偏微分方程（第二版 英文）
作者：（德）约斯特 著
出版时间：2011年版
内容简介
《偏微分方程（第2版）》是一部讲述偏微分方程理论的入门书籍。全书以椭圆偏微分为核心，系统讲述了相关内容，涉及到不少非线性问题，如，最大值原理方法，抛物方程和变分法。书中讲述了椭圆方程解的估计的主要方法，sobolev空间理论，弱解和强解，schauder估计，moser迭代。展示了椭圆，抛物和双曲解以及布朗运动，半群之间的关系。《偏微分方程（第2版）》可以作为一年级的教程，在这新的版本中增加了反应-扩散方程和系统，新材料有neumann边值问题，poincaré不等式，以及一个新的证明，poisson方程解的hlder规则等。目次：以拉普拉斯方程为原型的二阶椭圆偏微分方程；最大值原理；存在性技巧ⅰ：基于最大值原理的方法；存在性技巧ⅱ：抛物方法.热方程；反应-扩散方程和系统；波方程以及与laplace的关系和热方程；热方程，半群和布朗运动；dirichlet原理，pde解的变分法；sobolev空间和l2规范性理论；强解；schauder规范理论和连续性方法；moser迭代法和de giorgi和nash规范性定理。 读者对象：数学专业高年级的本科生，研究生和相关科研人员。
目录
introduction： what are partial differential equations？
1. the laplace equation as the prototype of an elliptic partial differential equation of second order
1.1 harmonic functions. representation formula for the solution of the dirichlet problem on the ball （existence techniques 0）
1.2 mean value properties of harmonic functions. subharmonic functions. the maximum principle
2. the maximum principle
2.1 the maximum principle of e. hopf
2.2 the maximum principle of alexandrov and bakelman
2.3 maximum principles for nonlinear differential equations
3. existence techniques i： methods based on the maximum principle
3.1 difference methods： discretization of differential equations
3.2 the perron method
3.3 the alternating method of h.a. schwarz
3.4 boundary regularity
4. existence techniques ii： parabolic methods. the heat equation
4.1 the heat equation： definition and maximum principles
4.2 the fundamental solution of the heat equation. the heat equation and the laplace equation
4.3 the initial boundary value problem for the heat equation
4.4 discrete methods
5. reaction-diffusion equations and systems
5.1 reaction-diffusion equations
5.2 reaction-diffusion systems
5.3 the turing mechanism
6. the wave equation and its connections with the laplace and heat equations
6.1 the one-dimensional wave equation
6.2 the mean value method： solving the wave equation through the darboux equation
6.3 the energy inequality and the relation with the heat equation
7. the heat equation， semigroups， and brownian motion
7.1 semigroups
7.2 infinitesimal generators of semigroups
7.3 brownian motion
8. the dirichlet principle. variational methods for the solu- tion of pdes （existence techniques iii）
8.1 dirichlet's principle
8.2 the sobolev space w1，2
8.3 weak solutions of the poisson equation
8.4 quadratic variational problems
8.5 abstract hilbert space formulation of the variational prob- lem. the finite element method
8.6 convex variational problems
9. sobolev spaces and l2 regularity theory
9.1 general sobolev spaces. embedding theorems of sobolev， morrey， and john-nirenberg
9.2 l2-regularity theory： interior regularity of weak solutions of the poisson equation
9.3 boundary regularity and regularity results for solutions of general linear elliptic equations
9.4 extensions of sobolev functions and natural boundary con- ditions
9.5 eigenvalues of elliptic operators
10. strong solutions
10.1 the regularity theory for strong solutions
10.2 a survey of the lp-regularity theory and applications to solutions of semilinear elliptic equations
11. the regularity theory of schauder and the continuity method （existence techniques iv）
11.1 ca-regularity theory for the poisson equation
11.2 the schauder estimates
11.3 existence techniques iv： the continuity method
12. the moser iteration method and the regularity theorem of de giorgi and nash
12.1 the moser-harnack inequality
12.2 properties of solutions of elliptic equations
12.3 regularity of minimizers of variational problems
appendix. banach and hilbert spaces. the lp-spaces
references
index of notation
index
