LECTURES VALUE PROBLEMS ON ELLIPTIC BOUNDARY SHMUEL AGMON AMS CHELSEA PUBLISHING

Transcription

1 LECTURES ON ELLIPTIC BOUNDARY VALUE PROBLEMS SHMUEL AGMON AMS CHELSEA PUBLISHING

2 Lectures on Elliptic Boundary Value Problems

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4 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 PUBLISHING American Mathematical Society Providence, Rhode Island AMERICAN MATHEMATICAL ΑΓΕΩΜΕ ΤΡΗΤΟΣ ΜΗ Ε Ι Σ Ι Τ Ω SOCIETY FOUNDED 1888

5 2000 Mathematics Subject Classification. Primary 35J40; Secondary 35P10. For additional information and updates on this book, visit 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, Includes bibliographical references. ISBN (alk. paper) 1. Differential equations, Elliptic Congresses. 2. Boundary value problems Congresses. I. Agmon, Shmuel, 1922 II. Title. QA377.S dc Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island USA. Requests can also be made by to reprint-permission@ams.org. c 1965 held by the American Mathematical Society. All rights reserved. Reprinted by the American Mathematical Society, 2010 The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at

6 Preface to the AMS Chelsea Edition This is a new edition of a book published in The book, based on notes of a summer course in 1963, is an introduction to the theory of higher-order elliptic boundary value problems, a theory developed in the 50s of the last century. The book contains a detailed study of basic problems of the theory, such as the problem of existence and regularity of solutions of higher-order elliptic boundary value problems. It also contains a study of spectral properties of operators associated with elliptic boundary value problems. Thus, Weyl s law on the asymptotic distribution of eigenvalues is studied in the book in a great generality. The new edition contains few changes. A number of explanatory remarks were added. Several inaccuracies and various misprints were corrected. Some notations were changed to conform to present day standard notation. A list of references was added. I would like to thank the American Mathematical Society for republishing this book, and in particular thank the Publisher Dr. Sergei Gelfand for his help in the preparation of this edition. Finally, I am deeply indebted to Professor Yehuda Pinchover who has given generously of his time, helping me throughout all stages of preparing this edition for publication. Jerusalem October, 2009 Shmuel Agmon v

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8 Preface This book reproduces with few corrections notes of lectures given at the Summer Institute for Advanced Graduate Students held at the William Rice University from July 1, 1963, to August 24, The Summer Institute was sponsored by the 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. In recent years considerable advances have been made in developing a general theory for such problems. It is the purpose of these lectures to present some selected topics of this theory. We consider elliptic problems only in the framework of the L 2 theory. This approach is particularly simple and elegant. The hard core of the theory is certain fundamental L 2 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 problems is perhaps less standard and differs in some important details from that given in the literature. This approach yields a very general form of the theorem on the asymptotic distribution of eigenvalues of elliptic operators. Only a few references are given throughout the text. The literature on elliptic differential equations is very extensive. A comprehensive bibliography on elliptic and 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 them both. Professor Jones also took upon himself the trouble of inserting explanatory and 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 the publication of these lectures. Jerusalem Shmuel Agmon vii

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10 Contents Preface to the AMS Chelsea Edition Preface v vii Chapter 0. Notations and Conventions 1 Chapter 1. Calculus of L 2 Derivatives Local Properties 3 Chapter 2. Calculus of L 2 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. Gårding 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 125 Part 1. Elementary spectral theory 125 Part 2. Operators of finite double-norm on an abstract Hilbert space 132 Part 3. Hilbert-Schmidt kernels 142 Chapter 13. Spectral Theory of Abstract Operators 147 Chapter 14. Eigenvalue Problems for Elliptic Equations; The Self-Adjoint Case 163 Part 1. Preliminary results on fundamental solutions 163 Part 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

