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

TL;DR

This paper proves gradient estimates and convergence results for anisotropic mean curvature flow with contact angle and Neumann boundary conditions in arbitrary dimensions, advancing understanding of geometric flows in convex domains.

Contribution

It establishes a priori gradient estimates and convergence for anisotropic mean curvature flow with boundary conditions in arbitrary dimensions, addressing degeneracy issues.

Findings

Solutions converge to time-translation invariant states.

Gradient estimates are obtained under degeneracy considerations.

Results apply to arbitrary-dimensional convex domains.

Abstract

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.