Mean Curvature Motion of Graphs with Constant Contact Angle and Moving Boundaries

TL;DR

This paper studies the evolution of graphs under mean curvature flow with a fixed contact angle and free boundary, proving local existence, continuation criteria, and behavior for concave initial data.

Contribution

It establishes local existence and continuation criteria for mean curvature flow with constant contact angle and free boundary, including concavity preservation and finite-time existence results.

Findings

Proves local existence of solutions for the mean curvature motion with free boundary.

Derives a continuation criterion based on the second fundamental form.

Shows concavity is preserved and solutions exist only for finite time 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 triple junction motion of hypersurfaces by mean curvature, with constant angles at the junction.