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Other titles in the Cambridge Tracts in Mathematics series:
Cambridge Tracts in Mathematics #0128: An Introduction to Maximum Principles and Symmetry in Elliptic Problemsby L. E. Fraenkel
Synopses & Reviews
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.
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 new work on non-linear elliptic equations. 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.
Advanced text on differential equations, with plentiful supply of exercises all with detailed hints.
Includes bibliographical references (p.332-336) and index.
Table of Contents
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 W. 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.
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