### An Introduction to Maximum Principles and Symmetry in Elliptic Problems (Cambridge Tracts in Mathematics) by L. E. Fraenkel

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**An Introduction to Maximum Principles and Symmetry in Elliptic Problems (Cambridge Tracts in Mathematics) by L. E. Fraenkel**

English | Apr 13, 2000 | ISBN: 0521461952 | 351 Pages | PDF | 2 MB

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.

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