,

I shall not discuss applications to engineering, physics, etc., at all, although I shall mention some mathematical applications. In general, I shall consider integrals of the form (1.1.1)

I [^,G-) = ff[^,z{x),

V

z{x)]dx

G

where G is a domain, (1.1.2) X = (A;1, . . ., x"), z = (^1, . . ., z^), dx = dx^ . . . dx\ z{x) is a vector function, V^ denotes its gradient which is the set of functions {zioc}, where z^^ denotes dz'^ldx°', and f{x, z, p) [p = {pi}) is generally assumed continuous in all its arguments. The integrals b

f l/l + {dzldx)^ dx and j y [ ( ^ ) ' + ( ^ , ) ' ] a

dxidx^

G

are familiar examples of integrals of the form (1.1.1.) in which iV = 1 in both cases, i' = 1 in the first case, i^ = 2 in the second case and the corresponding functions / are defined respectively by

f{x,z,p) = yi +p^,

f{^,z,p)

=pi+pi

where we have omitted the superscripts on z and p since N =^ \, The second integral is a special case of the Dirichlet integral which is defined in general by (1.1.3)

D{z,G) = j\\Jz\^dx,f{x,z,p) G

^\p\^

=

2[pi)'^. i,(x

Another example is the area integral

we denote its derivatives duldx^" by u ,«. If ^ ^ C2(G), then V^u denotes the tensor u ,oc^ where oc and /5 run independently from t o r . Likewise V ^ ^ = {^,a/Sy}, etc., and \\/^u\^ — 2^\u,ocp\^, etc. If G is also of class C^, then Green's theorem becomes (in our notations) f u ,oc(x) dx = J uUocdS = f udx'^. G

dG

dG

Sometimes when we wish to consider u as a function of some single x'^, we write x = (%*, x^) and u(x) = u(x^, x^) where x'^ denotes the remaining x^. One dimensional or {v — 1)-dimensional integrals are then indicated as might be expected. We often let oc denote a ''multi-index'\ i.e. a vector [oci, . . .,av) in which each oct is a non-negative integer. We

1.3- Very brief historical remarks

then define d l«i u

(^,!),c« = ^ , f - = (fir-..(i^j^ Using this notation |a|=m

We shall denote constants by C or Z with or without subscripts. These constants will, perhaps depend on other constants; in this case we may write C = C{h, jbt) if C depends only on h and //, for example. However, even though we may distinguish between different constants in some discussion b y inserting subscripts, there is no guarantee that C2, for example, will always denote the same constant. We sometimes denote the support of u by spt w. We denote by C^iG), Q ( G ) , and C^, {G) the sets of functions in C°°(G), C^(G), or CJJ(G), respectively, which have support in G (i.e. which vanish on and near dG). But it is handy to say t h a t u has support in GR \J GR 2 , they found it necessary to use a gradient line method which led to a non-linear system of parabolic equations which they then solved; the curvature restriction was essential in their work. 1.4. The Euler equations After a number of special problems had been solved, E U L E R deduced in 1744 the first general necessary condition, now known as EULER'S equation, which must be satisfied b y a minimizing or maximizing arc. His derivation, given for the case N = v = 1, proceeds as follows: Suppose t h a t the function z is of class C^ on [a, b] (= G) minimizes (for example) the integral I{z, G) among all similar functions having the same values a t a and h. Then, if C is any function of class C^ on [^, h] which vanishes a t a and h, the function 2: + AC is, for every A, of class C^ on [a, h] and has the same values as ^ at a and h. Thus, if we define

6 (1.4.1)

w[X)=I[z+Xl:,G)=jf[x,z{x)

+Xl:{x)^z'[x)+XC[x)]dx

a

(p must take on its minimum for A = 0. If we assume t h a t / i s of class C^ in its arguments, we find b y differentiating (I.4.I) and setting X = 0 t h a t

h (1.4.2)

/{C'(^) 'U[x,z{x),z'{x)]

+ (:{x)f,[x,z{x),z'{x)]]dx

=0

(fv = The integral in (1.4.2) is called the first variation of the integral / ; it is supposed to vanish for every f of class C^ on [a, 6] which vanishes at a and h. If we now assume that f and z are of class C^ on [a, H] (EULER had no compunctions about this) we can integrate (1.4.2) b y parts to obtain

h (1.4.3)

jC{x)-{fz-~h]dx

= 0,

U=U[x,z[x),\/z{x)],etc.

8

Introduction

Since (1.4-3) holds for all C as above, it follows that the equation

(1.4.4)

y,=h

must hold. This is Euler's equation for the integral / in this simple case. If we write out (1.4.4) in full, we obtain (1.4.5)

fvv'^" +fvz^' +fpx=fz

which shows that Euler's equation is non-linear and of the second order. I t is, however, linear in z"; equations which are linear in the derivatives of highest order are frequently called q^iasi-linear. The equation evidently becomes singular whenever/^^ = 0. Hence regular variational problems are those for which fpp never vanishes; in that case, it is assumed that fpp > 0 which turns out to make minimum problems more natural than maximum problems. It is clear t h a t this derivation generalizes to the most general integral (1.1.1) provided t h a t / a n d the minimizing (or maximizing, etc.) function z is of class C2 on the closed domain G which has a sufficiently smooth boundary. Then, if z minimizes / among all (vector) functions of class C^ with the same boundary values and C is any such vector which vanishes on the boundary or G, it follows t h a t 2: + AC is a "competing'' or ''admissible" function for each A so t h a t if 99 is defined by (1.4.6)

(p{^) =I{z

+

U,G)

then (p'{0) = 0. This leads to the condition that (1-4.7)

ll\^C:Jpi

+ C'fz^dx=0

Q i=-l l a = l

J

for all C as indicated. The integral in (1.4.7) is the first variation of the general integral (1.1.1). Integrating (1.4-7) by parts leads to

Since this is zero for all vectors C, it follows t h a t

(^•4-8)

2i^M=f^'' '-''

.N

which is a quasi-linear system of partial differential equations of the second order. In the case A^ = 1, it reduces to

(1.4.9)

V

"-i

V

1.4. The EuLER equations

9

The equation (1.4-9) is evidently singular whenever the quadratic form (1-4.10)

2'^a2.,(^,^>^)^a^^*

in A is degenerate. We notice from (1.4-5) t h a t ii N = v = \ and / depends only on p and the problem is regular, then Euler's equation reduces to / ' = 0. In general, if / depends only onp(== pi), Euler's equation has the form

and every linear vector function is a solution. In particular, if A^ = 1 a n d / = 1^12, Euler's equation is just Laplaces equation a

In c a s e / = (1 + |_/>|2)i/4 as in the first example in § 1.3, we see t h a t

4fvP = {2-p^){^

+^2)-7/4

which is not always > 0. On the other hand fpp > 0 if | ^ | < ]/2 so classical results which we shall discuss later (see § 1.6) show that the linear function z{x) = x minimizes the integral among all arcs having \z-(x)\,y ca,5 + 2'hy co'(D,y + 'c(jo^]dy > 0. Now, choose 0 0 so that hlH -> 0, we conclude t h a t ^ ^^

^-^

-^ ^

'all (0) = ^a^ (^^) ^1 ^1 ^ a''^{xo) >^a A^ > 0 which is the stated result. This is called the Legendre condition.

1.5. Other classical necessary conditions

11

If we repeat this derivation for the case of the general integral (1.1.1), a s s u m i n g / ^ C^ of course, we obtain

^"(0) =I{ZI 0

< f (^) == hi 4 [^> ^ (^), V ^ [x)], 6?,. = /g^ pi^,

c^ J- = /^i 2?.

Making the change of variables (1.5-2) and (1.5-3) and setting co^(yi, . . ., y") = f^a)(yi, . . ., y"),

{i=

i, . . .,N)

where f is an arbitrary constant vector and co is defined by (1.5-4), and letting h and ^ -> 0 as above, we obtain (^•5.5)

2

fp^ paxo.z{xo),Vz(xo)]Ao:Apii&'^0

for a l U , f ,

which is known as the Legendre-Hadamard condition (HADAMARD [1]). In this case, we say that the integral (1.1.1) or the i n t e g r a n d / i s regular if the inequality holds in (1.5.5) for all A 9^ 0 and f 9^ 0. It turns out that the system (1.4-8) of Euler's equations is strongly elliptic in the sense defined by NIRENBERG [2].

