Partial Differential Equations III
Partial Differential Equations III
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DOI：
10.1007/978-1-4419-7049-7
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发表时间：
1996
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影响因子：
--
通讯作者：
M. Taylor
中科院分区：
文献类型：
--
作者：
M. Taylor
偏微分方程是一个多方面的课题。它的创建是为了描述物体的力学行为，如振动的弦和吹风，它已经发展成为一个与许多数学分支相互作用的材料体，如微分几何、复分析和谐波分析，以及在数学物理问题的描述和阐明中无处不在的因素。这项工作的目的是提供PDE的一些主要方面的研究课程。它是写给具有美国大学基本入门级研究生数学课程背景的读者的：初等实分析和复分析，微分几何和测量理论。第一章提供了关于常微分方程（ODE）理论的背景资料。这包括非常基本的材料——关于ODE解的存在性和唯一性、常系数方程的显式解以及与线性代数的关系——以及更复杂的结果——关于向量场产生的流、与微分几何的联系、微分形式的微积分、力学中的固定作用原理以及它们与哈密顿系统的关系。我们讨论相对论运动方程和经典牛顿力学方程。拓扑结果也有应用，如度理论、browwer不动点定理和jordan - browwer分离定理。在本章中，我们还通过Hamilton-Jacobi理论处理标量一阶偏微分方程。第2-6章是对基本线性偏微分方程的概述。第二章以类似于第一章力学中ODE的推导的变分原理的连续介质力学的一些方程的推导开始。我们得到了弦和膜振动的方程；这些方程不一定是线性的，因此，当非线性偏微分方程被采用时，它们也会提供问题的来源。第二章的进一步内容围绕拉普拉斯算子展开，它在欧几里德空间rn上是
Partial differential equations are a many-faceted subject. Created to describe the mechanical behavior of objects such as vibrating strings and blowing winds, it has developed into a body of material that interacts with many branches of mathematics, such as differential geometry, complex analysis, and harmonic analysis, as well as a ubiquitous factor in the description and elucidation of problems in mathematical physics.This work is intended to provide a course of study of some of the major aspects of PDE. It is addressed to readers with a background in the basic introductory graduate mathematics courses in American universities: elementary real and complex analysis, differential geometry, and measure theory. Chapter 1 provides background material on the theory of ordinary differential equations (ODE). This includes both very basic material–on topics such as the existence and uniqueness of solutions to ODE and explicit solutions to equations with constant coefficients and relations to linear algebra–and more sophisticated results–on flows generated by vector fields, connections with differential geometry, the calculus of differential forms, stationary action principles in mechanics, and their relation to Hamiltonian systems. We discuss equations of relativistic motion as well as equations of classical Newtonian mechanics. There are also applications to topological results, such as degree theory, the Brouwer fixed-point theorem, and the Jordan-Brouwer separation theorem. In this chapter, we also treat scalar first-order PDE, via the Hamilton–Jacobi theory. Chapters 2–6 constitute a survey of basic linear PDE. Chapter 2 begins with the derivation of some equations of continuum mechanics in a fashion similar to the derivation of ODE in mechanics in Chap. 1, via variational principles. We obtain equations for vibrating strings and membranes; these equations are not necessarily linear, and hence they will also provide sources of problems later, when nonlinear PDE is taken up. Further material in Chap. 2 centers around the Laplace operator, which on Euclidean space R n is