LECTURES ON ELLIPTIC BOUNDARY

VALUE PROBLEMS

SHMUEL AGMON

AMS CHELSEA PUBLISHING# $% &'()*+%),)

Lectures on eLLiptic Boundary

VaLue proBLems

Lectures on eLLiptic Boundary

VaLue proBLems

shmueL agmon

Professor Emeritus The Hebrew University of Jerusalem

Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr.

AMS CHELSEA PUBLISHINGAmerican Mathematical Society • Providence, Rhode Island

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http://dx.doi.org/10.1090/chel/369.H

2000 Mathematics Subject Classification. Primary 35J40; Secondary 35P10.

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Library of Congress Cataloging-in-Publication Data

Summer Institute for Advanced Graduate Students (1963 : Rice University)Lectures on elliptic boundary value problems / Shmuel Agmon.

p. cm.Originally published: Lectures on elliptic boundary value problems. Princeton, N.J. : Van

Nostrand, 1965.Includes bibliographical references.ISBN 978-0-8218-4910-1 (alk. paper)1. Differential equations, Elliptic—Congresses. 2. Boundary value problems—Congresses.

I. Agmon, Shmuel, 1922– II. Title.

QA377.S925 1963515′.3533—dc22

2009047651

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Preface to the AMS Chelsea Edition

This is a new edition of a book published in 1965. The book, based on notes ofa summer course in 1963, is an introduction to the theory of higher-order ellipticboundary value problems, a theory developed in the 50s of the last century. Thebook contains a detailed study of basic problems of the theory, such as the problemof existence and regularity of solutions of higher-order elliptic boundary value prob-lems. It also contains a study of spectral properties of operators associated withelliptic boundary value problems. Thus, Weyl’s law on the asymptotic distributionof eigenvalues is studied in the book in a great generality.

The new edition contains few changes. A number of explanatory remarks wereadded. Several inaccuracies and various misprints were corrected. Some notationswere changed to conform to present day standard notation. A list of references wasadded.

I would like to thank the American Mathematical Society for republishing thisbook, and in particular thank the Publisher Dr. Sergei Gelfand for his help in thepreparation of this edition.

Finally, I am deeply indebted to Professor Yehuda Pinchover who has givengenerously of his time, helping me throughout all stages of preparing this editionfor publication.

Jerusalem Shmuel AgmonOctober, 2009

v

Preface

This book reproduces with few corrections notes of lectures given at the Sum-mer Institute for Advanced Graduate Students held at the William Rice Universityfrom July 1, 1963, to August 24, 1963. The Summer Institute was sponsored bythe National Science Foundation and was directed by Professor Jim Douglas, Jr.,of Rice University.

The subject matter of these lectures is elliptic boundary value problems. Inrecent years considerable advances have been made in developing a general theoryfor such problems. It is the purpose of these lectures to present some selected topicsof this theory. We consider elliptic problems only in the framework of the L2 theory.This approach is particularly simple and elegant. The hard core of the theory iscertain fundamental L2 differential inequalities.

The discussion of most topics, with the exception of that of eigenvalue problems,follows more or less along well-known lines. The treatment of eigenvalue problemsis perhaps less standard and differs in some important details from that given inthe literature. This approach yields a very general form of the theorem on theasymptotic distribution of eigenvalues of elliptic operators.

Only a few references are given throughout the text. The literature on ellipticdifferential equations is very extensive. A comprehensive bibliography on ellipticand other differential problems is to be found in [14].

These lectures were prepared for publication by Professor B. Frank Jones, Jr.,with the assistance of Dr. George W. Batten, Jr. I am greatly indebted to themboth. Professor Jones also took upon himself the trouble of inserting explanatoryand complementary material in several places. I am particularly grateful to him.I would also like to thank Professor Jim Douglas for his active interest in thepublication of these lectures.

Jerusalem Shmuel Agmon

vii

Contents

Preface to the AMS Chelsea Edition v

Preface vii

Chapter 0. Notations and Conventions 1

Chapter 1. Calculus of L2 Derivatives—Local Properties 3

Chapter 2. Calculus of L2 Derivatives—Global Properties 11

Chapter 3. Some Inequalities 17

Chapter 4. Elliptic Operators 33

Chapter 5. Local Existence Theory 35

Chapter 6. Local Regularity of Solutions of Elliptic Systems 39

Chapter 7. Garding’s Inequality 53

Chapter 8. Global Existence 65

Chapter 9. Global Regularity of Solutions of Strongly Elliptic Equations 75

Chapter 10. Coerciveness 95

Chapter 11. Coerciveness Results of Aronszajn and Smith 107

Chapter 12. Some Results on Linear Transformations on a Hilbert Space 125Part 1. Elementary spectral theory 125Part 2. Operators of finite double-norm on an abstract Hilbert space 132Part 3. Hilbert-Schmidt kernels 142

Chapter 13. Spectral Theory of Abstract Operators 147

Chapter 14. Eigenvalue Problems for Elliptic Equations; The Self-AdjointCase 163

Part 1. Preliminary results on fundamental solutions 163Part 2. Eigenvalue problems for elliptic equations 168

Chapter 15. Non-Self-Adjoint Eigenvalue Problems 185

Chapter 16. Completeness of the Eigenfunctions 197

Bibliography 205

ix

x CONTENTS

Notation Index 207

Index 209

Bibliography

[1] R. A. Adams, and J. J. F Fournier, Sobolev Spaces, Second edition, Pure and Applied Math-ematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam, 2003.

