Table Of Content

Partial Differential Operators of Elliptic Type Recent Titles in This Series 99 Norio Shimakura, Partial differential operators of elliptic type, 1992 98 V. A. Vassiliev, Complements of discriminants of smooth maps: Topology and applications, 1992 97 Itiro Tamura, Topology of foliations: An introduction, 1992 96 A. I. Markushevich, Introduction to the classical theory of Abelian functions, 1992 95 Guangchang Dong, Nonlinear partial differential equations of second order, 1991 94 Yu. S. I1'yashenko, Finiteness theorems for limit cycles, 1991 93 A. T. Fomenko and A. A. Tuzhilin, Elements of the geometry and topology of minimal surfaces in three-dimensional space, 1991 92 E. M. Nikishin and V. N. Sorokin, Rational approximations and orthogonality, 1991 91 Mamoru Mimura and Hirosi Toda, Topology of Lie groups, I and II, 1991 90 S. L. Sobolev, Some applications of functional analysis in mathematical physics, third edition, 1991 89 Valerii V. Kozlov and Dmitrii V. Treshchev, Billiards: A genetic introduction to the dynamics of systems with impacts, 1991 88 A. G. Khovanskii, Fewnomials, 1991 87 Aleksandr Robertovich Kemer, Ideals of identities of associative algebras, 1991 86 V. M. Kadets and M. I. Kadets, Rearrangements of series in Banach spaces, 1991 85 Mikio Ise and Masaru Takeuchi, Lie groups I, II, 1991 84 Dao Trong Thi and A. T. Fomenko, Minimal surfaces, stratified multivarifolds, and the Plateau problem, 1991 83 N. I. Portenko, Generalized diffusion processes, 1990 82 Yasutaka Sibuya, Linear differential equations in the complex domain: Problems of analytic continuation, 1990 81 I. M. Gelfand and S. G. Gindikin, Editors, Mathematical problems of tomography, 1990 80 Junjiro Noguchi and Takushiro Ochiai, Geometric function theory in several complex variables, 1990 79 N. I. Akhiezer, Elements of the theory of elliptic functions, 1990 78 A. V. Skorokhod, Asymptotic methods of the theory of stochastic differential equations, 1989 77 V. M. Filippov, Variational principles for nonpotential operators, 1989 76 Phillip A. Griffiths, Introduction to algebraic curves, 1989 75 B. S. Kashin and A. A. Saakyan, Orthogonal series, 1989 74 V. I. Yudovich, The linearization method in hydrodynamical stability theory, 1989 73 Yu. G. Reshetnyak, Space mappings with bounded distortion, 1989 72 A. V. Pogorelev, Bendings of surfaces and stability of shells, 1988 71 A. S. Markus, Introduction to the spectral theory of polynomial operator pencils, 1988 70 N. I. Akhiezer, Lectures on integral transforms, 1988 69 V. N. Salii, Lattices with unique complements, 1988 68 A. G. Postnikov, Introduction to analytic number theory, 1988 67 A. G. Dragalin, Mathematical intuitionism: Introduction to proof theory, 1988 66 Ye Yan-Qian, Theory of limit cycles, 1986 65 V. M. Zolotarev, One-dimensional stable distributions, 1986 64 M. M. Lavrent'ev, V. G. Romanov, and S. P. Shishat-skiff, I11-posed problems of mathematical physics and analysis, 1986 63 Yu. M. Berezanskii, Selfadjoint operators in spaces of functions of infinitely many variables, 1986 (Continued in the back of this publication) Translations of MATHEMATICAL MONOGRAPHS Volume 99 Partial Differential Operators of Elliptic Type Norio Shimakura Translated by Norio Shimakura American Mathematical Society Providence, Rhode Island f `ff f lla M f) rF,3 f 7t DAENKEI HENBIBUN SAYOSO (Elliptic Partial Differential Operators) by Norio Shimakura Copyright Q 1978 by Norio Shimakura Originally published in Japanese by Kinokuniya Company Ltd., Publishers, Tokyo in 1978 Translated from the Japanese by Norio Shimakura 1991 Mathematics Subject Classification. Primary 3 5-02; Secondary 35A, 35B, 35J. ABSTRACT. This book gives a comprehensive study of the theory of elliptic partial differential operators. Beginning with the definitions of ellipticity for higher order operators, it dis- cusses the Laplacian in Euclidean spaces, elementary solutions, smoothness of solutions, Vishik- Sobolev