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

This paper studies minimizing solutions to a two dimensional Allen-Cahn system on the upper half plane, subject to Dirichlet boundary conditions, \begin{equation*} Δu-\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: \ma