Two dimensional anisotropic mean curvature flow with contact angle condition

TL;DR

This paper investigates the evolution of surfaces under anisotropic mean curvature flow with contact angle boundary conditions in two dimensions, establishing gradient estimates and proving convergence to steady states.

Contribution

It introduces a new gradient estimate technique for anisotropic mean curvature flow with contact angle conditions and demonstrates convergence to translation-invariant solutions.

Findings

Established a prior gradient estimate for smooth solutions.

Proved convergence of solutions to translation-invariant states.

Extended the approach to Dirichlet boundary conditions in higher dimensions.

Abstract

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 $\mathbb{R}^n$, $n\geq 2$. For both problems, we prove that the solutions converge to one that is translation invariant in time.