导语

  

内容提要

  

    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。

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