　　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.

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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 Chiaracteristics
2．1．1 The Symbol of a Differential Expression
2．1．2 Scalar Equations of Second Order
2．1．3 Higher—Order Equations 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
……
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

前言/序言

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　　这次科学出版社购买了版权，一次影印了23本施普林格出版社出版的数学书，就是一件好事，也是值得继续做下去的事情。大体上分一下，这23本书中，包括基础数学书5本，应用数学书6本与计算数学书12本，其中有些书也具有交叉性质。这些书都是很新的，2000年以后出版的占绝大部分，共计16本，其余的也是1990年以后出版的。这些书可以使读者较快地了解数学某方面的前沿，例如基础数学中的数论、代数与拓扑三本，都是由该领域大数学家编著的“数学百科全书”的分册。对从事这方面研究的数学家了解该领域的前沿与全貌很有帮助。按照学科的特点，基础数学类的书以“经典”为主，应用和计算数学类的书以“前沿”为主。这些书的作者多数是国际知名的大数学家，例如《拓扑学》一书的作者诺维科夫是俄罗斯科学院的院士，曾获“菲尔兹奖”和“沃尔夫数学奖”。这些大数学家的著作无疑将会对我国的科研人员起到非常好的指导作用。
　　当然，23本书只能涵盖数学的一部分，所以，这项工作还应该继续做下去。更进一步，有些读者面较广的好书还应该翻译成中文出版，使之有更大的读者群。总之，我对科学出版社影印施普林格出版社的部分数学著作这一举措表示热烈的支持，并盼望这一工作取得更大的成绩。
国外数学名著系列（影印版）75：偏微分方程引论（第二版） [An Introduction to Partial Differential Equations(Second Edition)] 下载 mobi epub pdf txt 电子书格式