Stable transition layer for the Allen–Cahn equation when the spatial inhomogeneity vanishes on a nonsmooth hypersurface in ℝⁿ

Abstract

In this paper, we consider an inhomogeneous Allen–Cahn problem where ( ) is a bounded set with smooth boundary and is a non‐negative Lipschitz‐continuous function in . Let be an ‐dimensional hypersurface that divides into two disjoints components ( ) such that on and in . Using the variational concept of ‐convergence, we prove the existence of stable stationary solutions developing a transition layer on as .