On Stable Critical Points for a Singular Perturbation Problem

Abstract

We study a singular perturbation problem arising in the scalar two-phase field model. Assuming only the stability of the critical points for ε-problems, we show that the interface regions converge to a generalized stable minimal hypersurface as ε → 0. The limit has an L 2 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.

Numerical implementation of the variational formulation for quasi-static brittle fracture

“…We do not attempt to address the second problem. A similar issue has been settled in [Ton05] for the Modica-Mortola approximation of the perimeter functional.…”

Finite Morse Index Implies Finite Ends

“…Analyzing clustering interfaces is one of main difficulties in the study of singularly perturbed Allen-Cahn equations; see, e.g., [44,[68][69][70]. Analyzing clustering interfaces is one of main difficulties in the study of singularly perturbed Allen-Cahn equations; see, e.g., [44,[68][69][70].…”

“…x/ is the second fundamental form of the level set fu " D u " .x/g and r T denotes the tangential derivative along the level set fu " D u " .x/g; see [67,68].…”

“…It turns out the uniform second-order regularity does not always hold true, and the obstruction is associated to the existence of nontrivial entire solutions to the Toda system, This connection between the Allen-Cahn equation and the Toda system was previously used in [21,23,25] to construct solutions to the Allen-Cahn equation with clustering interfaces. In [68,70], the convergence of clustering interfaces as well as regularity of their limit varifolds were studied. It is shown that the energy at the clustered interfaces is quantized.…”

Regularity of stable minimal hypersurfaces: Recent advances in the theory and applications

“…[MM77], [Mod87], [Ste88], [KS89]) in which case the limit-interface is locally area-minimizing. Combined work of Hutchinson-Tonegawa [HT00] and Tonegawa [Ton05] had previously established that in the stable case, the limit interface is an n-dimensional stable integral varifold V on Ω but gave little information on the regularity of V except when n = 1, in which case it was shown in [Ton05] that spt V locally consists of finitely many disjoint straight line segments.…”