11 x CONTENTS Notation Index 207 Index 209

12 Bibliography [1] R. A. Adams, and J. J. F Fournier, Sobolev Spaces, Second edition, Pure and Applied Mathematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam, [2] S. Agmon, Remarks on self-adjoint and semi-bounded elliptic boundary value problems, in Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960), pp. 1 13, Jerusalem Academic Press, Jerusalem; Pergamon, Oxford, [3] S. Agmon, On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems, Comm. Pure Appl. Math. 15 (1962), [4] S. Agmon, On kernels, eigenvalues, and eigenfunctions of operators related to elliptic problems, Comm. Pure Appl. Math. 18 (1965), [5] N. I. Akhiezer, and I. M. Glazman, Theory of Linear Operators in Hilbert Space, translated from 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, [6] A. P. Calderón, and A. Zygmund, On singular integrals, Amer. J. Math. 78 (1956), [7] N. Dunford, and J. T. Schwartz, Linear Operators, Part II, Spectral Theory, Interscience, New York, [8] L. Gårding, Dirichlet s problem for linear elliptic partial differential equations, Math. Scand. 1 (1953), [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), [10] L. Hörmander, The spectral function of an elliptic operator, Acta Math. 121 (1968), [11] J. Karamata, Neuer Beweis und Verallgemeinerung der Tauberschen Sätze, welch die Laplacesche und Stieltjessche Transformation betreffen, Journal für die reine und angewandte Mathematik 164 (1931), [12] P. D. Lax, and A. N. Milgram, Parabolic equations. in Contributions to the Theory of Partial Differential Equations, pp Annals of Mathematics Studies, no. 33, Princeton University Press, Princeton, N. J., [13] H. Lewy, An example of a smooth linear partial differential equation without solution, Ann. of Math. (2) 66 (1957), [14] J. L. Lions, E quations Différentielles Opérationelles et Problèmes aux Limites, Die Grundlehren der Mathematischen Wissenschaften, Bd. 111 Springer-Verlag, Berlin, [15] N. G. Meyers, and J. Serrin, H = W, Proc. Nat. Acad. Sci. U.S.A. 51 (1964), [16] L. Nirenberg, Remarks on strongly elliptic partial differential equations, Comm. Pure Appl. Math. 8 (1955), [17] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa (3) 13 (1959), [18] K. T. Smith, Inequalities for formally positive integro-differential forms, Bull. Amer. Math. Soc. 67 (1961), [19] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J [20] A. E. Taylor, Introduction to Functional Analysis, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, [21] E. C. Titchmarsh, The Theory of Functions, Second edition, Oxford University Press, New York, [22] B. L. van der Waerden, Modern Algebra. Vol. II, Frederick Ungar Publishing Co., New York,

13 206 BIBLIOGRAPHY [23] H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlarumstrahlung), Math. Ann. 71 (1912), [24] A. C. Zaanen, Linear Analysis, First edition, Noordhoff, Groningen, [25] A. Zygmund, On singular integrals, Rendiconti di Matematica 16 (1957),

14 Notation Index A 39, 125 A 125 A λ 126 A 125 A 133 A k 148 A k 148 A(x, D) 33 A (x, D) 33 ( A α (ξ) 14 α β) 1 B(x, r) 7 B(u) 57 B (u) 57 B[u, v] 68 B [u, v] 72 C m (Ω) 3 C (Ω) 3 C# m 21 C# 21 C0 m (Ω) 57 C m L 2 (Ω) 3 Cb m (Ω) 86 γ s 148 D(A) 125 D α 1 D t δh i 30 u/ n 103 G, G R, G, G 77 G R 86 H m (Ω) 3 Hloc m (Ω) 7 H0 m (Ω) 69 H# m 21 I(λ) 131 J ε 5 j ε (x) 5 k u 108 M(λ) 128 N(A) 73 N(λ) 149 N + (λ) 176 N (λ) 176 N k 66 R(A) 125 R n 1 ρ(a) 125 ρ m (A) 126 R(λ) 33 S n 1 26 supp(u) 1 sp(a) 199 sp (A) 199 Σ r 32 σ(a) 125 tr(ab) 136 û 53 u m,ω 3 u m 3 u m,ω 3 u m 3 u # m 22 (u, v) m 3 u k u 29 W m (Ω) 4 Wloc m Ξ(θ, a) 129 Ω 1 Ω

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16 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 Calderón s extension theorem, 121 Calderón-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 C k,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, 53 Fredholm alternative, 73 fundamental solution, 163, 165 Gårding inequality, 53, 57, 61, 62 GDP (generalized Dirichlet problem), 70 generalized characteristic vector, 128 generalized Dirichlet problem, 70 generalized eigenspace, 128 generalized eigenvector, 128, 200 generalized 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, 113 Hilbert-Schmidt kernel, , 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, 49 linear functional, 71 local existence, 35 local regularity, 39 mixed boundary-value problems, 104 modified resolvent,

17 210 INDEX modified resolvent set, 126 mollifier, 5 multiplicity of eigenvalue, 128 natural boundary conditions, 101, 102 non-self-adjoint problems, 185 norm, 3 normal derivative, 65, 103 null space, 73 weak derivative, 4 weak limit, 30 weak solution, 14, 40 Weyl s law, 177 width, 54 oblique derivative, 96, 195 ordinary differential equations, 183 over-determined elliptic system, Parseval s identity, 53, 57 partition of unity, 7 perturbation, 188 Phragmen-Lindelöf theorem, 197 Poincaré s inequality, 54, 55 principal part, 33, 35 projection, 139 quadratic form, 57 range of operator, 125 regularity theorems, 43 regularity up to the boundary, 75 Rellich theorem, 24, 71, 147 resolvent operator, 126 resolvent set, 125 Riesz-Fréchet representation theorem, 70 Riesz-Schauder theory, 73, 199 right j-smooth, 86 s-smooth differential operator, 39 segment property, 11 self-adjoint, 163, 176 semi-norm, 3 Sobolev constant, 148, 149 Sobolev inequality, 25, 29, 147 Sobolev representation formula, 110, 114 spectral theory, 125 spectrum, 125 strong derivative, 4 strongly coercive form, 100 strongly elliptic operator, 33 subordinate operator, 14 support, 1 symmetric operator, 69 Tauberian theorem, 174 test function, 3 trace of function, 28 trace of operator, 136 uniformly elliptic, 33 Vandermonde determinant, 82 weak compactness, 29 weak convergence, 29

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21 CHEL/369.H