Let us suppose, now, t h a t / ^ C^ everywhere and that ^ ^ C^ on G and minimizes I[z, G), as given by (1.1.1), among all such functions with the same boundary values. A simple approximation argument shows t h a t z minimizes / among all LIPSCHITZ functions with the same boundary values. Let us choose XQ^G and a unit vector X, and let us introduce the y coordinates as in (1.5-2) and let us define (using part of the notation of (1.5.4)

I

,

(y^ + h)(p(rh-^l^)

/^i/2(/ji/2 _ yi) .(p{rh-^l^),

0

,

—h 0

which is t h e Weierstrass condition (see GRAVES). I n case N = 1, (1.5-8) yields the following more familiar form of this condition:

E{x,z,Vz,P)

==f(x,z,P)

—f{x,z,Vz)

— [Poc —

z,^)f^^[x,z,\Jz)^0

(1.5.9) for all P and all x. The function E(x, z,p,P) here defined is known as the Weierstrass E-function. HESTENES and MACSHANE studied these general integrals in cases where v — 2. H E S T E N E S and E. H O L D E R studied the second variation of

these integrals. DEDECKER studied the first variation of very general problems on manifolds. 1.6. Classical sufficient conditions A detailed account of classical and recent work in this field is given in the recent book by FUNK, pp. 410—433) where other references are given. I shall give only a brief introduction to this subject. It is clear that the positiveness of the second variation along a function z guarantees that z furnishes a relative minimum to I[Z, G) among all Z ( = z on dG) in any finite dimensional space. However, if TV = 1, a great deal more can be concluded, namely t h a t z furnishes a strong relative minimum to / , i.e. minimizes I[Z, G) among all Z^ C^{G-) with Z = z on dG ior which \Z{x) — z(x)\ 0 regardless of the values of the derivatives. WEIERSTRASS was the first to prove such a theorem b u t his proof was greatly simplified by the use of HILBERT'S invariant integral. Of course, the original proof was for the case N = v = 1; we present briefly an extension to the case N = \, v arbitrary. Suppose G is of class Q , z^ C^ (G), a n d / and fp are of class C^ in their arguments, 0 < ju < i (see § 1.2), and suppose that the second variation, as defined in (1.5.1), > 0 for each C$ Q(G) (compact support). By a straightforward approximation, it follows t h a t the second variation is defined for all f ^ Hl^ (G) (see § 1.8). If we call the integral (I.5.I) h{z\^]G) we see from the theorems of § 1.8 below that 7-2 is lower-semicontinuous with respect to weak convergence in H\Q {G) . Moreover, from the assumed positive definiteness of the form (1.4.10), it follows from the continuity of the a'^P {x) (they ^ Q {G) in fact) t h a t there exist w i > 0 and M l such t h a t (1.6.1)

^«^ {x)?ioc?i^ > mi |A|2,

J [^'"^ W]^ ^ ^ 1 •

1.6. Classical sufficient conditions

13

Then, from the SCHWARZ and CAUCHY inequahties, we conclude t h a t there is a i^ such t h a t (1.6.2)

h{z\l:\G)>'^ j\\/^\^dx G

~ K j l:^dx. G

Since weak convergence in HIQ{G) imphes strong convergence in L2{G) (RELLICH'S theorem, Theorem 3-4.4), it follows t h a t there is a fo in (actually C^ (G)) which minimizes 12 among all I^^H\Q[G)

H\Q{G)

for

which f C^ dx = 1. Since we have assumed /2 > 0 for every f 9^ 0, it G

follows t h a t (1.6.3)

l2(z;C]G)

^AijC^dx^Ai

> 0.

G

From the theory of §§ 5-2—5.6, it follows t h a t there is a unique solution f of Jacohi's equation (1.6.4)

^^^i^(«"^f

- ) + iK -o)C = 0

with given smooth boundary values. I t is to be noted t h a t JACOBI'S equation is just the Euler equation (z fixed) corresponding to 12. I t is also the equation of variation of the Euler equation for the original / , i.e. (1.6.5)

^ f = ^ { ^Jpc. [^.^ + eC, Vz + QVCl-fz

[same] }^^^.

It follows from Theorems 6.8.5 and 6.S.6 t h a t there is a unique solution of the Euler equation for all sufficiently near (in C^ (dG)) boundary values, in particular for the boundary values z -{- Q, and t h a t z = z (Q) satisfies an ordinary differential equation (1.6.6)

p^^Fiz)

in theBanach space (C^^ [G)), where F{z) denotes the solution f of Jacobi's equation (1.6.4) with z = Z{Q) for which C = 1 on dG. We shall show below that this solution f cannot vanish on G for Q sufficiently small; it is sufficient to do this for ^ = 0, when Z{Q) = our solution z, on account of the continuity. So, let fi be this solution. If Ci(^) < 0 anywhere, then the set where this holds is an open set D and fi = 0 on ^ D ( C G). Since Ci is a solution on D, I(Ci, D) ^0 since Ci is minimizing on D. {D m a y not be smooth, b u t see Chapters 3 — 5)- But if we set ^ = ^i on D and C = ^, otherwise, f ^ HIQ {G) SO (1.6.3) holds and we must have C = 0. Hence Ci{^) > 0 everywhere. Now, suppose Ci(^o) = 0. From Theorem 6.8.7, it follows t h a t we m a y choose R so small that B(xo, R) G G and there is a non-

14

Introduction

vanishing solution co of (1.6.4) on B(xo,R). t h a t V satisfies the equation

Letting fi = coy, we see

and v(xo) = 0. But from the maximum principle as proved by E. H O P F [1] (see § 2.8), it follows t h a t v^ cannot have a minimum interior t o G. Accordingly Ci ^ 0 anywhere in G. Therefore it is possible to embed our solution z in s. field of extremals. That is, there is a 1 parameter family Z{X,Q) of solutions of Euler's equation where Z{x, 0) == z{x), our given solution. Z(X,Q)^ C^{GX [—^O>^O]) and ^ C^ (G) as a function of x for each Q with ^^ > 0. Consequently there are functions Po: {x, z) on the set F, where F: x^ G, Z{x, ~QO) < ^ < Z{x, go),

(\.6.7)

which act as slope-functions for the field, i.e. (1.6.8)

Z,oc{x, Q) = Pa [x, Z{x, Q)].

By virtue of the facts t h a t Z[x, q) satisfies Euler's equation for each Q, t h a t (1.6.8) holds, and t h a t if {x, z) $ F, then z = Z(x, Q) for a unique Q on [—^0, ^o], we find t h a t ^ ' ' ^ f,=fz\x^ z, P{x, z)l etc., (^, z) ^ F. Let us define /*(^, G) = ff*{x,z,Vz)dx, (1.6.10) G' / * {x. z,p) =f^, [X, z, P [x, z)]' \_p. - P« [x, z)\ + / [ ^ , z, P {X, z)\. We observe that Sl = [P- -

^^''^^^

Poc {X, Z)] ' {fp^^ + / p ^ p^ Ppz } -

Poczfvu

+fz+fv,Po.z; f;^=fvJ^>z,P(x,z)].

Thus, ii z^ Gi(G) and {x, z{x)) ^F for x^G, we see that (1.6.12)

/ : [X, z (x), Vz(x)]

- ^j;^

[X, z {x), Vz{x)]

= 0.

Accordingly the integral I*(z, G) has the same value for all such z which have the same boundary values. Moreover, if z{x) = Z[x, q) for some q, then ^

'

I*{z,G)=I{z.G).

This integral 7 * (^, G) is known as Hilbert's invariant integral. Therefore, if z^ Ci(G) and [x, z{x)) ^ F for all x^ G, and z{x) = zo[x) on dG, then (1.6.14)

I{z, G) - I(zo, G) = I(z, G) - I*{zo, G) = I(z, G) - I*{z, G) = f E[x, z{x), P{x, z{x)}, Vz{x)] dx

1.7. The direct methods

15

where E(x, z, P, p) is the Weierstrass £^-function defined in (1.5-9). Thus ZQ minimizes I{z, G) among all such z, and hence furnishes a strong relative minimum to / , provided that (1.6.15)

E[x, z, P{x, z),p] > 0, [x, z)^r,p

arbitrary.

