內容簡介

This revision of the 1983 second edition of”Elliptic Partial Differential Equations of Second Order” corresponds to the Russian edition, published in 1989, in which we essentially updated the previous version to 1984. The additional text relates to the boundary H61der derivative estimates of Nikolai Krylov, which provided a fundamental component of the further development of the classical theory of elliptic (and parabolic), fully nonlinear equations in higher dimensions. In our presentation we adapted a simplification of Krylov’’s approach due to Luis Caffarelli.
 

目錄

Chapter 1. Introduction

Part Ⅰ Linear Equations
　Chapter 2 Laplace』s Equation
　　2.1 The Mean Value Inequalities
　　2.2 Maximum and Minimum Principle
　　2.3 The Harnack Inequality
　　2.4 Green』s Representation
　　2.5 The Poisson Integral
　　2.6 Convergence Theorems
　　2.7 Interior Estimates of Derivatives
　　2.8 The Dirichlet Problem； the Method of Subharmonic Functions
　　2.9 Capacity
　　Problems
　Chapter 3 The Classical Maximum Principle
　　3.1 The Weak Maximum Principle
　　3.2 The Strong Maximum Principle
　　3.3 Apriori Bounds
　　3.4 Gradient Estimates for Poisson』s Equation
　　3.5 A Harnack Inequality
　　3.6 Operators in Divergence Form
Notes
　　Problems
Chapter 4 Poisson’’s Equation and the Newtonian Potential
4.1 Holder Continuity
4.2 The Dirichlet Problem for Poisson’’s Equation
4.3 Holder Estimates for the Second Derivatives
4.4 Eximates at the Boundary
4.5 Holder Estimates for the First Derivatives
Notes
　　Problems
Chapter 5 Banach and Hilbert Spaces
5.1 The Contraction Mapping Principle
5.2 The Method of Continity
5.3 The Fredholm Alternative
5.4 Dual Spaces and Adjoints
5.5 Hilbert Spaces
5.6 The Projection Theorem
5.7 The Riesz Represenation Theorem
5.8 The Lax-Milgram Theorem
5.9 The Fredholm Alternative in Hilbert Spaces
5.10 Weak Compactness
Notes
　　Problems
Chapter 6 Calssical Solutions; the Schauder Approach
Chapter 7 Sobolev Spaces
Chapter 8 Generalized Solutiona and regularity
Chapter 9 Strong Solutions

Part Ⅱ Quasilinear Equations
Chapter 10 Maximum and Comparison Principles
Chapter 11 Topological Fixed Point Theorems and Their Application
Chapter 12 Equation in Two Varables
Chapter 13 Holder Extimates for the Cradient
Chapter 14 Boundary Gradient Estimates
Chapter 15 Global and Interior Gradient Bounds
Chapter 16 Equations of Mean Curvature Type
Chapter 17 Fully Nonlinear Equations
Bibliography
Epilogue
Subject Index
Notation Index
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