A considerable amount of new material has been added to this edition.There is an extensive discussion of real analytic functions of several variables in Chapter 3. This permits estimation of the size of the domain of existence in the Cauchy-Kowalevski theorem. A first application of these estimates consists in a rigorous proof of a new version of Holmgren's uniqueness theorem for linear analytic partial differential equations (only sketched in the earlier editions).

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)Duhamel's 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 H~ 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

H. Lewy's Example of a Linear Equation without Solutions

 Problems

Bibliography

Glossary

Index