The Limit of the Fully Anisotropic Double-Obstacle Allen--Cahn Equation in the Nonsmooth Case

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 $N,\ V_N, \mbox{ and } R$ denote the normal, the normal velocity, and the second fundamental form of the interface, respectively, and $B := \sqrt{2A}$.