problems, general boundary value problems, the Schauder theory, and degenerate elliptic operators. The Appendix consists of preliminaries: ordinary differential equations, Sobolev spaces, etc. Taking account of the trend in mathematics that elliptic operators appear in several branches, the book collects for the users' benefit many fundamental techniques for treating elliptic operators. Library of Congress Cataloging-in-Publication Data Shimakura, Norio, 1940- [Daenkei henbibun sayoso. English] Partial differential operators of elliptic type/Norio Shimakura; translated by Norio Shi- makura. p. cm.-(Translations of mathematical monographs; v. 99) Rev. translation of: Daenkei henbibun sayoso. 1978. Includes bibliographical references and index. ISBN 0-8218-4556-X 1. Hypoelliptic operators. I. Title. II. Series. QA329.42.S5513 1992 92-2953 515'.7242-dc20 CIP Copyright @ 1992 by the American Mathematical Society. All rights reserved. Translation authorized by the Kinokuniya Company Ltd. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America Information on Copying and Reprinting can be found at the back of this volume. The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. This publication was typeset using AMS-TEX, the American Mathematical Society's TEX macro system. 10987654321 979695949392 To My Parents and Fusako Contents Preface to the Japanese Edition xi Preface to the English Translation xiii Chapter I. Partial Differential Operators of Elliptic Type 1 § 1. Notation 1 §2. Defintions of elliptic operators 6 §3. Elementary solutions and parametrices 11 §4. Method of Levi 14 §5. Elliptic systems 17 Chapter II. The Laplacian in Euclidean Spaces 21 §1. Laplacian 0 and its elementary solutions 21 §2. Harmonic polynomials and Gegenbauer polynomials 27 §3. The polar coordinate system 30 §4. The Laplace-Beltrami operator on the unit sphere 35 §5. Green functions in half spaces and rectangles 39 Chapter III. Constructions and Estimates of Elementary Solutions 45 §1. Elementary solutions of John 45 §2. Parametrices as pseudodifferential operators 50 §3. Estimates of parametrices (1) 56 §4. Estimates of parametrices (2) 59 §5. Elementary solutions of Hadamard 65 Chapter IV. Smoothness of Solutions 73 § 1. Garding's inequality 73 §2. Interior L2-estimates and hypoellipticity 77 §3. Analytic hypoellipticity 80 §4. Interior Schauder estimates 84 §5. A theorem of de Giorgi, Nash, and Moser 91 Chapter V. Vishik-Sobolev Problems 93 § 1. Vishik-Sobolev problems 93 §2. Smoothness of solutions 98 §3. Friedrichs extensions 102 vii viii CONTENTS §4. Green operators 104 §5. The Dirichlet problem for the Laplacian 107 §6. Asymptotic distribution of eigenvalues 111 Chapter VI. General Boundary Value Problems 119 § 1. Method of continuity 119 §2. L2 a priori estimates 124 §3. Existence and uniqueness of solutions 132 §4. Green functions and Poisson kernels 140 §5. Indices and some comments 148 §6. General boundary value problems of Vishik-Sobolev type 151 Chapter VII. Schauder Estimates and Applications 163 § 1. Poisson kernels 163 §2. Schauder estimates 169 §3. Quasilinear elliptic equations (1) 172 §4. Quasilinear elliptic equations (2) 177 Chapter VIII. Degenerate Elliptic Operators 183 § 1. Degenerate elliptic operators 183 §2. Weighted Sobolev spaces 189 §3. Models of ordinary differential operators (1) 195 §4. Models of ordinary differential operators (2) 201 §5. Dirichlet problem for second order equations 208 §6. General boundary value problems 211 §7. Supplements 214 §8. Examples of elementary solutions 217 Appendix 225 §A. Maximum principles 225 §B. Stokes formula and systems of boundary operators 228 §C. Preliminaries from ordinary differential equations 235 §D. Fredholm operators 243 §E. Sobolev spaces 246 §F. Holder spaces and Schauder spaces 260 §G. Geodesic distance 263 §H. Lemma for approximation of domains 268 §I. A priori estimates of Talenti 270 Bibliography 275 Subject Index 283 Notation 287