This same proof shows t h a t if (1.6.15) holds for all {x, z, p) in some domain Ry where all the (x, z) involved ^ F, then I(zo, G) < I{z, G) for all z for which [x, z[x), \7 z{x)'] ^Rfor all x^G. In the cases v = \, the Jacobi condition is frequently stated in terms of ''conjugate points". A corresponding condition for )^ > 1 is t h a t the JACOBI equation has no non-zero solution which vanishes on the boundary dD oi any sub-domain D G G; D m a y coincide with G or may not be smooth, in which case we say u vanishes ondD on G; we have seen above t h a t this is implied by the positivity of the second variation. If we then set ^ = cou, where u = 0 on dG, then the reader may easily verify t h a t I2{^',^',G)

= J[w^a°'^u,ocU,p

— u^a)L(o]dx

> 0

G

for all u(^ H\Q (G), since Leo = 0. In cases where v > 1 and A^ > 1 it is still true (if we continue to assume the same differentiabihty for G , / , and z) t h a t (1.6.2) holds with I2 defined by (1.5.1') even in the general regular case where (1.5-5) holds with the inequality for A 7^ 0 and f 7^ 0. This is proved in § 5.2. So it is still true t h a t if /2 > 0 for all f, the Euler equations have a unique solution for sufficiently nearby boundary values. I t is more difficult (but possible) to show t h a t there is an A^-parameter field of extremals and then it turns out t h a t such a field does not lead so easily to an invariant integral. By allowing slope functions P* {x, z) which are not "integrable" (i.e. there may not be zs such that z^^ = P^ {x, z)), W E Y L [1] (see also D E BONDER) developed a comparatively simple field theory and showed the existence of his types of fields under certain conditions. His theory is succesful i f / i s convex in all the p'l. To treat more general cases, CARATHEODORY [1], BOERNER, and L E PAGE have introduced more

general field theories. The latter two noticed t h a t exterior differential forms were a natural tool to use in forming the analog of Hilbert's invariant integral. However, the sufficient conditions developed so far are rather far from the necessary conditions and many questions remain to be answered. 1.7. The direct methods The necessary and sufficient conditions which we have just discussed have presupposed the existence and differentiability of an ex-

16

Introduction

tremal. I n the cases v = i, this was often proved using the existence theorems for ordinary differential equations. However, until recently, corresponding theorems for partial differential equations were not available so the direct methods were developed t o handle this problem and to obtain results in the large for one dimensional problems. As has already been said, H I L B E R T [1] and L E B E S G U E [2] h a d solved

the Dirichlet problem b y essentially direct methods. These methods were exploited and popularized b y TONELLI in a series of papers and a book ([1], [2], [3], [4], [5], [7], [8]), and have been and still are being used b y many others. The idea of the direct methods is to show (i) t h a t the integral to be minimized is lower-semicontinuous with respect t o some kind of convergence, (ii) t h a t it is bounded below in the class of ''admissible functions," and (iii) that there is a minimizing sequence (i.e. a sequence {zn] of admissible functions for which I[zny G) tends to its infimum in t h e class) which converges in the sense required to some admissible function. Tonelli applied these methods t o many single integral problems and some double integral problems. I n doing this he found it expedient to use uniform convergence (at least on interior domains) and to allow absolutely continuous functions (satisfying the given boundary conditions) as admissible for one dimensional problems; and he defined what he called absolutely continuous functions of two variables ([6]) to handle certain double integral problems (see the next section). I n the double integral problems (iV == 1, r = 2), he found it expedient to require t h a t f[x, z, p) satisfy conditions such as (1.7.1) (1.7.2)

m\p\^ — K 2, m>0, where f{x, z,p)'>0 and f{x, ^, 0) = 0 if k = 2,

in order to obtain equicontinuous minimizing sequences. However, Tonelli was not able t o get a general theorem to cover the case where / satisfies (1.7-1) with 1 < ^ < 2. Moreover, if one considers integrals in which r > 2, one soon finds t h a t one would have to require ^ > i^ in order to ensure that the functions in any minimizing sequence would be equicontinuous, at least on interior domains (see Theorem 3.5-2). To see this, one needs only to notice t h a t the functions loglog(1 + 1/|:v|), 1/|A;|^

0 o o . The convexity assumption for all (x, z,p) is suggested by the conditions in §§ 1.4—1.6. In the existence theorems mentioned above, the author considered integrals of the form (1.1.1) in which v and N are arbitrary but in w h i c h / is convex in all the pl^; with this convexity assumption, no difficulties were introduced in the proofs by allowing AT > 1. The results have been extended and the old proofs greatly simplified by SERRIN ([1], [2]); we shall present (in § 1.8) a simple lower-semicontinuity proof based on some of his ideas and on some ideas of TONELLI. However, for iV > 1, the proper condition would be to assume t h a t (1.5-8) and/or (1.5-5) held for all (x, z, p, A, f). The author has studied these general integrals ([9]) and found that if / satisfies the conditions mV^

- K 1

F = (1 + 1^12)1/2

then a necessary and sufficient condition that I[z, G) be lower semicontinuous on the space H\ (G) with respect to uniform convergence is t h a t / be quasi-convex as a function oi p. A function/(^), p = {pl^} is quasiconvex if and only if it is continuous and jflPo

+ VC(«)] dx >f{po)

• \G\,

C€ Q ( G ) ;

G

t h a t is, linear vectors give the absolute minimum to I{z, G) among all z with such boundary values (note that linear functions always satisfy EULER'S equation if f^C^). A necessary condition for quasi-convexity is just (1.5.5)- The author showed t h a t (1.5-5) is sufficient for quasi-convexity if f{p) is of one of the two following forms Wm

= 1 is not convex in all the ^ j , b u t is a regular integrand in the general sense; for v = N = 2, the integral is

I(z,G) = JJl/l + (4)' + (4)' + (4)^ + (4)' + (TlJ^J'i^dy. G

f{x,z,p) = [1 + iPl)^ + (Pl)^ + iPl)^ + (Pl)^ + iPlPl - PlPiW^. The integrand of a necessarily degenerate (1.5.5) and (1.5.8) with speak about parametric

parametric problem is never regular since it is b u t a large class of such integrands do satisfy the equality allowed (whenever/^ C^). We shall problems in Chapter 9-

1.8. Lower semicontinuity We begin with a brief discussion of the spaces of admissible functions which we shall use; a more extended discussion including complete proofs is given in Chapter 3. I t is convenient to call these spaces SOBOLEV spaces; in addition to the brevity of this designation, it is appropriate since he proved some important results concerning these functions [1] and popularized them in his book [2]. However, he was by no means the first person to use these functions. Beppo LEVI was probably the first to use (in (I906) admissible functions which required the use of the Lebesgue integral to express I{z, G); his functions (of two variables) were continuous, absolutely continuous in each variable for almost all values of the other, and their first partial derivatives were in L2. In discussing the area of surfaces, TONELLI [6] introduced his notion of absolutely continuous functions (ACT) in 1926. His definition was identical with that of Levi except that the partial derivatives were required only to be in Li; he used these functions to discuss the double integral problems mentioned above. But meanwhile in I92O, G. C. EVANS {W> [2]) had encountered more general functions, essentially those we now use, in his study of potential theory. RELLICH proved the compactness in L2 of bounded sets in HI. SOBOLEV proved his well known results on theLp-properties of these functions in 1938 [1]. I n 1940, J . W. CALKIN and the author ([5]) proved many of the fundamental properties of these functions stated below. Since the war ARONSZAJN and SMITH have studied these functions in great detail [1]. No doubt many others have studied these functions. Recently these functions have been used by many people in m a n y different connections (see, for instance, D E N Y , F R I E D R I C H S [1], [2], [3], FuBiNi, J O H N [2], L A X , MORREY [1], [10], MORREY and E E L L S , 2*

20

Introduction

NiKODYM, SHIFFMAN, SIGALOV ([2]); their use is now standard in partial differential equations (see FRIEDMAN [2], HORMANDER [1], LIONS [1]).

Definition 1.8.1. We say that a function z is of class Hl[G), r > 1, if and only if z is of class Lr[G) and there exist functions pi, . . ., pv, also of class Lr{G), such that (1.8.1)

jg[x)Pa,[x)dx= G

-lg,oc{x)z{x)dx,

g^Cl{G),oc=

\,...,v.

