Half-space minimizing solutions of a two dimensional Allen-Cahn system

TL;DR

This paper classifies and analyzes minimizing solutions to a two-dimensional Allen-Cahn system on the upper half plane with boundary conditions, focusing on their asymptotic behavior and blow-down limits.

Contribution

It provides a complete classification of half-space minimizing solutions and characterizes their asymptotic behavior near sharp interfaces.

Findings

Classification of solutions via blow-down limits

Asymptotic behavior near sharp interfaces characterized

Complete description of solutions on the upper half plane

Abstract

This paper studies minimizing solutions to a two dimensional Allen-Cahn system on the upper half plane, subject to Dirichlet boundary conditions, \begin{equation*} \Delta u-\nabla_u W(u)=0, \quad u: \mathbb{R}_+^2\to \mathbb{R}^2,\ u=u_0 \text{ on } \partial \mathbb{R}_+^2, \end{equation*} where $W: \mathbb{R}^2\to [0,\infty)$ is a multi-well potential. We give a complete classification of such half-space minimizing solutions in terms of their blow-down limits at infinity. In addition, we characterize the asymptotic behavior of solutions near the associated sharp interfaces.