An Introduction to Maximum Principles and Symmetry in Elliptic Problems [Hardcover]

Advanced text, originally published in 2000, on differential equations, with plentiful supply of exercises all with detailed hints.This is the first book to present the basic theory of the symmetry of solutions to second-order elliptic equations by means of the maximum principle. It proceeds from elementary facts about the linear case to recent results, 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 a plentiful supply of exercises, with detailed hints, some of which contain new results.This is the first book to present the basic theory of the symmetry of solutions to second-order elliptic equations by means of the maximum principle. It proceeds from elementary facts about the linear case to recent results, 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 a plentiful supply of exercises, with detailed hints, some of which contain new results.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 anlsš