Partial Differential Equations II: Qualitative Studies of Linear Equations
Partial Differential Equations II: Qualitative Studies of Linear Equations
复制标题
DOI：
10.1007/978-1-4419-7052-7
复制
发表时间：
1996-06
期刊：
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问题。为了描述物体的机械行为，如振动的弦和吹动的风，它已经发展成为一种与数学的许多分支相互作用的材料，如微分几何，复分析和谐波分析，以及一个无处不在的因素在描述和阐明问题的数学物理。这项工作的目的是提供一个过程中的一些主要方面的研究PDE。它是给读者的背景，在基本介绍研究生数学课程，在美国大学：小学真实的和复杂的分析，微分几何和测量理论。第一章提供了关于常微分方程理论的背景材料。这包括两个非常基本的材料上的主题，如存在性和唯一性的解决方案，常微分方程和明确的解决方案，常系数和关系的线性代数和更复杂的结果，对流动产生的向量场，连接与微分几何，微积分的微分形式，固定行动的原则，在力学，以及它们之间的关系，以哈密顿系统。我们讨论相对论运动方程以及经典牛顿力学方程。也有应用拓扑结果，如度理论，布劳威尔不动点定理，和乔丹-布劳威尔分离定理。在这一章中，我们也处理标量一阶偏微分方程，通过Hamilton-Jacobi理论。第2-6章是对基本线性偏微分方程的概述。第二章首先推导了连续介质力学的一些方程，其方法类似于第二章中力学中常微分方程的推导。1，通过变分原理。我们得到振动弦和膜的方程;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，它们也将提供问题的来源。更多的材料在Chap。2中心周围的拉普拉斯算子，这对欧几里德空间R n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