Partial Differential Equations, Volume 2

Author: M.E. Taylor
Publisher:
Publication Date: 1999-06-01
Features: This book is in English.
Excerpt: Introduction Partial differential equations is 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 Hamiltonian 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 Hamilton-Jacobi theory. Chapters 2 through 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 Chapter 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 Chapter 2 centers around the Laplace operator, which on Euclidean space \(\mathbb{R}^n\) is and the linear wave equation. We also consider the Laplace operator on a general Riemannian manifold and the wave equation on a general Lorentzian manifold. We discuss basic consequences of Green's formula, including energy conservation and finite propagation speed for solutions to linear wave equations. We also discuss Maxwell's equations for electromagnetic fields and their relation with special relativity. Before we can establish general results on the solvability of these equations, it is necessary to develop some analytical techniques. This is done in the next couple of chapters. Chapter 3 is devoted to Fourier analysis and the theory of distributions. These topics are crucial for the study of linear PDE. We give a number of basic applications to the study of linear PDE with constant coefficients. Among these applications are results on harmonic and holomorphic functions in the plane, including a short treatment of elementary complex function theory. We derive explicit formulas for solutions to Laplace and wave equations on Euclidean space, as well as the heat equation. We also produce solutions on certain subsets, such as rectangular regions, using the method of images. We include material on the discrete Fourier transform, germane to the discrete approximation of PDE, and on the fast evaluation of this transform, the FFT. Chapter 3 is the first chapter to make extensive use of functional analysis. Basic results on this topic are compiled in Appendix A, Outline of Functional Analysis. Sobolev spaces have proven to be a very effective tool in the existence theory of PDE, and in the study of regularity of solutions. In Chapter 4 we introduce Sobolev spaces and study some of their basic properties. We restrict attention to \(L^s\)-Sobolev spaces, such as \(H^k(\mathbb{R}^n)\), which consist of \(L^s\)-functions whose derivatives of order \(k\) (defined in a distributional sense, in Chapter 3) belong to \(L^2(\mathbb{R}^n)\), when \(k\) is a positive integer. We also replace \(k\) by a general real number. The \(L^s\)-Sobolev spaces, which are very useful for nonlinear PDE, are treated later, in Chapter 13. Chapter 5 is devoted to the study of the existence and regularity of solutions to linear elliptic PDE on bounded regions. We begin with the Dirichlet problem for the Laplace operator, and then treat the Neumann problem and various other boundary problems, including some that apply to electromagnetic fields. We also study general boundary problems for linear elliptic operators, giving a condition that guarantees regularity and solvability (perhaps given a finite number of linear conditions on the data). Also in Chapter 5 are some applications to other areas, such as a proof of the Riemann mapping theorem, first for smooth simply connected domains in the complex plane \(\mathbb{C}\), then, after a treatment of the Dirichlet problem for the Laplace operator on domains with rough boundary, for general simply connected domains in \(\mathbb{C}\). We also develop Hodge theory and apply it to De Rham cohomology, extending the study of topological applications of differential forms begun in Chapter 1.