Boundary asymptotics for two-dimensional vectorial Allen-Cahn systems

Abstract

We study asymptotic properties of critical points of the two-dimensional vectorial Allen-Cahn energy with finitely many non-degenerate wells, subject to a homogeneous Neumann boundary condition. For any sequence with uniformly bounded energy, we prove that the limiting full and potential measures are supported on a closed countably 1-rectifiable set up to the boundary and satisfy the discrepancy relations. The potential measure defines a free-boundary stationary rectifiable varifold. Boundary mass may occur: weights are constant on regular boundary arcs, finite-type junctions obey projected balance, and the sole non-boundary branch meets the boundary orthogonally. This provides a Neumann-boundary extension of Bethuel's planar interior theory. The key idea is to read the boundary geometry from the limiting stress-energy tensor, which identifies the potential-energy measure as the stationary interfacial measure even in the presence of boundary concentration.