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 \begin{displaymath} \varepsilon \beta(\nabla \varphi) \partial_t \varphi - \varepsilon \nabla A'(\nabla \varphi) - \frac{1}{\varepsilon} \varphi = \frac{\pi}{4} u \quad \mbox{ in } [|\varphi| < 1], \end{displaymath} where $\ A\ $ is a convex function, homogeneous of degree two, and $\ \beta\ $ depends only on the direction of $\ \nabla \varphi\ $, converge to an anisotropic mean-curvature flow \begin{displaymath} \beta(N) V_N = - \mbox{tr}(B(N) D^2 B(N) R) - B(N) u. \end{displaymath} Here $\ V_N \mbox{ and } R\ $ respectively denote the normal velocity and the second fundamental form of the interface, and $\ B := \sqrt{2A}\ $. 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 $\ \varphi\ $ and the interface of the flow is of order $\ O(\varepsilon^2)\ $.