二階橢圓偏微分方程

內(nèi)容概要

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.

作者簡介

作者:( )David Gilbarg，( )Neil S.Trudinger[著]

書籍目錄

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 SolutionsPart 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 EquationsBibliographyEpilogueSubject IndexNotation Index

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