Two dimensional anisotropic mean curvature flow with contact angle condition

In this paper, we study surfaces which evolve by anisotropic mean curvature flow with contact angle boundary condition over a strictly convex domain in ${\mathbb{R}}^2$. We establish a prior gradient estimate for smooth solutions to this boundary value problem. The same approach can also handle Dirichlet boundary condition in ${\mathbb{R}}^n$, $n\ge 2$. For both problems, we prove that the solutions converge to one that is translation invariant in time.