The limit of the anisotropic double-obstacle Allen–Cahn equation

Abstract

In this paper, we prove that solutions of the anisotropic Allen–Cahn equation in doubleobstacle formwhere A is a strictly convex function, homogeneous of degree two, converge to an anisotropic mean-curvature flowwhen this equation admits a smooth solution in ℝn. Here VN and R respectively denote the normal velocity and the second fundamental form of the interface, and More precisely, we show that the Hausdorff-distance between the zero-level set of φ and the interface of the above anisotropic mean-curvature flow is of order O(ε2).