G

Remarks. I t is clear that the functions px are uniquely determined up to null functions and that if z is of class Hj{G) and z'^{x) = z{x) almost everywhere, then ^* is of class Hl{G) and the same functions po^ will do for 2:*. Definition 1.8.2. As in the case of the L^-spaces, the elements of the space Hl[G) are classes of equivalent functions of class Hj(G). We denote the classes of equivalent functions px by z^« and call them the distribution derivatives of the element z. An element z^ space H'^[G) if and only if z and its distribution derivatives u p to order m — 1 are successively seen to ^ Hj (G). Remark. Naturally we may regard an element z in H^ (G) as a distribution and then the distribution corresponding to z^« would be the derivative of z in the distribution sense. Definition 1.8.3. 99 is a Friedrichs mollifier if and only if 99$ ^^{Rv), q)(x) > 0, spt 99 (i.e. the support of 99) C ^(0,1), and

f (p(x) dx = \. B{0,1)

If ^ is locally summable on an open set G, we define its cp-mollified function UQ b y u,{x)= I u{i)(pf{^-x)di, X^G, = {X^G\B(X,Q)CG}, B(X,Q)

The following theorems are almost evident and are proved in Chapter 3 (in fact, Theorem B is evident): Theorem A. The space H^ (G) is a Banach space binder the norm lUlli _ _

IMlo

4- V l l ^ !lo

k ,'°'\\v,G \\v,G — \r\\v,G^ ^2 ' l\r a=l

Theorem B. (a) If u^Lp{G) and UQ denotes its (p-mollified then UQ ^ C'^(Gg) and UQ ->u in Lp(D) for each D CCG. (b) If u^ ^KQy

^^^^ ^e ,a(^) = {u ,OC)Q(X) for X^GQ SO that UQ ->u

and UQ ^oc ->u ^oc in Lp(D) for each D C C G. (c) The convergence in {a) and {b) holds for almost all x. The lower semicontinuity theorems in this section depend on some well-known theorems on convex functions which we now define:

1.8. Lower semicontinuity

21

Definition 1.8.4. A set S in a linear space is said to be convex ^ the segment P1P2C. S whenever P i and P2^S. A function 99 is said to be convex on the convex set S ^[(1 - A) f i + U2] < (1 - >^) (fih) + M^2), fi, f2€ 5 ,

0 < A < 1.

Remark. Evidently cp is convex on the convex set 5 99(1) is convex. We now state the characteristic property of convex functions: Lemma 1.8.1. A necessary and sufficient condition that cp he convex on the open convex set S G Rp is that for each ^ in S there exists a linear function ap ^^ -\- b such that (1.8.2)

cp{0=apC^

+ b,

(p(^)^apS^

+ b,

^^S.

If (p is of class C^ on S, this condition is equivalent to (1.8.3)

£(^,l)-95(|)-9,(a-(|2>_fj.)9,^(^)>0,

| , f 6 5.

If Cp is of class C2 on S, this condition is equivalent to for all ^ and all rj. Remark. The function E(^, f) in (1.8.3) is seen to be the WEIERSTRASS £^-f unction. Our first lower-semicontinuity theorem depends on Jensen's inequality which we now state: Lemma 1.8.2 (Jensens's inequality). Suppose cp is convex on Rp, Sis a set, pL is a non-negative bounded measure on S, and the functions ^^ ^ Li(S,pi),p = 1 , . . . , P . Then cp{i\ . . . , F ) < [f^{S)]-^f(p[^Hx),..

(1.8.4)

, i:p =

.,iP{x)]dii,

^ r [i,(S)r^f^Pdpt.

s Remark. I.e. 99 (average) < average of 99. Proof. Choose ap (Lemma 1.8.1) so that ^(C) + cipii^ - C^) < ^(f) for all f and then average over 5. We can now state and prove our first lower semicontinuity theorem; in this general form, it is due to SERRIN [2]: Theorem 1.8.1. Suppose thatf = f[p) is non-negative and convex for all p = {py] and suppose that z and each Zn^ ^\[^) ^'^^ t^^t z^—^z (i.e. tends weakly to) z in Li{D) for each DCGG. Then I{z, G) and I[ZnG) are each finite or + 00 and I{z, G) < lim inf I{zn, G)

Introduction

22

Proof. The first conclusion is obvious. Let Z) C C (^; we may suppose t h a t D C Ga for some a^ 0. Let 99 be a moUifier and, iox 0 / [ ^ , Zn{x), V z{x)] + fp[x, z(x), V z(x)] ' [V Zn{x) - \7z(x)] +{fp[x,Zn(x),Vz{x)] -fj,[x,z{x),\7z{x)]} • [VZn{x) - Vz{x)]. (1.8.9) The weak convergence imphes (see Theorem 3-2.4 (a)) the weak convergence oi VZn to Vz in L\{D) which imphes, in turn, that (1.8.10)

Hm (fp[x, z{x), \/z[x)] • [\/Zn[x)

— \/z{x)] dx = 0.

The uniform convergence of ^^^ t o ^ on S, together with the uniform boundedness of the Li norms of V Zn and V z, imphes that (1.8.11)

hm f {fj)[x,Zn, Vz] —fp[x,z,

Vz]} • [V^^^ — \/z] dx = 0.

Hence, from (1.8.8—1.8.11), we conclude that I{z, D) — s 1, an existence theorem can easily be deduced from the lower-semicontinuity theorems of § 1.8 and Theorems 3-4.4 and 3-4-5- The following simple lemma enables us to prove easily a more general existence theorem (Theorem 1.9-1 below) Lemma 1.9.1. Suppose that fQ{p) is continuous and that (1.9.1)

lim|^i-i/o(i^) = + 0 0 .

39^00

Then, for each M, there exists a function cp such that cplg) '^ 0 for ^ >- 0 and (P{Q) -^0 as Q -^0 such that (1.9.2)

l\p{x)\dx 0 a s ^ - ^ 0 , H an £0 > 0 and sequences {cn} and {pn} such that \en\ - > 0 and r \pn{x) \ dx > £Q. We define ip{G) = inf \p\~^fo{p):

for \p\ > a,

* We often use 1^1 to denote the measure of the set e.

24

Introduction

Clearly ^(cr) -> + oo as c - ^ + c>o. For each n, let gn be the subset of en where \pn{oc)\ > cr^ = eol2\en\. Obviously j

\pn(x)dx

| ^ | • 'y^(o') for | ^ | > (X, we obtain (1.9.4)

/ \pn{x)\dx > £0/2,

f Mpni^)] dx > tp{an)'£ol2.

gn

gn

Since or^ -> + 00 and ^'(o'^) -> + "^^^^ (1.9.4) contradicts (1.9.3)We now state and prove our principal existence theorem. Theorem 1.9.1. We suppose that (i) / and the fpi are continuous in their arguments; (ii) / is convex in p for each {x, z); (iii) there is a function /o satisyfing the conditions of Lemma 1.9.1 such that f{x, z, p) >fo(p) for all (x, z, p); (iv) F * is a (non-empty) family of vector functions which is compact with respect to weak convergence in H\[G)\ (v) F is a family, closed under weak convergence in H\[G), such that each z in F coincides on dG with a fimction z"^ in F"^ (i.e. z ~ z'^^B.\^{G), see §3-2); (vi) ^(-2^0, G) < + ^>o for some ZQ^ F; (vii) G is bounded. Then I{z, G) takes on its minimum for some z in F. Remarks. Since we have not made any assumptions on G other than boundedness and since the admissible functions are not continuous, the most convenient way to specify the boundary values of a function is t o state t h a t z ~ z"^^ •^io(Q ^^^ some given ,^*. Thus the family i^* defines, so to speak, the class of boundary values being allowed. Of course, F * could consist of a single function ^*. Proof of the theorem. Let {zn} be a minimizing sequence; we may assume t h a t I{zny G) 0, ^ > 1, F = (1 + |^|2 + |^|2)i/2

(1.10.7'") The same as (1.10.7") with A = A{x,p), V = {\ + 1:^12)1/2,

k>\,

\B\^ + \B,\^ +

(1.10.8")

B = 0,

mi{R)V^-^\7i\^ 0 , F = (1 + |^p)i/2. In all cases, it is assumed that the A^ and 5 ^ ^ CJ if n = 2 or the ^ ? € ^^""^ and Bi^ CJJ~^ iin > } . And, of course, it is not assumed that Morrey, Multiple I n t e g r a l s

3

34

Introduction

Theorem 1.10.4. With these assumptions, the results (ii), (iii), (v), and (vi) hold if the assumption that z be an extremal he replaced hy that that it he a weak solution in H\{G) of (1.10.13) cind satisfy the additional conditions mentioned in the cases (iii) and (v). LADYZENSKAYA and URAL'TSEVA ([1], [2], [3]) and GILBARG and

others have obtained existence theorems for equations of the type (1.10.13). B u t in the last two or three years, V I S I K , MINTY, B R O W D E R ([3])

and LERAY-LIONS {[6^)'\ have developed an existence theory for nonlinear equations in Banach spaces which covers a wide class of equations including many of higher order and all equations of the type (1.10.14) in which the Af and Bi satisfy (1.10.7") or (1.10.7'")- Their theory yields solutions in H\{G) (or corresponding spaces in the higher order cases) rather than classical solutions and therefore takes the place of the existence theory rather than the differentiability theory. However, combined with our regularity results we now have an existence theorem for any equation of the form (1.10.13) which satisfies the (1.10.7") or (1.10.7"') conditions with N = \. The relevant abstract theorem of Leray and Lions is stated and proved in § 5.12 where this existence theorem is proved. 1.11. Differentiability; reduction to linear equations In this section we wish to give some indication as to how one goes about proving the results on differentiability which we stated in Theorem 1.10.4. We begin b y applying a difference quotient process to the equations (1.10.14) in which we regard the solution as known. We shall assume first that the A^ and Bi satisfy (1.10.7") with ^ > 2; we shall indicate the modifications for the case (1.10.8"). To do this we choose an integer y, 1 < y < i^, let ^y be the unit vector in the x^ direction and define ^l (x) = h~^ \p {x-h

