An Introduction to Maximum Principles and Symmetry in Elliptic Problems

This is the first book to present 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 non-linear elliptic equations. Gidas, Ni and Nirenberg, building on work of Alexandrov and of 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. Recent and important results are presented with minimal prerequisites in a style suited to graduate students. Two long and leisurely appendices give basic facts about the Laplace and Poisson equations. There is a plentiful supply of exercises, with detailed hints, some of which contain new results.