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

In this paper, we prove that solutions of the anisotropic\r\nAllen-Cahn equation in double-obstacle form with kinetic term\r\n\begin{displaymath}\r\n \varepsilon \beta(\nabla \varphi) \partial_t \varphi -\r\n \varepsilon \nabla A'(\nabla \varphi) -\r\n \frac{1}{\varepsilon} \varphi\r\n = \frac{\pi}{4} u \quad \mbox{ in } [|\varphi| < 1],\r\n\end{displaymath}\r\nwhere $\ A\ $ is a convex function, homogeneous of degree two,\r\nand $\ \beta\ $ depends only on the direction of $\ \nabla \varphi\ $,\r\nconverge to an anisotropic mean-curvature flow\r\n\begin{displaymath}\r\n \beta(N) V_N = - \mbox{tr}(B(N) D^2 B(N) R) - B(N) u.\r\n\end{displaymath}\r\nHere $\ V_N \mbox{ and } R\ $ respectively denote the normal\r\nvelocity and\r\nthe second fundamental form of the interface, and $\ B := \sqrt{2A}\ $.\r\nWe prove this in the case when the above flow admits a smooth\r\nsolution, and we establish that the Hausdorff-distance\r\nbetween the zero-level set of $\ \varphi\ $ and the interface\r\nof the flow is of order $\ O(\varepsilon^2)\ $.