Cy) - C* {x)],

(1.11.1)

zi {x) = h~^ [z^ [x + h ey) - z^ {x)],

0 < \h\< a

where C bas support in a domain D' (Z C Ga. If we replace f by ^h in (1.10.14), make a change of coordinates x in the terms containing C{x — hCy) or V C(^ — ^ ^v)y we obtain the equation (1.11.2) AAl =A'^[x-{-

fh-HCU^^f

+ C'ABi]dx

= 0,

hey, z(x + hcy), \7z(x + he-y)] — Af[x, z(x),

S7z(x)]

and A Bi is given b y a similar formula. Now for each fixed h,0 V^-^ in Lr(D'), Zfi ->z^y in L]c[D') and P;^ - ^ F in L]c{D') so that the right side of (1.11.8) is bounded independently of h. A straightforward argument, given in § 5-9, shows t h a t we may lei h ^0 (through a subsequence) to obtain the following theorem: Theorem 1.11.1. Suppose that z ^ Hl{G) and is a weak solution of (1.10.14), where the Af and Bi satisfy (I.IO.7") or (1.10.7'") with ^ > 2. Then z and U = V^l^^ Hl(D) for each D (Z d G and the vectors py satisfy

I

D

(1 .11. 9) (1 .11. 10) / Vfc--^vp \^dx 3, i.e. i f / a n d / ^ ^ Cl~^, in the variational case, or if ^ ^ CJJ"^ and B ^ C'^~^ in general, it follows t h a t the a'^^ and g^ C^^ and the differentiability of z stated in Theorem 1.10.4 follows from a repeated application of Theorem 5-6.3 as follows: Since z^ Cl{D) for each Z) C C G, it follows t h a t a""^ and g^ Cl(D) for such D so t h a t z^ Cl(D) for such D by Theorem 5-6.3. Then a^"^ and g^ Cl{D) for such D and hence z^ C"*(Z)) for such D. The result follows by induction. The C°^ result follows b u t the analyticity requires a separate proof. The analyticity proof for this case {N = 1) is presented in § 5.8 where references are given. It is to be noticed t h a t the first theorem above which does not hold for systems is Theorem 1.11.2 (see the remark after the proof of Theorem 1.11.2 in § 5.9)- This enables us to show t h a t solutions are Lipschitz on interior domains. But even if this could be proved, the D E GIORGI-NASHMosER theory (presented in §§5.3 and 5-4) has not been extended to systems so we would still be unable to prove the Holder continuity of the py, or even their continuity. In case a part of ^G is smooth and the boundary values are smooth along t h a t part, there is a gap in the differentiability results. By making a correspondingly smooth (CJJ, n^ 1) change of variables and subtracting off a smooth function having the given boundary values, one reduces the equation (1.10.14) (or (1.10.13)) to one of the same type in which one works in a hemisphere and the solution is supposed to vanish (in some Hl^ sense) on the flat part GR {X^ = 0) of the hemisphere Gn (see § 1.2). The transformed equation or system has the same properties as the original except possibly for the bounds. One then carries out the difference quotient procedure of Theorem 1.11.1 in the tangential directions and shows that the py ^ HI (Gr) foTr 0 in (2.4-3), we obtain (2.4-4)

u{xo) =J[^J^

SG

- ^ ^ ) ^S + f^oi^

G

- ^o)f(pc) dx

[v(x) = Ko{x — xo)). Definition 2.4.1. The function KQ, defined in (2.4-2) is called the elementary function for Laplace's equation Au = 0. If f[x) = 0, then (2.4.4) expresses u{xo) in terms of its boundary values and those of its normal derivative. However, from the maximum principle, a harmonic function is completely determined by its boundary values alone. If, in (2.4-4), we have/(A;) = 0 and could take (2.4-5)

^(^) = Ko(x

— xo) + H[x,

XQ) -= GO{XQ, X)

44

Semi-classical results

where H is harmonic in x for each XQ and is so chosen that ?; = 0 on 3G, then (2.4.4) would express U{XQ) in terms of its boundary values. Such a function v, if it exists, is called a GREEN'S function for G with pole at XQ. By the maximum principle, the GREEN'S function is unique if it exists at all. From the discussion so far given, it follows that {a) if a GREEN'S function v exists for a given domain G and point XQ and [h) if u is of class C^ on G and harmonic on G, then (2.4-4) expresses U{XQ) in terms of its boundary values. If a GREEN'S function could be found and if it could be shown to be harmonic in xo for each x m G, then the function u defined by (2.4-4) w i t h / = 0 and u in the boundary integral replaced by a function u * would be harmonic; it would then remain to show t h a t u{x) ->^**(xo) d.s X -^XQ for each x^^ on dG. And, of course, proving the existence of the GREEN'S function requires proving the existence of harmonic functions having given boundary values. This problem is called the Dirichlet problem. Because of all the problems mentioned in the discussion above, the DIRICHLET problem is not usually solved by proving the existence of the GREEN'S function. However, there are two cases where this is possible, namely the case when G is a sphere which, obviously, may be assumed to have center at the origin and the other is a half-space which we may assume to be that where x"^ > 0. We now derive' the GREEN'S function for such a sphere. Let x^ BR — B(0, R) and let x' be the point inverse to x with respect to BR, that is, the point where

'x^^ = R^x^'llx]^. Using the spherical symmetry, it is easy to verify that the ratio \^ — x\l If — x'\ is the same for all ^ on OGR SO that (2.4.6)

H-x'

Thus we note that if we define ( ^ [ l o g | f - x \ - log\i - x'\ - log(|;^|/i^)], (2.4-7)

G(x, I) -

_ (^ _ 2)-i p-i [|| _ ;^|2-v ._ ||: _„

[

V= 2

x'\^-''{\x\IR)^-']

v> 2

then G(x, f) is of the form (2.4-5)- Moreover, by using the formulas for x' and f', we see that (2.4-8)

G{1 x) = G{x, I)

so that G is harmonic in x for each f and vanishes for f interior to BR and x^dBR. Thus the function u defined by (2.4-4) with / = 0 and v{^)

2.4. POISSON'S integral formula; elementary functions; GREEN'S functions

45

= G{x, f) is harmonic on BR, Finally, since the function w = 1 and G satisfy the hypotheses of t h e argument in the paragraph containing equations (2.4-3) and (2.4-4), we conclude that (2.4.9)

| ^ , S ( | )

= 1,

.^B,.

dBR

By computation from (2.4.7), we see t h a t dn{i)

R-^^^G^oc = i ^ - i r ^ - i f ^ p l - x\-'{^°^ - A;«) ;|:^|/i^)2-r||_^'|-v(|a_^^a)

For I on dBn, we may use (2.4-6) to obtain

(2-4-10)

^ ^ = (/;7?)-i if - x\-r{m - \x\^) > 0

and thus obtain Poisson's integral formula (2.4.11)

u{x) = {ryR)-'^ j\^-x\-''{R'^-

\x\^)u*(^)di

for the harmonic function u which takes on given values ^ * on SBR. To see t h a t u{x) ->w*(fo) as x->io if u* is continuous, X^BR, fo€ ^^i2, we note from (2.4.9), (2.4-10), and (2.4-11) that u(x) - M*(|o) = ( r . i ^ ) - i / | | - x\-^{R^

(2.4.12)

- |^|2)[^^*(f) -

u*(^o)]dS.

