Partial Differential Equations: An Accessible Route through Theory and Applications

The study of partial differential equations (PDEs) is fundamental in pure and applied mathematics. PDEs also play an important role in the modeling and understanding of physical phenomena. It is no wonder, then, that there are a large number of textbooks dedicated to helping students at a variety of levels, and from a variety of disciplines learn the subject. András Vasy’s Partial Differential Equations: An Accessible Route through Theory and Applications is an admirable contribution to the list of textbooks on PDEs. Vasy’s book is carefully written and engaging; I would certainly recommend this book. Before delving into the specifics, I will compare the aim and spirit of this text with that of some other books on PDEs with which I am familiar.

At the undergraduate level, there are books such as Haberman’s Applied Partial Differential Equations and Strauss’s Partial Differential Equations An Introduction, which require little or no background in real analysis from the student. At the graduate level, one can find texts such as Introduction to Partial Differential Equations by Folland and Partial Differential Equations by Evans, both of which require a background in real analysis at the graduate level. Vasy’s Accessible Route is at a level in between. While Vasy does not avoid analysis, his book does not expect or require as much background in analysis such as that of Evans or Folland. Specifically, the book makes use of the basics of metric spaces, different types of convergence, the inverse function theorem and some other concepts and techniques from elementary real analysis, but does not use or develop measure theory or functional analysis. It is true that distributions, the basics of Hilbert space theory, and the space of functions \(L^{1}\) are introduced, but all in such a way that does not require very much sophistication in analysis on the part of the reader. In fact, I believe that Partial Differential Equations: An Accessible Route through Theory and Applications could inspire a student to want to learn more analysis. A common complaint I hear from students regarding undergraduate real analysis is that it is too much like calculus, which they believe they already know very well. Vasy’s book shows many ways in which elementary real analysis can be used to obtain interesting results in territory that is unfamiliar and perhaps even exciting.

Even with regular appeal to some basic ideas from real analysis, An Accessible Route through Theory and Applications does not eschew the concrete. Students learning from this book will have plenty of opportunities to see closed form solutions of specific PDEs. Long story short, I think that Vasy strikes an really nice balance between the abstract and the concrete at a level that is accessible to a wide variety of students. The book does not risk giving the impression that writing down closed form solutions to specific problems is the main thrust of study in PDE, but at the same time does not use sophisticated machinery to derive very general abstract results.

Like each of the other texts mentioned, this textbook is a survey of PDEs in the sense that elliptic, parabolic, hyperbolic, linear and nonlinear equations are all studied. Furthermore, the book gives the reader a good sense of what types of results one typically wants to obtain in the study of partial differential equations. This gives the book a modern feel even in the treatment of classical results.

There are several aspects of Partial Differential Equations: An Accessible Route through Theory and Applications that I really enjoy. For instance, the second chapter of the book motivates PDEs as arising in the study of physical phenomena. I especially like that the author explains how many PDEs arise as the Euler-Lagrange equation corresponding to a variational problem. In general, the topics discussed in the book are well motivated by the author, who appeals to geometry or other ideas to help the reader gain an intuition for what is going on with the big picture in solving or analyzing solutions to PDEs. I also enjoy the author’s use of energy methods for the wave, heat and Laplace equations. The treatment of energy for Laplace’s equation includes a derivation for a special case of the Poincaré inequality. It is nice that a student could be introduced to the utility of working with such inequalities before embarking on a more general study of inequalities related to the function spaces occurring in the study of PDEs.

Each chapter ends with a series of exercises for the reader. In keeping with the spirit of the book, there are both “solve the PDE”, and “show that” type problems. In some cases, the exercises extend the discussion in the main part of the chapter, while in others they provide the reader an opportunity to test their understanding of the principal concepts and techniques. I would suggest that an appropriate use for this textbook is in a graduate course in PDEs for first year graduate students, maybe even senior undergraduates. Why should a student interested in applied mathematics or PDEs wait until after a year of graduate analysis to get a taste of modern PDEs? Partial Differential Equations: An Accessible Route through Theory and Applications is an ideal book to expose students to modern PDE with minimal background. It is likely that a first year graduate student could read the majority of this text on their own although I suspect that most undergraduates would find independent reading of the text difficult. Furthermore, this book is likely to encourage students to want to learn more analysis such as functional analysis and operator theory.

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Jason M. Graham is an assistant professor in the department of mathematics at the University of Scranton, Scranton, Pennsylvania. His current professional interests are in teaching applied mathematics and mathematical biology, and collaborating with biologists specializing in the collective behavior of groups of organisms.