Even Symmetry of Some Entire Solutions to the Allen-Cahn Equation in Two Dimensions
Abstract
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 finite Morse index and four ends must be evenly symmetric with respect to two orthogonal axes. A classification scheme of general entire solutions with finite Morse index is also presented using energy quantization.
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