Motion of sharp interface of Allen-Cahn equation with anisotropic nonlinear diffusion

Abstract

We consider the Allen-Cahn equation with nonlinear anisotropic diffusion and derive anisotropic direction-dependent curvature flow under the sharp interface limit. The anisotropic curvature flow was already studied, but its derivation is new. We prove both generation and propagation of the interface. For the proof we construct sub- and super-solutions applying the comparison theorem. The problem discussed in this article naturally appeared in the study of the interacting particle systems, especially of non-gradient type. The Allen-Cahn equation obtained from systems of gradient type has a simpler nonlinearity in diffusion and leads to isotropic mean-curvature flow. We extend those results to anisotropic situations.