Partial Differential Operators of Elliptic Type

308 Pages·1992·4.168 MB·English

by  Nirio Shimakura

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 differen-
tial 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 ellip-
tic operators. The Appendix consists of preliminaries: ordinary differential equations, Sobolev
spaces, etc. Taking account of the trend in mathematics that elliptic operators appear in sev-
eral 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