An Introduction to Partial Differential Equations

by Michael Renardy, Robert C. Rogers

This book is intended for a three or four semester course in "Partial Differential Equations". It is based on a four-semester course taught at Virginia Polytechnic Institute and State University. The goal of this class was to provide the background necessary to initiate work on a PhD thesis in partial differential equations. The book opens with an introduction to the subject matter and its characteristics which contain the Cauchy-Kovalevskaya Theorem and Holmgren's Uniqueness Theorem. Conversation laws and shocks are then covered, followed by maximum principles and function spaces. Linear elliptic equations and nonlinear elliptic equations are also included. The text concludes with energy methods for evolution problems and semigroup methods. In the standard graduate curriculum, the subject of partial differential equations is seldom taught with the same thoroughness as algebra or integration theory. This book is aimed at rectifying the situation, by going further than the competition. There are numerous other textbooks on partial differential equations, but few are directed at a beginning graduate audience as this one is. The level of the book is aimed at beginning graduate students. Prerequisites include a truly advanced calculus course and basic complex variables, but no knowledge is required of Lebesque integration theory or functional analysis. This book provides a thorough introduction to partial differential equations, bringing the students up to the level at which research can begin.

Genres: Mathematics

428 pages, Hardcover

First published May 1, 1993

Reviews

[Tomáš Ševček · Rating 5 out of 5]

An interesting book on PDEs, which covers way more than the usual introductory textbook on the subject. The book starts with a recap of the basic ODE concepts and an introduction of the three arguably best-known PDEs (Laplace's, heat and wave). More advanced topics are covered later, such as conservation laws and shocks, maximum principles, distribution theory, transform methods and Green's functions, weak solutions and variational and semigroup methods. The main text also covers some of the basic concepts from functional analysis, so even people with little exposure to this branch of analysis can follow the text. This is in contrast with Evans, where functional analysis is relegated to the appendix, and a deeper prior understanding thereof is assumed. While Evans covers more topics, R&R introduces the theory of distributions and its applications to PDEs (the omission of which is, in my opinion, a gaping hole in Evans). Moreover, more concepts from semigroups, shocks, and characteristics (mostly about classifying PDEs) appear in R&R than in Evans as well. Sometimes, certain concepts are introduced in a different way from Evans, which might pose some issues (for instance, some of the signs in the second-order elliptic operator are swapped, which results in the maximum principles being formulated with opposite inequalities from Evans). All in all, I recommend the book to any pure or applied mathematician who does not feel quite ready to tackle Evans, but still has solid foundations in analysis.

[Cody A · Rating 5 out of 5]

The best introduction to the analysis of PDEs out there. This book and its bibliography have been a guiding light in my research for years.