“国外数学名著系列”丛书是科学出版社组织学术界多位知名院士、专家精心筛选出来的一批基础理论类数学著作，读者对象面向数学系高年级本科生、研究生及从事数学专业理论研究的科研工作者。

本书为该丛书之《偏微分方程引论(第2版)》分册。全书共分十二章，主要内容包括：偏微分方程的特征线、守恒定律、最大值原理、广义函数、函数空间、算子理论、线性椭圆方程、非线性椭圆方程等。

Partial differential equations are fundamental to the modeling of natural phenomena, arising in every field of science. Consequently,the desire to understand the solutions of these equations has always had a prominent place in the efforts of mathematicians; it has inspired such diverse fields as complex function theory, functional analysis and algebraic topology. Like algebra, topology, and rational mechanics,partial differential equations are a core area of mathematics. This book aims to provide the background necessary to initiate work on a Ph.D. thesis in PDEs for beginning graduate students. Prerequisites include a truly advanced calculus course and basic complex variables.Lebesgue integration is needed only in Chapter 10, and the necessary tools from functional analysis are developed within the course. The book can be used to teach a variety of different courses. This new edition features new problems throughout and the problems have been rearranged in each section from simplest to most difficult. New examples have also been added. The material on Sobolev spaces has been rearranged and expanded. A new section on nonlinear variational problems with "Young-measure" solutions appears. The reference section has also been expanded.

Series Preface

Preface

1 Introduction

 1.1 Basic Mathematical Questions

1.1.1 Existence

1.1.2 Multiplicity

1.1.3 Stability

1.1.4 Linear Systems of ODEs and Asymptotic Stability

1.1.5 Well-Posed Problems

1.1.6 Representations

1.1.7 Estimation

1.1.8 Smoothness

 1.2 Elementary Partial Differential Equations

1.2.1 Laplace's Equation

1.2.2 The Heat Equation

1.2.3 The Wave Equation

2 Characteristics

 2.1 Classification and Characteristics

2.1.1 The Symbol of a Differential Expression

2.1.2 Scalar Equations of Second Order

2.1.3 Higher-Order Equatioas and Systems

2.1.4 Nonlinear Equations

 2.2 The Cauchy-Kovalevskaya Theorem

2.2.1 Real Analytic Functions

2.2.2 Majorization

2.2.3 Statement and Proof of the Theorem

2.2.4 Reduction of General Systems

2.2.5 A PDE without Solutions

 2.3 Holmgren's Uniqueness Theorem

2.3.1 An Outline of the Main Idea

2.3.2 Statement and Proof of the Theorem

2.3.3 The WeierstraB Approximation Theorem

3 Conservation Laws and Shocks

 3.1 Systems in One Space Dimension

 3.2 Basic Definitions and Hypotheses

 3.3 Blowup of Smooth Solutions

3.3.1 Single Conservation Laws

3.3.2 The p System

 3.4 Weak Solutions

3.4.1 The Rankine-Hugoniot Condition

3.4.2 Multiplicity

3.4.3 The Lax Shock Condition

 3.5 Riemann Problems

3.5.1 Single Equations

3.5.2 Systems

 3.6 Other Selection Criteria

3.6.1 The Entropy Condition

3.6.2 Viscosity Solutions

3.6.3 Uniqueness

4 Maximum Principles

 4.1 Maximum Principles of Elliptic' Problems

4.1.1 The Weak Maximum Principle

4.1.2 The Strong Maximum Principle

4.1.3 A Priori Bounds

 4.2 An Existence Proof for the Dirichlet Problem

4.2.1 The Dirichlet Problem on a Ball

4.2.2 Subharmonic Functions

4.2.3 The Arzela-Ascoli Theorem

4.2.4 Proof of Theorem 4.13

 4.3 Radial Symmetry

4.3.1 Two Auxiliary Lemmas

4.3.2 Proof of the Theorem

 4.4 Maximum Principles for Parabolic Equations

4.4.1 The Weak Maximum Principle

4.4.2 The Strong Maximum Principle

5 Distributions

6 Function Spaces

7 Sobolev Spaces

8 Operator Theory

9 Linear Elliptic Equations

10 Nonlinear Elliptic Equations

11 Energy Methods for Evolution Problems

12 Semigroup Methods

A References

Index