Interface estimates for the fully anisotropic Allen-Cahn equation and anisotropic mean curvature flow

In this paper, we prove that solutions of the anisotropic Allen-Cahn equation in double-obstacle form with kinetic term in {|φ|<1}, where A is a convex function, homogeneous of degree two, and β depends only on the direction of ∇φ, converge to an anisotropic mean-curvature flow Here VN and R denote respectively the normal velocity and the second fundamental form of the interface, and . We prove this in the case when the above flow admits a smooth solution, and we establish that the Hausdorff-distance between the zero-level set of φ and the interface of the flow is of order O(e2).