This book, which originally appeared in Japanese, was written for use in an undergraduate course or first year graduate course in partial differential equations and is likely to be of interest to researchers as well. This book presents a comprehensive study of the theory of elliptic partial differential operators. Beginning with the definitions of ellipticity for higher order operators, Shimakura discusses the Laplacian in Euclidean spaces, elementary solutions, smoothness of solutions, Vishik-Sobolev problems, the Schauder theory, and degenerate elliptic operators. The appendix covers such preliminaries as ordinary differential equations, Sobolev spaces, and maximum principles. Because elliptic operators arise in many areas, readers will appreciate this book for the way it brings together a variety of techniques that have arisen in different branches of mathematics.
Readership: First year graduate students specializing in partial differential equations, researchers in other fields of mathematics.
Author(s): Nirio Shimakura
Series: Translations of Mathematical Monographs, Vol. 99
Publisher: American Mathematical Society
Year: 1992
Language: English
Pages: C+xiv+288+B
Cover
S Title
Titles in This Series
Partial Differential Operators of Elliptic Type
Copyright
1992 by the American Mathematical Society
ISBN 0-8218-4556-X
QA329.42.S5513 1992 515'.7242-dc20
LCCN 92-2953
Dedication
Contents
Preface to the Japanese Edition
Preface to the English Translation
CHAPTER I Partial Differential Operators of Elliptic Type
§1. Notation
§2. Definitions of elliptic operators
§3. Elementary solutions and parametrices
§4. Method of Levi
§5. Elliptic systems
CHAPTER II The Laplacian in Euclidean Spaces
§1. The Laplacian O and its elementary solutions
§2. Harmonic polynomials and Gegenbauer polynomials
§3. The polar coordinate system
§4. The Laplace-Beltrami operator on the unit sphere
§5. Green functions in half spaces and rectangles
CHAPTER III Constructions and Estimates of Elementary Solutions
§1. Elementary solutions of John
§2. Parametrices as pseudodifferential operators
§3. Estimates of parametrices (1)
§4. Estimates of paramatrices (2)
§5. Elementary solutions of Hadamard
CHAPTER IV Smoothness of Solutions
§1. Garding's inequality
§2. Interior L2-estimates and hypoellipticity
§3. Analytic hypoellipticity
§4. Interior Schauder estimates
§5. A theorem of de Giorgi, Nash, and Moser
CHAPTER V Vishik-Sobolev Problems
§1. Vishik-Sobolev problems
§2. Smoothness of solutions
§3. Friedrichs extensions
§4. Green operators
§5. The Dirichlet problem for the Laplacian
§6. Asymptotic distribution of eigenvalues
CHAPTER VI General Boundary Value Problems
§1. Method of continuity
§2. L2 a priori estimates
§3. Existence and uniqueness of solutions
§4. Green functions and Poisson kernels
§5. Indices and some comments
§6. General boundary value problems of Vishik-Sobolev type
CHAPTER VII Schauder Estimates and Applications
§1. Poisson kernels
§2. Schauder estimates
§3. Quasilinear elliptic equations (1)
§4. Quasilinear elliptic equations (2)
CHAPTER VIII Degenerate Elliptic Operators
§1. Degenerate elliptic operators
§2. Weighted Sobolev spaces
§3. Models of ordinary differential operators (1)
§4. Models of ordinary differential operators (2)
§5. Dirichlet problem for second order equations
§6. General boundary value problems
§7. Supplements
§8. Examples of elementary solutions
Appendix
§A. Maximum principles
§B. Stokes formula and systems of boundary operators
§C. Preliminaries from ordinary differential equations
§D. Fredholm operators
§E. Sobolev spaces
§F. Holder spaces and Schauder spaces
§G. Geodesic distance
§H. Lemma for approximation of domains
§1. A priori estimates of Talenti
Bibliography
Subject Index
Notation. Sets, spaces of functions, and spaces of distributions
Titles in This Series
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