Tools and Problems in Partial Differential Equations

This book is a collection of 65 solved problems in the theory of partial differential equations. The first part of the book contains a brief review of some of the basic theory and accompanying problem statements. The second part of the book contains the solutions as well as an appendix which contains some basic results in the theory of partial differential equations, such as the change of variables formula in R d and the implicit function theorem.  This is not a typical book in the theory of partial differential equations (i.e. not a textbook); rather, the focus is on the problems related to the various topics. The basic theory which is provided collects some of the main results (e.g. the Sobolev Embedding Theorem), but

proofs are not provided.

 

The authors state this book is meant for graduate students who would like to assess their understanding through practice. Indeed, this book is quite appropriate for such students. While the book is perhaps not meant for students who have not seen the basics of partial differential equations, at the end of each chapter a number of references are provided.

 

Of course, since the theory of partial differential equations is so vast, the authors choose a subset of topics (and corresponding problems) to include. Besides the classical function spaces and classical partial differential equations (Laplace equation, heat equation, wave equation), the authors also include a chapter on Microlocal Analysis, which is an interesting addition.

 

The problems themselves are wide-ranging, from classical results such as showing the smoothing effect for the heat equation, to more detailed problems not necessarily seen in a first course in partial differential equations, such as determining the finite time blow-up for the Euler equations.

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Eric Stachura is currently an Assistant Professor of Mathematics at Kennesaw State University. He is generally interested in analysis and partial differential equations.