^^«

To show t h a t this difference -> 0, we break t h e integral on the right in (2.4-12) into integrals / i over SBR H ^ ( I O , Q) and I2 over ^^i? — B{^o, Q) where we m a y choose g so t h a t | ^ * ( | ) — w*(fo)| < e/2 for | ^ ^jBi? Pi B(^O,Q), S being given. The reader m a y complete t h e proof. Thus we have the following theorem: Theorem 2.4.1. There is a unique function u which is continuous on B(0, R), coincides with a given continuous function u* on dB(0, R), and is harmonic in B(0, R). It is given in B(0, R) by Poisson's integral formula (2.4.11). We can now prove the following important reflection principle for harmonic functions: Theorem 2.4.2 (Reflection principle). Suppose u is continuous on G and harmonic on the {possibly unbounded) domain G which lies in a halfspace bounded by the hyper plane 77, suppose Q n U = S, and suppose u{x) = Oforx^ S. Letr = G\J G' \J S^^\ where G' is the domain obtained by reflecting G in U and S^^) is the non-empty set of interior [with respect to IT) points ofS. Define U(x) = u{x)forx^ G and define U{x) = —u{x') for x^r — G, where x' is the point obtained by reflecting x in II. Then U is continuous on P and harmonic in P.

46

Semi-classical results

Proof. That U is continuous on T and harmonic near each XQ in r — S ^ |M,||v2i^||;.2;log(1 +zJ/^), if Qo. Now, if we write ^ = x -\- rrj where r > 0 and rj^I! and use (2./. 12), we conclude that g(x)=l

lA{f])h(x, ri)d^[ri), i h{x, f]) = lim r r-^f(x J^rr])dr.

(27.15)

\T\>Q

For each rj, let us write x = XQ + SYJ where XQ- rj = 0, Then = \im r^l^ii^^, ^-^^s-*i>. ' (p{t;xo,rj) =f(xo + trj)

h{x, + sri,n) (2.7.16)

From Theorem 2.7.1, we conclude that h(xo -\- srj,r))^Lp {xo, rj) with (2.7.17)

||v^||« < Cp • Ml.

yj(s; xo,rj) - /^(^o +

in s for each

srj;r]).

By raising (2.7.17) to the p - t h power and integrating, we obtain (2.7.18)

/ \h{x, fj) \Pdx < Cl{\\f\\l)P

[dx =

dx^ds).

The result (2.7.I4) follows from (2.7.15) and (2.7.18) by first applying the Holder inequality with measure -\A{y)\d^{y)

to (2.7.15) to estimate

|^(:v)|2? and then integrating with respect to x. Before we can prove the theorem corresponding to Theorem 2.7.2 when A satisfies (2.6.10) instead of (2.7.12), we need the following useful lemmas: Theorem 2.7.3. If u ^ Cl,(G), then u(x) = f Ko^oc{x ~ d

(2.7.19)

i)u^oc(S)di

=r-^ I \x - i\-^ {x- - t)u,4i)ds. G

Proof. We m a y assume G C B{x, A), u = 0 in B(x, A) — G. Taking polar coordinates at x, the integral is just /

f

0

Li:

-Ur(r,rj)d2^(ri)]dr.

Lemma 2.7.1. If xi and X2 $ B^, (2.7.20) / | | - : ^ i | l - ' ' * | f - ^ 2 p - ' ' ^ l BR

=

— ClKo{x2 — xi) + e{xi,X2,R), if V > 2 -{27z)^Ko{x2-xi) + 27tlogR'^e(xi,X2,

+ Ci + R), v = 2

Semi-classical results

58

where Co and Ci are constants and e{x\, X2, R) converges uniformly on BA X BAto 0 as R ^ o o . Proof. If r > 2, it follows t h a t as J? -> oo, the integral tends to a function oi \x2 — xi\ only which is homogeneous of degree 2 — r and so must be a negative multiple of KQ. Clearly also 8{Xi, X2, R) =

-f\S-Xi\^-'''\i-~X2\^-''dS By-BR

converges as stated. In the case v — 2,\etx R> A +Q. Then

= {xi + X2)l2 and Q = \x2 — xi\ and suppose

f \^' - Xi\-^'\S

- X2\-^dS < f \S - Xi\-^'\^

B(x,R-\x\)

(2.7.21)

-

X2\-^d^

BR

Bix,R+\x\)

Evaluating the integral on the left, we obtain C2 + 2 7rlog[(i? - \X\)IQ] + cp[{R -

\x\)lQ]

where l\i-Xi\-^'\i-X2\-^ «

C2 =

-/|c-I

ei

1 + ^.1

B(O.l)

Bix,Q)

o o . If we take Ci = C2 + 9^0, then 27r log(1 - l^l/i?) + (p[(R - \X\)IQ] -(po<s < 27rlog(1 + \x\IR) + 99[(i? + 1^1)/^] - 990. Theorem 2.7.4. Theorem 2.7.2 holds, with \\A\\^^ replaced by MQ + Mi in (2.7.14), if A satisfies (2.6.9) and (2.6.10) instead of (2.6.9) and {2.7 A2). Here MQ and M, are the respective maxima of \A{x)\ and \\/A{x)\ for x ondB{0,\). Proof. We suppose first t h a t / ^ C'^(Rv), the general case follows by approximations. Let us define (2.7.22)

h{x)==fR{x-

i)f(i)

di,

R(y) = Co-i I y | i -

59

2-7. The CALDERON-ZYGMUND inequalities

Co = 2 7Z if V = 2 and otherwise is the constant of Lemma 2.7.1. Then, by Theorem 2.6.4, A€ Q [Rv) with (2.7.23)

A aW = Hm fR,4^ - S)f(i)di

from which it follows t h a t h^ ^^(Rp). follows t h a t (2.7.24)

Vh(x) = 0(\x\-^),

==fR{x-i)f,4i)di

S i n c e / has compact support, it

Ah{x) = 0{\x\-''-^)

a t CXD.

Next, we define (2.7.25) k(x)=^-fR{x-^)Ah{i)di=-fR{x-^)Ah(i)di Ry

+ 8R(x}

BR

where SR converges uniformly t o 0 on any compact set on account of (2.7.24). Since t h e support of / is in some BA, we m a y write (since jAf(r))drj = 0) BA k{x) = en{x) - I \ J R ( X BA [BR =

£IR{X)

-

i)R(^

-

v)dS]Af{rj)

drj

J

+ fKo{x BA

—

ri)Af{rj)dri

= £1R + f KG,cc{x~r])f,a{r])d7] BA

£iit(x) = SR{X) — jei{x,

rjy

= SIR(X)

+f(x),

R)Af(rj)dr)

BA

for all r > 2. Since k and / are independent of R, SIR ^ 0. Hence, substituting the right side of (2.7.25) for/(:^) in (2.7.13), we obtain g{x) = lim JHm

J -- R(i - r])Ah(r])dr] d^

j A(x-^)\

Q-*0 \a^0 R^-B(x.Q)

(2.7.26)

=:lim|Hm

[R^-Bix.o)

jAh(7i)\

g^O ya-*0 B^-B(x,a)

J - A(x - S) R{i - rj)di] [R^-B{X,Q)

di].

\

If we now set f = ;t: — CO on the right in (2.7.26), we obtain lim jlim

fS(x — r]; Q)Ah[rj) drX

(2.7.27) ^{y'y Q) = fA(co)R{y

— a))da).

Rv-^Q

From the homogeneity, we conclude t h a t

(2.7.28)

S{y\Q) = \y\^-'S(rj;Ql\y\), ^i-yiQ)

rj = \y\-^y,

\rj\ = 1

= s{y>9)*

Now, if \rj\ = 1 a n d \co\ > 3/2, then \rj — co\ > \co\l}. If \7]\ = 1 and 1/2 < \(o\ < 3/2, then \A{co)\ < 2"Mo if h — co] < 1/2 and R(rj — co)

60

Semi-classical results

< Z"-!- Co if \r) -a>\> Finally, if 0 < ^ < 1 / 2 ,

1/2, MQ being the max. of |Zl(co)| for |co| = 1.

5(7-/; Q) = lA(co)R{f]

- co)dco + JA(a))[R{rj

Bv-BiOA/2)

- co) - R(7])]dco

B{0,li2)-B(0,Q)

(2.7.29) and as Q -^0, S{r}; Q) converges uniformly for ^y on i7 to S{rj). From the analysis above, we see t h a t (2.7.30)

\S(y; Q)\ < C{v) MQ | y | i -

limS(y; Q) = S(y)

so t h a t we may let o* - > 0 and then ^ - > 0 in (2.7.27) to yield (2.7.31)

g{^) = I S(x- rj) Ah(rj) drj.

