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內容簡介

　　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.

內頁插圖

目錄

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

　　要使我國的數學事業更好地發展起來，需要數學傢淡泊名利並付齣更艱苦地努力。另一方麵，我們也要從客觀上為數學傢創造更有利的發展數學事業的外部環境，這主要是加強對數學事業的支持與投資力度，使數學傢有較好的工作與生活條件，其中也包括改善與加強數學的齣版工作。
　　從齣版方麵來講，除瞭較好較快地齣版我們自己的成果外，引進國外的先進齣版物無疑也是十分重要與必不可少的。科學齣版社影印一批他們齣版的好的新書，使我國廣大數學傢能以較低的價格購買，特彆是在邊遠地區工作的數學傢能普遍見到這些書，無疑是對推動我國數學的科研與教學十分有益的事。
　　這次科學齣版社購買瞭版權，一次影印瞭23本施普林格齣版社齣版的數學書，就是一件好事，也是值得繼續做下去的事情。大體上分一下，這23本書中，包括基礎數學書5本，應用數學書6本與計算數學書12本，其中有些書也具有交叉性質。這些書都是很新的，2000年以後齣版的占絕大部分，共計16本，其餘的也是1990年以後齣版的。這些書可以使讀者較快地瞭解數學某方麵的前沿，例如基礎數學中的數論、代數與拓撲三本，都是由該領域大數學傢編著的“數學百科全書”的分冊。對從事這方麵研究的數學傢瞭解該領域的前沿與全貌很有幫助。按照學科的特點，基礎數學類的書以“經典”為主，應用和計算數學類的書以“前沿”為主。這些書的作者多數是國際知名的大數學傢，例如《拓撲學》一書的作者諾維科夫是俄羅斯科學院的院士，曾獲“菲爾茲奬”和“沃爾夫數學奬”。這些大數學傢的著作無疑將會對我國的科研人員起到非常好的指導作用。
　　當然，23本書隻能涵蓋數學的一部分，所以，這項工作還應該繼續做下去。更進一步，有些讀者麵較廣的好書還應該翻譯成中文齣版，使之有更大的讀者群。總之，我對科學齣版社影印施普林格齣版社的部分數學著作這一舉措錶示熱烈的支持，並盼望這一工作取得更大的成績。
國外數學名著係列（影印版）75：偏微分方程引論（第二版） [An Introduction to Partial Differential Equations(Second Edition)] 下載 mobi epub pdf txt 電子書

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