Anisotropic mean curvature flow with contact angle and Neumann boundary conditions in arbitrary dimensions

Over a bounded strictly convex domain in ${\mathbb{R}}^n$ with smooth boundary, we establish a priori gradient estimate for an anisotropic mean curvature flow with prescribed contact angle and Neumann boundary conditions. The estimates require careful analysis of the degeneracy property of the anisotropic mean curvature operator. As a result, for both problems, we can infer that the solutions converge to one that is translation invariant in time.