Now, in (2.7.29), we assume t h a t \r)\ is near 1 and 0 < ^ < 1/2. By defining Sl{V'> Qy'^) ^ J ^ (^) ^{'^ Rv-B(Q,l/2)-B{rj;r)

~~ 0 in B[xi, Ri). Finally (2.8.3) Let (2.8.4)

h{x) < 0 on Se,

h(xi) = 0.

v(x) = u{x) + dh(x),

(3 > 0,

where 6 is small enough so that v(x) < Af on 5^. From (2.8.3), we see t h a t v(x) < M on dB(xi, Ri), v{xi) = M so t h a t v has a maximum at a point xs in B{xi, Ri) while L(v) > 0 there. But this would imply that (since all v^oc{xs) = 0) (2.8.5)

a^'^ixs) v^oc,3{xs) > 0 but v^upixs) r)°'rj^ ^0

for all rj.

Now, we may define new variables y and w by the rotation yv = cl[x^ — xl),

C^ = clr)°',

w(y) = v{x)

62

The spaces H'^ and H^^

where the matrix c is chosen so that c"^ a(xs) c = d(0) is diagonal. Then (2.8.5) is equivalent t o (2.8.6)

2'^^^(0)'^\yy(0) > 0 b u t w^y6{0) C'^ C^ < 0 for all C

y=i

where all the d^y > 0. But the first inequality in (2.8.6) implies that some one w^yy(0) > 0 which contradicts the second. Corollary. / / u and the coefficients satisfy the conditions of Theorem 2.8.1 except that we require c[x) < 0, then u cannot have a positive maximum. If, also, f{x)-^E0, then u has neither a positive maximum nor a negative minimum.

Chapter 3

The spaces H"; and il^o 3.1. Definitions and first theorems In this chapter we collect statements and proofs of the theorems which we need concerning these functions. As we stated in Chapter 1, these a n d similar spaces have been discussed at great length by many authors and have certainly proved their worth in connection with the study of differential equations. Definition 3.1.1. A function u is said to be of class H^ on G iff u is of class Lp on G and there exist functions r^ of class Lp on G, a = oci, . . ., ^v, 0 < 1^1 < w, such t h a t (3.1.1)

fg(x)r4x)dx G

= {-iy-\fD-g{x)u{x)dx,

g^C-(G).

G

The functions ra (or rather the classes of equivalent functions determined by them) are called the distribution derivatives of u and r^ is hereafter denoted by D^" uovu,«; we make the convention that 0°" u = uii\(x\ = 0 . Remarks. I t is clear t h a t if u is of class H^ on G, its distribution derivatives are determined only u p t o additive null functions; moreover \i u* differs from w b y a null function, then ^ * is also of class H'^ on G and has the same distribution derivatives. The definitions above extend t o vector functions and t o complex-valued functions. The LEBESGUE derivatives (SAKS p . 106) of the set functions T^a(A;) dx

are called t h e

e

generalized derivatives of u at points where they exist. Theorem 3.1.1. The space H'^{G) of classes of equivalent vector functions of class H^ on G with norm defined by

(3.1.2)

||«i|» = { /

pl2

] lip

dx\

(u = u^, . . ., u^)

3.1. Definitions and first theorems

63

is a BANACH space. 7 / ^ = 2, the space is a H I L B E R T space if we define

(3.1.3)

(^,^)?= f Z

Z

Co^D'^u^Wvidx.

In (3-1-2) and (3.1-3), Co, denotes the multinomial coefficient \(x\\j(Xi\ . . .ocpl Proof. The only property requiring proof is the completeness. So suppose {un} is a CAUCHY sequence in H^. Then each of the sequences D°^ Ufi, with 0 < 1^1 < w, is a CAUCHY sequence in Lp(G) and so converges in Lp{G) to some vector function roc(=u if | ^ | = 0 ) . But, for each oc with 0 < | ^ | < w, it follows that (3.1.1) holds in the limit so that the r^ are the corresponding distribution derivatives of u. Remark 1. I t will be convenient to call these elements of H^ functions and to say that u is continuous, harmonic, etc., iff some representative of the class forming the element has these properties. Naturally, also, there are many different topologically equivalent norms which could be used. Remark 2. The spaces H^{G) have been defined for all real m (see, for example LIONS [1]) and interesting results in the theory of differential equations have been obtained using these and other spaces such as those introduced b y ARONSZAJN and SMITH [1], [2], CALDERON and others.

A discussion of these matters, though interesting, would lead us rather far afield into functional analysis and is not really relevant to our discussion. Accordingly, we shall not define them here. However, a special class is used in Chapter 8. Theorem 3.1.2. (a) Ifu^ H^(Q and D C G, then U\B ^ H^(D). (b) Suppose u is defined on the whole of G and each point of G is in a domain D such that U\D^ H^{D). Then there are functions r^ defined on G such that {0°^ U\D) (X) = ro:{x) for almost all x on D, and any D C C G. (c) (d) (e) (f)

/ / , in (b), the roc^Lp{G), then u^H^(G). Ifu^ H^(G), then V^u^ H^-^(G), 1 < ^ < m. Ifu^H^{G) and V'^u^Hl{G), then u ^ H^+'{G). Ifu^H^iG) andg^ CZ'^G) (i. e. g^ CrHG)

and all its derivatives of order < m — 1 are LIPSCHITZ, then (3.1-1) holds with roc = D'^ u ^ u^oc. (g) If u^Lp(G), u is absolutely continuous in each variable (on segments in G) for almost all values of the other variables, and if its first partial derivatives {which consequently exist a.e. and are measurable) ^Lp(G), then u^Hl{G) and its partial and generalized derivatives coincide almost everywhere. Proof. Parts (a) — (e) are obvious and part (f) follows b y a straightforward approximation of the function g. To prove (g), we notice t h a t if g^ C^(G), then g[x) • u{x) has the absolute continuity properties of u

64

The spaces H ^ and H^^^

and has compact support in G. From FUBINI'S theorem, it follows that we may take the ra as the partial derivatives dujdx^" in (3.1-1)We recall the definitions of a moUifier 99 and the 99-mollified functions UQ which were given in Chapter 1 (Definition 1.8.3). Theorem 3.1.3. Suppose u^H^(G), cp is a mollifier, and UQ is the (p-mollified function of u. Then UQ ->U in H'^{D) for each D C. d G and (3.1.4)

IP(XQ(X) = D°'Ue(x)

if y)a = u^oc,

1 1 ) of H onto G, u^H^(G), and v{y) ^ u[x(y)]. Then v ^ H^(H) and if all the generalized derivatives 0°^ U[XQ) exist and all those of the xy at yo exist, and if XQ = x(yo), then all the generalized derivatives D'^ v{yo) exist and are given by their usual formulas. Proof. Since x = x{y) is of class C^-^, v $ H^-^[D) on each D CCG. If m > 1, we conclude from t h a t theorem t h a t any derivative D^^ v of order \(x\ < m — 1 is given b y a formula of the form (3.1.5)

D^v{y) =

j:A^^{y)u,p[x{y)]

where 1 < /5 < m — 1 and the AQ are polynomials in the derivatives of the xy of order | ^ | + 1 — | ^ | < m — 1 and hence are LIPSCHITZ. Since the u^^ all ^ H^{G) at least, the theorem will follow from Theorem 3.1.2(g) and the special case m — \. If UQ is a mollified function of u, defined on G^, if Z) C C the counter image of GQ, and v*{y) = Ue[x{y)], then v* is LIPSCHITZ and (3.1.6)

vl4y)

= u,,p[x(y)] ^x^Jy)

(a.e.)

It is clear that we m a y let ^ - > 0 and conclude v^ Hl[D) with v^OL given by the limit of the right side of (3.1.6). If, now the generalized derivatives u ,^(XQ) and x^o,{yo) all exist and e runs through a regular family of sets Morrey, Multiple Integrals

c

66

The spaces H^ and H^Q

at yo and E is t h e image of e, these forming a regular family a t XQ, we conclude:

11 H " V [^ Ay) - ^ ,i8 (^o) ^"fa(yo)] ^y I I

(3.1.7)

I

e

D{u, G) G

unless C = 0. (c) Let {un} be a minimizing sequence and let f/j = ^* — Un. Each Cn^HlQ(G) and D(Cn, G) is uniformly bounded. By POINCARE'S inequality IICwIII,^? and hence l^wfLc? is uniformly bounded. Thus a subsequence, still called Un—yu. Since the norm of an element in a BANACH space {L2{G) in this case) is lower semicontinuous with respect to weak convergence, it follows that the DIRICHLET integral (=^(|i^,a||2.^)^) has this property so that D{u, G) < lim inf D{uny G). But u^H\ [G) and f%-7f = ^* — ^ s o that C€ Hloi^) ^^^ ^ = '^** on dG. Thus u minimizes the DIRICHLET integral. So if f ^ HloiQ, u + ^C = u* on dG for all Z and D{u + ZC,G) =D(u,G) + 2A/C,a^,a^A; + A2 7)(C,G). G

Since A = 0 gives the minimum, (3.3.1) holds and the results follow.