[2] S. Agmon, Remarks on self-adjoint and semi-bounded elliptic boundary value problems, inProc. Internat. Sympos. Linear Spaces (Jerusalem, 1960), pp. 1–13, Jerusalem Academic Press,Jerusalem; Pergamon, Oxford, 1960.

[3] S. Agmon, On the eigenfunctions and on the eigenvalues of general elliptic boundary valueproblems, Comm. Pure Appl. Math. 15 (1962), 119–147.

[4] S. Agmon, On kernels, eigenvalues, and eigenfunctions of operators related to elliptic problems,Comm. Pure Appl. Math. 18 (1965), 627–663.

[5] N. I. Akhiezer, and I. M. Glazman, Theory of Linear Operators in Hilbert Space, translatedfrom the Russian and with a preface by Merlynd Nestell, reprint of the 1961 and 1963 translations,two volumes bound as one, Dover Publications, Inc., New York, 1993.

[6] A. P. Calderon, and A. Zygmund, On singular integrals, Amer. J. Math. 78 (1956), 289–309.[7] N. Dunford, and J. T. Schwartz, Linear Operators, Part II, Spectral Theory, Interscience, New

York, 1963.[8] L. Garding, Dirichlet’s problem for linear elliptic partial differential equations, Math. Scand.1 (1953), 55–72.

[9] G. H. Hardy, and J. E. Littlewood, Notes on the theory of series (XI). On Tauberian theorems,Proc. London Math. Soc., Second Series, 30 (1930), 23–37.

[10] L. Hormander, The spectral function of an elliptic operator, Acta Math. 121 (1968), 193–218.[11] J. Karamata, Neuer Beweis und Verallgemeinerung der Tauberschen Satze, welch dieLaplacesche und Stieltjessche Transformation betreffen, Journal fur die reine und angewandteMathematik 164 (1931), 27–39.

[12] P. D. Lax, and A. N. Milgram, Parabolic equations. in Contributions to the Theory of PartialDifferential Equations, pp. 167–190. Annals of Mathematics Studies, no. 33, Princeton UniversityPress, Princeton, N. J., 1954.

[13] H. Lewy, An example of a smooth linear partial differential equation without solution, Ann.of Math. (2) 66 (1957), 155–158.

[14] J. L. Lions, Equations Differentielles Operationelles et Problemes aux Limites, Die Grund-lehren der Mathematischen Wissenschaften, Bd. 111 Springer-Verlag, Berlin, 1961.

[15] N. G. Meyers, and J. Serrin, H = W , Proc. Nat. Acad. Sci. U.S.A. 51 (1964), 1055–1056.[16] L. Nirenberg, Remarks on strongly elliptic partial differential equations, Comm. Pure Appl.Math. 8 (1955), 649–675.

[17] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa (3) 13(1959), 115–162.

[18] K. T. Smith, Inequalities for formally positive integro-differential forms, Bull. Amer. Math.Soc. 67 (1961), 368–370.

[19] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math-ematical Series, No. 30, Princeton University Press, Princeton, N.J. 1970.

[20] A. E. Taylor, Introduction to Functional Analysis, John Wiley & Sons, Inc., New York;Chapman & Hall, Ltd., London, 1958.

[21] E. C. Titchmarsh, The Theory of Functions, Second edition, Oxford University Press, NewYork, 1939.

[22] B. L. van der Waerden, Modern Algebra. Vol. II, Frederick Ungar Publishing Co., New York,1953.

205

206 BIBLIOGRAPHY

[23] H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differen-tialgleichungen (mit einer Anwendung auf die Theorie der Hohlarumstrahlung), Math. Ann. 71(1912), 441–479.

[24] A. C. Zaanen, Linear Analysis, First edition, Noordhoff, Groningen, 1953.[25] A. Zygmund, On singular integrals, Rendiconti di Matematica 16 (1957), 479–481.

