偏微分方程——方法及应用/天元基金影印系列丛书

    This book has evolved from a two-term graduate course in partial differential equations which I have taught at Northeastern University many times since 1980. The first term is intended to give the student a basic and classical introduction to the subject, including first-order equations by the method of characteristics and linear second-order equations which arise in mathematical physics: the wave equation, Laplace equation, and heat equation.
All of this material is more than adequately covered by many textbooks which are readily available. The second term, however, is intended to introduce the student to a wide variety of more modern methods, especially the use of functional analysis, which has characterized much of the recent development of partial differential equations. This latter material is not as readily available, except in a number of specialized reference books.
This textbook is intended to bridge this gap by providing the student with a basic introduction to the subject and an exposure to some of the more modern methods.

Chapter 1 First |Order Equations
1.1 The Cauchy Problem for Quasilinear Equations
1.2 Weak Solutions for Quasilinear Equations
1.3 General Nonlinear Equations
1.4 Concluding remarks on First-Order Equations
Chapter 2 Principles for Higher |Order Equations
2.1 Trhe Cauchy Problem
2.2 Second-Order Equations in Two Varibales
2.3 Linear Equations and Generalized Solutions
Chapter 3 The Wave Equation
3.1 The One-Dimensional Wave Equation
3.2 Higher Dimensions
3.3 Energy Methods
3.4 Lower-order Terms
Chapter 4 The Laplace Equation
4.1 Introudiction to the Laplace Equation
4.2 Potential Theory and Green's Functions
4.3 General Existence Theory
4.4 Eigenvalues of the Laplacian
Chapter 5 The Heat Equation
5.1 The Heat Equation in a Bounded Domain
5.2 The Pure Initial Value Problem
5.3 Regularity and Similarity
Chapter 6 Linear Functional Analysis
6.1 Function Spaces and Linear Operators
6.2 Application to the Dirichlet Problem
6.3 Duality and Compactness
6.4 Sobolev Imbedding Theorems
6.5 Generalizations and Refinements
Chapter 7 Differential Calculus Methods
……
Chapter 8 Linear Elliptic Theory
Chapter 9 Two Additional Methods
Chapter10 Systems of Conservation Laws
Chapter11 Linear and Nonlinear Diffusion
Chapter12 Linear and Nonlinear Waves
Chapter13 Nonlinear Elliptic Equations