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A Compact Course on Linear PDEs

Alberto Valli
Publication Date: 
Number of Pages: 
[Reviewed by
Bill Satzer
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This book arose from a course the author taught in Italy for Masters-degree level students. In the preface, the author describes his decisions for what to include in a course on partial differential equations (PDEs) at that level. He decides to focus on linear PDEs. He notes that while most applications involve nonlinear PDEs, linear approximations often give quite reasonable results.
The resulting book is more advanced and more abstract than a classical treatment of linear PDEs. The main difference in the author’s treatment compared to similar texts is his concentration on what is often called the weak formulation of PDEs where weak derivatives are used to satisfy integral versions of the PDEs. Broadly speaking, this means that solving a linear PDE means solving a problem where a linear operator operates between suitably chosen infinite-dimensional vector spaces.
The elements of the author’s more abstract formulation include weak derivatives, weak solutions, Sobolev spaces, and corresponding elements of functional analysis. Sobolev spaces are Banach spaces having norms that combine that of \( L^{p} \) functions and weak derivatives up to a certain order. The topology is defined in such a way that the resulting space is indeed complete.
Implementation of the weak formulation amounts to transforming the original PDE into a set of infinitely many integral equations, one for each test function in an appropriate vector space. The author motivates this approach by beginning with finite dimensional linear systems and then passing into the infinite dimensional setting. He devotes much of a chapter to the differences between operating in finite and infinite dimensional spaces.
The weak formulation for linear PDEs is developed first for elliptic PDEs. This is followed by a collection of technical results and a variety of other topics including the Fredholm alternative, spectral theory for elliptic operators and Sobolev embedding theorems. Linear parabolic and hyperbolic PDEs are treated at the end.
Exercises are provided at the end of each chapter, and they are immediately followed by solutions. Some of the exercises ask for deeper analysis of results presented in the text. There is a modest set of references and an index that is rather sparse.
The book is designed to be compact, and it is – very. The author treats many topics very quickly, and the connecting links are also much abbreviated. A lot is demanded of the reader, even one starting with a background of classical PDEs at the undergraduate level.

Bill Satzer ([email protected]), now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications ranging from network modeling and speech recognition to materials science and optical films. He did his PhD work in dynamical systems and celestial mechanics.