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.
Categories: math.AP
