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An Introduction to Maximum Principles and Symmetry in Elliptic Problems

L. E. Fraenkel
Cambridge University Press
Publication Date: 
Number of Pages: 
Cambridge Tracts in Mathematics 128
[Reviewed by
Fernando Gouvêa
, on

OK, true confession first: I know nothing about the subject treated in Introduction to Maximum Principles and Symmetry in Elliptic Problems. But how could I resist? The introduction, speaking of the course at the University of Bath that led to this book, says, after describing a pamphlet that specified the goal of the course:

Naturally, the pamphlet did not state how this goal was to be reached in twenty lectures to students who could not be assumed to have any experience whatever of partial differential equations. Nor were detailed suggestions issued to me when, in the autumn of 1988, I joined the University of Bath and was ordered to give these lectures...

"No experience whatever of partial differential equations" fits me pretty well...

The author goes on to say that "the word Introduction in the title of the book is no gloss." The book tries to make good on the promise to lead the student quickly into the theory with minimal prerequisites by including a lot of background material in five appendices. The writing is lively and has a touch of humor. While it's certainly not easy, it strives to be friendly and to help the reader along. All in all, this seems a worthwhile introduction to some interesting material.

 Fernando Gouvêa ([email protected]) is the editor of MAA Online. He teaches both "History of Mathematics" and "Number Theory", among others, at Colby College. He is a number theorist whose main research focus is on p-adic modular forms and Galois representations.

Preface; 0. Some notation, terminology and basic calculus; 1. Introduction; 2. Some maximum principles for elliptic equations; 3. Symmetry for a non-linear Poisson equation; 4. Symmetry for the non-linear Poisson equation in RN; 5. Monotonicity of positive solutions in a bounded set Ω. Appendix A. On the Newtonian potential; Appendix B. Rudimentary facts about harmonic functions and the Poisson equation; Appendix C. Construction of the primary function of Siegel type; Appendix D. On the divergence theorem and related matters; Appendix E. The edge-point lemma; Notes on sources; References; Index.