72

The spaces Hi' and H^^

3.4. Boundary values In this section we introduce the class of strongly LIPSCHITZ domains and prove that if G is strongly LIPSCHITZ and u ^ H^ (G) for some arbitrary w > 1, then u can be approximated in H^ (G) by functions each of class ^^{r) for some Fz^G. We then prove a variant CALDERON'S extension theorem for such domains and discuss bundary values on the boundaries of smoother regions. For the case m = \, some of the results are extended to ordinary LIPSCHITZ domains i.e. domains of class C\. An example is given of a LIPSCHITZ domain which is not strongly LIPSCHITZ. Definition 3.4.1. A domain G is said to be strongly LIPSCHITZ iff G is bounded and each point XQOI dG is in a neighborhood 3? which is the image under a rotation and translation of axes of a domain \y[,\ '*'!< 2L 2^ in which XQ corresponds to the origin, "St H dG corresponds to the locus of y^ = f (y^) where / satisfies a LIPSCHITZ condition with constant L, and 31 0 G corresponds to the set of y where \y!^\ 0 such that \\gh — g\\< si} for \h\ :2LR. Clearly, the function Wjn defined by Wjn{y) = Vj{y + n~^ Cp) € H'^i^^) where 9^+ is the part of 9^^ where >»' >/(y;) - n-^ and also '^jn ->^j- in H^{^+) where 3^+ is where y^ >/(yi). There is a sequence Qn ->0 such that Q^ > 1, the right side of (3.4.2) is bounded by a constant times || / | | J ; Up = 1, it follows t h a t the right side ^ Lr for every r with r norm bounded by a constant times ]| / | | ^ . So, let us assume t h a t m — \oc\ — v = —s, 0 < s < r . Let Ux denote the right side of (3.4.2). Then, for almost all x, \Uo:{x)\<M^,J\x-y\-^\f(y)\dx (34.4)

f\x-~y

\U4x)\p<Mf,^

{"^dyY ^ / I ^ — y i"^ \f{y) \^dy

IG

<Mf^i • CP-^{V, S) • 1 G 1(2^-1) (i-^/»')Jj x — y\-' \f{y) \Pdy G

by Lemma 3.4.3- From (3.44) and the preceding discussion, it follows t h a t II Uoc ||p,2) < C'll/JlJ for 0 < |(:x| < w — 1. So, suppose D is a fixed bounded domain and we approximate / strongly in Lp b y functions fn^ C^(G). Then all the Unoc->Ux in Lp(D). Moreover, from the CALDERON-ZYGMUND inequalities, it follows t h a t the D^" Un converge strongly in Lp{R) to some limits f/a when | ^ | = m. The results follow. Theorem 3.4.3. (Variant of CALDERON's Extension Theorem) Suppose G is strongly LIPSCHITZ and G (ZG D. There is a linear hounded extension operator @ which carries H^{G) into H^Q{D) and which has the property that if V = (^u, then v(x) = u(x) for x^G. Proof. We begin as in the proof of Theorem 3.4.1 assuming, as we may, t h a t each 3^^ C Z). For each j for which Cj has support C G, we define (Bj u as the extension of Uj{x) t o D obtained b y setting Uj(x) = Oior x^D — G. For a j for which ^j has support in a boundary neighborhood ?ft, we define ©^ u first for u^C^ [G) and show t h a t it is bounded. Then @ u = 2J®J '^' To define ©j u for such j and u, we define Vj{y) — Uj\_x[y)'] for y $ 9 t + (notation of the proof of Theorem 3.4-1), extend the function / to the whole space t o have LIPSCHITZ constant L and extend Vj{y) = 0 for y* >- fiy'p) where it is not already defined; clearly Vj{y) ^ C^ in t h a t domain U. Now, let y ^ C7 and let f be a unit vector with | f^ | 1. Then bounded subsets of H^'{G) are conditionally (sequentially) compact as subsets of H^^-^G). If Un-7U in H^{G), then Un->u in H^'^{G). The theorem is true for any bounded domain G if we replace the spaces H'^(G) and H^-HG) by H^^iQ ^^^ H^o^{G), respectively. Proof .J Suppose GdC D CdRvy Gc DQQ, (£ is the extension operator of Theorem 3-4.3» and Vn — &Un. Then, from Theorem ')AAO we conclude t h a t (3-47)

II Vn, - Vn WZa' 1. Then, according to Theorem 3.2.4(e), g a subsequence, still called {vn], such t h a t Vn—zvm H^{D). It follows from the formula in Definition 1.8.3 for VnQ and VQ and the weak convergence that D°'VnQ converges uniformly to 0°" VQ on G, for 0 < | ^ | < < w — 1. Then, taking norms on H^~'^{G), we obtain ^ Un — '^ II ^

II '^n — '^TiQ II +



II ^n Q — ^ e II + || ^ e — '^ ||

-\-\\UnQ — Ue\\(u

= V\G, Un = Vn\G)y

'^Q = VQ\G)

using (3.4.7). The result follows easily. If ^ = 1, all the D^^ Vnq are equicontinuous and uniformly bounded so t h a t a subsequence, still called {un} exists so t h a t the Z)* Unq converge uniformly to some functions cpocQ for each of a sequence of ^ - > 0 . Since (3.4-7) still holds the sequences

76

The spaces H^ and H^^

1 and there is a bounded operator B from H^{G) into H^'^dG) such that B u = U\QQ whenever u^ Cf-i(G). If Un-^u in H^{G), then Un -^u in H';-^{dG). If p > 1, the mapping B is compact. Proof. To prove the first statement, we select a finite covering of G by neighborhoods 9^^, each of which is a cell with closure interior to G or is a boundary neighborhood in the sense of the definition in § 1.2, Notations; in the latter case we suppose that 9^^ is mapped onto / \ U 0*1 by a regular map x = Xi(y) of class Cf"^, A being the set 0 < y" < 1, |y^| < 1. We select a partition of unity {fs}, s = 1, . . ., 5, of class Q - ^ ( G ) , each Cs having support in some 9^^. Now, suppose u^H^(G), let Cs ^ = i^s, and let Vs{y) == Us[Xi{y)] for y^Fim case 3^^ is a boundary neighborhood. For those s for which 3^^ C G, we can approximate to Ug,

3.4. Boundary values

77

by {uns}, in which each Uns^ ^TiQ- ^ ^ r the other s, we can approximate to Vs by similar Vns on / i U cfi, using Theorem 3-4.1; clearly each v^s niay be chosen to vanish near G D dFi since v does. The first statement follows b y setting Un ==^Uns where, of course Uns[xi{y)] = Vs(y) when y ^ A and Uns{^) = 0 elsewhere. Next, suppose u^ Cf~^(G) and let % and Vs be defined as above. We define BsU = Us\eQ. Then, clearly Bu = u\Qg. For each s for which ^iG G, BsU = 0. So let s be one of the remaining indices; evidently Vs = Us u, Us being bounded. Since Vg^ Cf~^(A), we have yl) - V^'Vs(0, yl) \^\ ' dyl

l\l\'7^vs(y\ (3.4.10)

< {yn^-^fflz\'^'^s(v'> lim£(^) = : 0 ,

yl)

0Us in H^~^{G) if % C G and Vqs ->^s in H^~Hri) and hence Uqs-^Us in H^-HG). li Un-ru in /f^(G), then Uns->Us and Vns ->Vs- The theorem follows. An example. That not every LIPSCHITZ domain is strongly LIPSCHITZ is seen by the following example of a bi-LiPSCHiTZ map in the case v = 2. Let {r, 6) and {R, 0) be polar coordinates in R^. We define the mapping T:R = r, 0 == 0 — logr, O < 0 < 7 7 : , 0 v, then u is continuous. We now present a "DIRICHLET growth'' theorem guaranteeing continuity (MoRREY [4] and [7]): Theorem 3.5.2. Suppose u ^ Hl[B(xo, R)], 1 < ^ < r, and suppose (?-5-4)

f \\/ u\p dx < LPirldy-p+Pf",

0 dl2 from dB{xQ, R) and then using

80

The spaces H^ and H^^

(3.5-4) and the HOLDER inequality we obtain (3.57)

l\Vu(y)\dy
\ can be proved b y applying the inequality (3-5.9) for ^ = 1 to the function v defined b y v[x) = \u(x)\*. For, if this is done, we obtain (3.5.10)

f\u{x)\rdx=

f \v{x)\^dx

->

''