An Introduction to Partial Differential Equations (Texts in Applied Mathematics Vol. 13) - Hardcover

Michael Renardy; Robert C. Rogers

Hardcover

ISBN 10: 3540979522 ISBN 13: 9783540979524

Publisher: Springer Verlag

Synopsis

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 differntial 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 should provide a thorough introduction to partial differential equations, bringing the students up to the level at which research can begin.

About this title

From the Back Cover

  Partial differential equations (PDEs) 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, PDEs 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 coarse. 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.

Review

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)

Edition details

• Publisher: Springer Verlag
• ISBN 10: 3540979522
• ISBN 13: 9783540979524
• Binding: Hardcover
• Language: English