An Introduction to Partial Differential Equations

Partial differential equations are fundamental to the modeling of natural phenomena, arising in every field of science. Consequently, the desire to understand the solutions of these equations has always had a prominent place in the efforts of mathematicians; it has inspired such diverse fields as complex function theory, functional analysis and algebraic topology. Like algebra, topology, and rational mechanics, partial differential equations are a core area of mathematics.

This book aims to provide the background necessary to initiate work on a Ph.D. thesis in PDEs for beginning graduate students. Prerequisites include a truly advanced calculus course and basic complex variables. Lebesgue integration is needed only in Chapter 10, and the necessary tools from functional analysis are developed within the course. The book can be used to teach a variety of different courses. This new edition features new problems throughout and the problems have been rearranged in each section from simplest to most difficult. New examples have also been added. The material on Sobolev spaces has been rearranged and expanded. A new section on nonlinear variational problems with "Young-measure" solutions appears. The reference section has also been expanded.

From the reviews of the second edition:

"This second edition of the book from 1993 is still one of the most modern books about PDEs. There is a number of new examples and exercises, or new sections about Sobolev spaces and nonlinear variational problems. The book is written very well and can be strongly recommended as a textbook for a number of PDE courses. The presentation of a number of subjects is the best available in the literature." (G. Schneider, Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 84(10-11), 2004)

"An Introduction to Partial Differential Equations (2nd ed.) is a very careful exposition of functional analytic methods applied to PDEs. a self-contained text that can be used as the basis of an advanced course in PDEs or as an excellent guide for self-study by a motivated reader. acts and feels like a standard book in a specific area of mathematics. ... The Renardy and Rogers text contains a large number of problems . For students, these problems will be challenging and interesting. is a great book." (Ronald B. Guenther, Enrique A. Thomann, SIAM Reviews, Vol. 47(1), 2005)

"The purpose of this book is to put the topic of differential equations on the same footing in the graduate curriculum as algebra and analysis. the authors describe it as a book for three or four semesters. they succeed admirably. The book is extremely well-written with lots of examples and motivation for the theory." (Gary M. Lieberman, Zentralblatt MATH, Vol. 1072, 2005)

Autorentext

Inhalt
Characteristics.- Conservation Laws and Shocks.- Maximum Principles.- Distributions.- Function Spaces.- Sobolev Spaces.- Operator Theory.- Linear Elliptic Equations.- Nonlinear Elliptic Equations.- Energy Methods for Evolution Problems.- Semigroup Methods.