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  3. An Introduction to Maximum Principles and Symmetry in Elliptic Problems
  • Cited by 126
Publisher:
Cambridge University Press
Online publication date:
February 2010
Print publication year:
2000
Online ISBN:
9780511569203
DOI:
https://doi.org/10.1017/CBO9780511569203
Subjects:
Mathematics (general), Mathematics, Differential and Integral Equations, Dynamical Systems and Control Theory, Real and Complex Analysis
Series:
Cambridge Tracts in Mathematics (128)
Information

Book description

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 its 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.

Reviews

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.'

Source: Mathematical Reviews

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