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An Introduction to Maximum Principles and Symmetry in Elliptic Problems
Author(s): L. E. Fraenkel (Author)
- Publisher: Cambridge University Press
- Publication Date: April 14, 2011
- Edition: Illustrated
- Language: English
- Print length: 352 pages
- ISBN-10: 0521172780
- ISBN-13: 9780521172783
Book Description
This book presents the basic theory of the symmetry of solutions to second-order elliptic partial differential equations by means of the maximum principle. It proceeds from elementary facts about the linear case to recent results about positive solutions of nonlinear elliptic equations. Gidas, Ni and Nirenberg, building on the work of Alexandrov and Serrin, have shown that the shape of the set on which such elliptic equations are solved has a strong effect on the form of positive solutions. In particular, if the equation and its boundary condition allow spherically symmetric solutions, then, remarkably, all positive solutions are spherically symmetric. These recent and important results are presented with minimal prerequisites, in a style suited to graduate students. Two long appendices give a leisurely account of basic facts about the Laplace and Poisson equations, and there is an abundance of exercises, with detailed hints, some of which contain new results.
Editorial Reviews
Review
“The originality of this book mainly consists in new proofs and new extensions of quite well known results concerning maximum principles and symmetry properties related to semilinear elliptic equations.” Mathematical Reviews
Book Description
Advanced text, originally published in 2000, on differential equations, with plentiful supply of exercises all with detailed hints.
