On Stable Critical Points for a Singular Perturbation Problem

摘要

We study a singular perturbation problem arising in the scalar two-phase field model. Assuming only the stability of the critical points for epsilon-problems, we show that the interface regions converge to a generalized stable minimal hypersurface as epsilon goes to 0. The limit has an L2 generalized second fundamental form and the stability condition is expressed in terms of the corresponding inequalities satisfied by stable minimal hypersurfaces. We show that the limit is a finite number of lines with no intersections when the dimension of the domain is 2.