Mean Curvature Motion of Graphs with Constant Contact Angle at a Free Boundary

TL;DR

This paper studies the mean curvature flow of graphs with a fixed contact angle at a free boundary, proving local existence, a continuation criterion, and analyzing concavity preservation and finite-time existence.

Contribution

It establishes local existence and continuation criteria for mean curvature flow with free boundary conditions and explores concavity preservation and finite-time singularity formation.

Findings

Local existence for the quasilinear parabolic equation with free boundary.

Concavity of the initial graph is preserved during evolution.

Finite-time existence occurs for concave initial graphs.

Abstract

We consider the motion by mean curvature of an $n$-dimensional graph over a time-dependent domain in $\mathbb{R}^n$, intersecting $\mathbb{R}^n$ at a constant angle. In the general case, we prove local existence for the corresponding quasilinear parabolic equation with a free boundary, and derive a continuation criterion based on the second fundamental form. If the initial graph is concave, we show this is preserved, and that the solution exists only for finite time. This corresponds to a symmetric version of mean curvature motion of a network of hypersurfaces with triple junctions, with constant contact angle at the junctions.