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 $W:{\mathbb{R}}^2\to [0,\mathrm{\infty })$ 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.