Partial Differential Equations (eBook)

This book offers a self-contained introduction to partial differential equations (PDEs), primarily focusing on linear equations, and also providing perspective on nonlinear equations. The treatment is mathematically rigorous with a generally theoretical layout, with indications to some of the physical origins of PDEs. The Second Edition is rewritten to incorporate years of classroom feedback, to correct errors and to improve clarity. The exposition offers many examples, problems and solutions to enhance understanding. Requiring only advanced differential calculus and some basic Lp theory, the book will appeal to advanced undergraduates and graduate students, and to applied mathematicians and mathematical physicists.

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This is a revised and extended version of my 1995 elementary introduction to partial di?erential equations. The material is essentially the same except for three new chapters. The ?rst (Chapter 8) is about non-linear equations of ?rst order and in particular Hamilton-Jacobi equations. It builds on the continuing idea that PDEs, although a branch of mathematical analysis, are closely related to models of physical phenomena. Such underlying physics in turn provides ideas of solvability. The Hopf variational approach to the Cauchy problem for Hamilton-Jacobi equations is one of the clearest and most incisive examples of such an interplay. The method is a perfect blend of classical mechanics, through the role and properties of the Lagrangian and Hamiltonian, and calculus of variations. A delicate issue is that of identifying "e;uniqueness classes. "e; An e?ort has been made to extract the geometrical conditions on the graph of solutions, such as quasi-concavity, for uniqueness to hold. Chapter 9 is an introduction to weak formulations, Sobolev spaces, and direct variationalmethods for linear and quasi-linearelliptic equations. While terse, the material on Sobolev spaces is reasonably complete, at least for a PDEuser. Itincludesallthebasicembeddingtheorems,includingtheirproofs, and the theory of traces. Weak formulations of the Dirichlet and Neumann problems build on this material. Related variational and Galerkin methods, as well as eigenvalue problems, are presented within their weak framework.