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An Introduction to Maximum Principles and Symmetry in Elliptic Problems : Cambridge Tracts in Mathematics - L. E. Fraenkel

An Introduction to Maximum Principles and Symmetry in Elliptic Problems

Cambridge Tracts in Mathematics


Published: 14th April 2011
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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.

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

Some notation, terminology and basic calculus
Some maximum principles for elliptic equations
Symmetry for a non-linear Poisson equation
Symmetry for the non-linear Poisson equation in RN
Monotonicity of positive solutions in a bounded set O. Appendix A. On the Newtonian potential
Rudimentary facts about harmonic functions and the Poisson equation
Construction of the primary function of Siegel type
On the divergence theorem and related matters
The edge-point lemma
Notes on sources
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780521172783
ISBN-10: 0521172780
Series: Cambridge Tracts in Mathematics
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 352
Published: 14th April 2011
Publisher: Cambridge University Press
Country of Publication: GB
Dimensions (cm): 22.86 x 15.24  x 2.54
Weight (kg): 0.52