Unconditional static convergence for vector-valued Allen–Cahn energies

Prove unconditional convergence of stationary vector-valued Allen–Cahn critical points to generalized minimal surfaces without assuming convergence of the energies in the limit.

Background

The paper discusses the static, or steady-state, analogue of the Allen–Cahn singular-limit problem. Hutchinson and Tonegawa established convergence of scalar critical points to generalized minimal surfaces, while a conditional vector-valued result applies when the energies converge in the limit.

The authors explicitly note that the corresponding unconditional vector-valued convergence problem remains unresolved. This is distinct from the paper’s time-dependent convergence theorem, which concerns a specially constructed system and does not settle the general static vector-valued problem.