An Introduction to Maximum Principles and Symmetry in Elliptic Problems

ISBN-13: 9780521461955 / Angielski / Twarda / 2000 / 352 str.

L. E. Fraenkel; B. Bollobas; W. Fulton

Opis

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.

Recenzje

Review of the hardback: '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

Spis treści

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 O. 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.