Even Symmetry of Some Entire Solutions to the Allen-Cahn Equation in Two Dimensions

Details Even Symmetry of Some Entire Solutions to the Allen-Cahn Equation in Two Dimensions Even Symmetry of Some Entire Solutions to the Allen-Cahn Equation in Two Dimensions

2011-02-19

arxiv.org/abs/1102.4022v1

Mathematics

Analysis of PDEs

Scientific paper

In this paper, we prove even symmetry and monotonicity of certain solutions of Allen-Cahn equation in a half plane. We also show that entire solutions with {\it finite Morse index} and {\it four ends} must be evenly symmetric with respect to two orthogonal axes. A classification scheme of general entire solutions with {\it finite Morse index} is also presented using energy quantization.

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