Salsa Sandro, Verzini Gianmaria. Partial Differential Equations in Action. Complements and Exercises

Springer, 2015. — 433 p. — ISBN: 978-3-319-15415-2.This book is designed for advanced undergraduate students from various disciplines, including applied mathematics, physics, and engineering. It evolved during the PDE courses that both authors have taught during recent decades at the Politecnico di Milano, and consists of problems of various types and difficulties.
In the first part of the book, while much emphasis is placed on the most common methods of resolution, such as separation of variables or the method of characteristics, we also invite the student to handle the basic theoretical tools and properties of the solutions to the fundamental equations of mathematical physics.
The second part is slightly more advanced and requires basic tools from functional analysis. A small number of exercises aims to familiarize the student with the first elements of the theory of distributions and of the Hilbertian Sobolev spaces. The focus then switches to the variational formulation of the most common boundary value problems for uniformly elliptic equations. A substantial number of problems is devoted to the use of the Riesz representation and the Lax-Milgram theorems together with Fredholm alternative to analyse well posedness or solvability of those problems. Next, a number of problems addresses the analysis of weak solutions to initial-boundary value problems for the heat or the wave equation.
The text is completed by two short appendixes, the first dealing with Sturm-Liouville problems and Bessel functions and the second listing frequently used formulas.
Each chapter begins with a brief review of the main theoretical concepts and tools that constitute necessary prerequisites for a proper understanding. The text Partial Differential Equation in Action [18], by S. Salsa, is the natural theoretical reference.
Within each chapter, the problems are divided into two sections. In the first one we present detailed solutions and comments to provide the student with a reasonably complete guide. In the second section, we propose a set of problems that each student should try to solve by him- or herself. In each case, a solution can be found at the end of the chapter.
Some problems are proposed as theoretical complements and may prove particularlychallenging; this is especially true of those marked with one or two asterisks.