Notation Index

A∗ 39, 125A∗ 125Aλ 126‖A‖ 125|||A||| 133‖A‖k 148

|A|k 148A(x,D) 33A′(x,D) 33Aα(ξ) 14(αβ

)1

B(x, r) 7B(u) 57B′(u) 57

B[u, v] 68B∗[u, v] 72Cm(Ω) 3C∞(Ω) 3Cm

# 21

C∞# 21

Cm0 (Ω) 57

CmL2 (Ω) 3

Cmb (Ω) 86

γs 148D(A) 125Dα 1Dt 191∆ 34∆2 69 46δih 30∂u/∂n 103G,GR,G′,G′′ 77GR′′′ 86Hm(Ω) 3Hm

loc(Ω) 7Hm

0 (Ω) 69

Hm# 21

(λ) 131Jε 5jε(x) 5k ∗ u 108

M(λ) 128

N(A) 73N(λ) 149N+(λ) 176N−(λ) 176Nk 66R(A) 125

Rn 1

ρ(A) 125ρm(A) 126(λ) 33Sn−1 26supp(u) 1sp(A) 199sp′(A) 199Σr 32σ(A) 125tr(AB) 136u 53‖u‖m,Ω 3‖u‖m 3|u|m,Ω 3|u|m 3

‖u‖#m 22(u, v)m 3uk u 29Wm(Ω) 4Wm

loc(Ω) 7

Ξ(θ, a) 129Ω1 Ω2 1

207

Index

a priori estimates, 48, 93, 104

Abelian theorem, 175

adjoint form, 72

Aronszajn’s theorem, 121

biharmonic operator, 69, 105, 183

boundary value problems, 95

bounded width, 54

Calderon’s extension theorem, 121

Calderon-Zygmund kernel, 107

Cauchy-Riemann operator, 34

characteristic boundary, 95

characteristic direction, 95

characteristic value, 128, 151

classical solution, 66

coercive bilinear form, 100

coercive quadratic form, 112

coerciveness results, 111

compact operator, 134

compact support, 3

completeness, 197

cone property (ordinary and restricted), 11

conormal derivative, 103

convolution, 108

counting function, 149, 176

difference operator, 30

direction of minimal growth, 129, 149

Dirichlet (bilinear) form, 68, 86

Dirichlet boundary conditions, 65

Dirichlet data, 66, 67

Dirichlet integral, 66

Dirichlet principle, 66

Dirichlet problem, 65, 75, 193, 204

discrete spectrum, 176

domain of class Ck, 91

domain of operator, 125

double-norm, 133

eigenfunctions, 197

eigenvalue, 128

elliptic operator, 33

ellipticity constant, 57

formal adjoint, 39, 68

formally self-adjoint, 69

Fourier series, 21

Fourier transform, 53Fredholm alternative, 73

fundamental solution, 163, 165

Garding inequality, 53, 57, 61, 62GDP (generalized Dirichlet problem), 70

generalized characteristic vector, 128

generalized Dirichlet problem, 70

generalized eigenspace, 128

generalized eigenvector, 128, 200generalized solution, 67, 70

global existence, 65

global regularity, 75, 89

Green’s formula, 66, 95, 101

Hardy-Littlewood’s Tauberian theorem, 174

Hermitian, 69

Hilbert space, 3, 125

Hilbert’s Nullstellensatz, 113Hilbert-Schmidt kernel, 142–144, 153

homogeneous function, 107

hypoelliptic, 48

inner product, 3

integral operator, 143

interior regularity, 80

interpolation inequality, 20, 27, 94

interpolation theorem, 20

Karamata’s Tauberian theorem, 174

Laplace equation, 66, 67

Laplacian, 34

Lax-Milgram theorem, 70, 151

Leibnitz rule, 9

Lewy’s example, 49linear functional, 71

local existence, 35

local regularity, 39

mixed boundary-value problems, 104

modified resolvent, 126

209

210 INDEX

modified resolvent set, 126mollifier, 5

multiplicity of eigenvalue, 128

natural boundary conditions, 101, 102non-self-adjoint problems, 185norm, 3

normal derivative, 65, 103null space, 73

oblique derivative, 96, 195ordinary differential equations, 183over-determined elliptic system, 46–48

Parseval’s identity, 53, 57

partition of unity, 7perturbation, 188Phragmen-Lindelof theorem, 197Poincare’s inequality, 54, 55

principal part, 33, 35projection, 139

quadratic form, 57

range of operator, 125regularity theorems, 43

regularity up to the boundary, 75Rellich theorem, 24, 71, 147resolvent operator, 126resolvent set, 125

Riesz-Frechet representation theorem, 70Riesz-Schauder theory, 73, 199right j-smooth, 86

s-smooth differential operator, 39segment property, 11

self-adjoint, 163, 176semi-norm, 3Sobolev constant, 148, 149

Sobolev inequality, 25, 29, 147Sobolev representation formula, 110, 114spectral theory, 125spectrum, 125

strong derivative, 4strongly coercive form, 100strongly elliptic operator, 33

subordinate operator, 14support, 1symmetric operator, 69

Tauberian theorem, 174test function, 3

trace of function, 28trace of operator, 136

uniformly elliptic, 33

Vandermonde determinant, 82

weak compactness, 29weak convergence, 29

weak derivative, 4weak limit, 30weak solution, 14, 40Weyl’s law, 177width, 54

CHEL/369.H