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偏微分方程引论（第二版）

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版次：1
页数：
字数：
印刷时间：2011年06月01日
开本：12k
纸张：胶版纸
包装：精装
是否套装：否
国际标准书号ISBN：9787030313881

作者:Michael Renardy出版社:科学出版社出版时间:2011年06月

内容简介

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.

目　　录

Series PrefacePreface1 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 Equation2 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 Theorem3 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 Uniqueness4 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 Principle5 Distributions6 Function Spaces7 Sobolev Spaces8 Operator Theory9 Linear Elliptic Equations10 Nonlinear Elliptic Equations11 Energy Methods for Evolution Problems12 Semigroup MethodsA ReferencesIndex

<div id="ml" style="word-wrap: break-word; word-break: break-all;">
<pre>
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
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