An Introduction to Maximum Principles and Symmetry in Elliptic Problems

Erscheinungsjahr 2010
ISBN: 978-0-521-17278-3
Verlag: Cambridge University Press

Buch, Englisch, Band 128, 352 Seiten, Format (B × H): 152 mm x 229 mm, Gewicht: 572 g

Reihe: Cambridge Tracts in Mathematics

Originally published in 2000, this was 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 ist boundary condition allow spherically symmetric solutions, then, remarkably, all positive solutions are spherically symmetric. 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.

Autoren/Hrsg.

Fraenkel, L. E.

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen

Inhaltsverzeichnis

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.