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| 內容簡介： |
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本书是一部非常优秀的介绍偏微分方程的入门书籍，可以作为研究生阶段学习的基石。本书详尽地介绍了偏微分方程理论的重要方面，并从数学分析的角度做了进一步的探讨。本书是第4版，增加了全新的一章讲述无解线性方程的Lewy例子。
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| 關於作者： |
Fritz JohnF.
约翰，美国，著名数学家，曾获得伯克霍夫奖（Birkhoff Prize）和斯蒂尔奖（Steele
Prize）等多个奖项。
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| 目錄：
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Chapter 1 The Single First-Order Equation
?1.Introduction
?2.Examples
?3.Analytic Solution and Approximation Methods in a Simple Example
?Problems
?4.Quasi-linear Equations
?5.The Cauchy Problem for the Quasi-linear Equation
?6.Examples
?Problems
?7.The General First-Order Equation for a Function of Two Variables
?8.The Cauchy Problem
?9.Solutions Generated as Envelopes
?Problems
Chapter 2 Second-Order Equations: Hyperbolic Equations for Functions of Two Independent Variables
?1.Characteristics for Linear and Quasi-linear Second-order Equations
?2.Propagation of Singularities
?3.The Linear Second-Order Equation
?Problems
?4.The One-Dimensional Wave Equation
?Problems
?5.Systems of First-Order Equations
?6.A Quasi-linear System and Simple Waves
?Problem
Chapter 3 Characteristic Manifolds and the Cauchy Problem
?1.Notation of Laurent Schwartz
?Problems
?2.The Cauchy Problem
?Problems
?3.Real Analytic Functions and the Cauchy-Kowalevski Theorem
?(a) Multiple infinite series
?Problems
?(b) Real analytic functions
?Problems
?(c) Analytic and real analytic functions
?Problems
?(d) The proof of the Cauchy-Kowalevski theorem
?Problems
?4.The Lagrange-Green Identity
?5. The Uniqueness Theorem of Holmgren
?Problems
?6.Distribution Solutions
?Problems
Chapter 4 The Laplace Equation
?1.Green''s Identity, Fundamental Solutions, and Poisson''s Equation
?Problems
?2.The Maximum Principle
?Problems
?3.The Dirichlet Problem, Green''s Function, and Poisson''s Formula
?Problems
?4.Proof of Existence of Solutions for the Dirichlet Problem Using Subharmonic Functions ("Perron''s Method")
?Problems
?5.Solution of the Dirichlet Problem by Hilbert-Space Methods
?Problems
Chapter 5 Hyperbolic Equations in Higher Dimensions
?1.The Wave Equation in n-Dimensional Space
?(a) The method of spherical means
?Problems
?(b) Hadamard''s method of descent
?Problems
?(c) Duhamers principle and the general Cauchy problem
?Problem
?(d) Initial-boundary-value problems ("Mixed" problems)
?Problems
?2.Higher-Order Hyperbolic Equations with Constant Coefficients
?(a) Standard form of the initial-value problem
?Problem
?(b) Solution by Fourier transformation
?Problems
?(c) Solution of a mixed problem by Fourier transformation
?(d) The method of plane waves
?Problems
?3.Symmetric Hyperbolic Systems
?(a) The basic energy inequality
?Problems
?(b) Existence of solutions by the method of finite differences
?Problems
?(c) Existence of solutions by the method of approximation by analytic functions (Method of Schauder)
Chapter 6 Higher-Order Elliptic Equations with Constant Coefficients
?1.The Fundamental Solution for Odd n
?Problems
?2. The Dirichlet Problem
?Problems
?3.More on the Hilbert Space Hg and the Assumption of Boundary Values in the Dirichlet Problem
?Problems
Chapter 7 Parabolic Equations
?1.The Heat Equation
?(a) The initial-value problem
?Problems
?(b) Maximum principle, uniqueness, and regularity
?Problem
?(c) A mixed problem
?Problems
?(d) Non-negative solutions
?Problems
?2.The Initial-Value Problem for General Second-Order Linear Parabolic Equations
?(a) The method of finite differences and the maximum principle
?(b) Existence of solutions of the initial-value problem
?Problems
Chapter 8 H.Lewy''s Example of a Linear Equation without Solutions
?Problems
Bibliography
Glossary
Index
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