内容简介
本书专为希望了解现代偏微分方程理论基础的读者而写，这些理论对应用很重要，但不必使用大多数教科书中所需的大量分析工具。读者仅需多元微积分和基本度量空间的知识背景，而后者与本书的内展密切相关。 本书的主要目标是不让读者在数学上不知所措，同时用研究人员的思考方式来介绍偏微分方程理论。一个具体的例子是，书中较早介绍了分布理论和弱解的概念，因为虽然这些概念需要学生花一些时间适应，但它们本质上很简单，另一方面，它们都在该领域发挥着核心作用。然后，本书介绍了在后来发展中重要的Hilbert空间，在无须了解测度论的前提下，基本提供了人们想要的所有特征。 除核心内容外，本书还为想要学内容的读者提供了额外材料，所配的大量巩固对内容的理解。本书适合工程或科学领域的高年级本科生或低年级研究生阅读参考。
目录
PrefaceChapter 1. Introduction 1. Preliminaries and notation 2. Partial differential equations itional material: More on normed vector spaces and metric spaces ProblemsChapter 2. Where do PDE come from 1. An example: Maxwell's equations 2. Euler-Lagrange equations ProblemsChapter 3. First order scalar semilinear equations itional material: More on ODE and the inverse function theorem ProblemsChapter 4. First order scalar quasilinear equations ProblemsChapter 5. Distributions and weak derivatives itional material: The space I ProblemsChapter 6. Second order constant coefficient PDE: Types and d'Alembert's solution of the wave equation S1. Classification of second order PDE S2. Solving second order hyperbolic PDE on R2 ProblemsChapter 7. Properties of solutions of second order PDE: Propagation, energy estimates and the maximum principle 1. Properties of solutions of the wave equation: Propagation phenomena 2. Energy conservation for the wave equation 3. The maximum principle for Laplace's equation and the heat equation 4. Energy for Laplace's equation and the heat equation ProblemsChapter 8. The Fourier transform:Basic properties,the inversion formula and the heat equation 1. The definition and the basics 2. The inversion formula 3. The heat equation and convolutions 4. Systems of PDE 5.Integral transforms itional material: A heat kernel proof of the Fourier inversion formula ProblemsChapter 9. The Fourier transform:Tempered distributions,the wave equation and Laplace's equation 1. Tempered distributions 2. The Fourier transform of tempered distributions 3. The wave equation and the Fourier transform 4. More on tempered distributions ProblemsChapter 10. PDE and boundaries 1. The wave equation on a half space 2. The heat equation on a half space 3. More complex geometries 4. Boundaries and properties of solutions 5. PDE on intervals and cubes ProblemsChapter 11. Duhamel's principle 1. The inhomogeneous heat equation 2. The inhomogeneous wave equation ProblemsChapter 12. Separation of variables 1. The general method 2. Interval geometries 3. Circular geometries ProblemsChapter 13. Inner product spaces, sy mmetric operators, orthogonality 1. The basics of inner product spaces 2. Symmetric operators 3. Completeness of orthogonal sets and of the inner product space ProblemsChapter 14. Convergence of the Fourier series and the Poisson formula on disks 1. Notions of convergence 2. Uniform convergence of the Fourier transform 3. What does the Fourier series converge to 4. The Dirichlet problem on the disk itional material: The Dirichlet kernel ProblemsChapter 15. Bessel functions 1. The definition of Bessel functions 2. The zeros of Bessel functions 3. Higher dimensions ProblemsChapter 16. The method of stationary phase ProblemsChapter 17. Solvability via duality 1. The general method 2. An example: Laplace's equation 3. Inner product spaces and solvability ProblemsChapter 18. Variational problems 1. The finite dimensional problem 2. The infinite dimensional minimization ProblemsBibliographyIndex