On Stable Critical Points for a Singular Perturbation Problem

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.

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Published 1 January 2005