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

TL;DR

This paper proves symmetry and monotonicity properties of certain solutions to the Allen-Cahn equation in two dimensions, especially those with finite Morse index and four ends, and provides a classification scheme for such solutions.

Contribution

It establishes symmetry results for solutions with finite Morse index and introduces a classification framework using energy quantization.

Findings

Solutions with finite Morse index and four ends are symmetric with respect to two axes.

Entire solutions with finite Morse index can be classified via energy quantization.

Proved symmetry and monotonicity properties for solutions in a half-plane.

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 {\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.