DiBenedetto Partial Differential Equations

Second Edition
2. Auflage 2010
ISBN: 978-0-8176-4552-6
Verlag: Birkhäuser Boston
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 402 Seiten, eBook

Reihe: Cornerstones

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 “uniqueness classes. ” 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.

Zielgruppe

Graduate

Autoren/Hrsg.

DiBenedetto, Emmanuele

Inhaltsverzeichnis

Preliminaries.- Quasi-Linear Equations and the Cauchy#x2013;Kowalewski Theorem.- The Laplace Equation.- Boundary Value Problems by Double-Layer Potentials.- Integral Equations and Eigenvalue Problems.- The Heat Equation.- The Wave Equation.- Quasi-Linear Equations of First-Order.- Non-Linear Equations of First-Order.- Linear Elliptic Equations with Measurable Coefficients.- DeGiorgi Classes.