A Mean Curvature Flow with Prescribed Contact Angles in a High Dimensional Cylinder

TL;DR

This paper studies a mean curvature flow with prescribed contact angles in a high-dimensional cylinder, establishing uniform gradient bounds and classifying long-term behavior based on geometric and boundary conditions.

Contribution

It introduces conditions under which uniform gradient bounds are achieved and provides a trichotomy for the asymptotic behavior of solutions in high-dimensional cylinders.

Findings

Established uniform-in-time gradient bounds for solutions.

Derived a trichotomy for long-term solution behavior.

Provided criteria based on geometric and boundary parameters.

Abstract

In this paper we consider a mean curvature flow $V=H+A$ in a high dimensional cylinder $\Omega\times \R$, where, $A$ is a constant, $\Omega$ is a bounded domain in $\R^n$, and, for a hypersurface $y=u(x,t)$ over $\Omega$, $V$ and $H$ denote its normal velocity and mean curvature, respectively. Assume the hypersurface contacts the cylinder boundary $\partial \Omega \times \R$ with prescribed angle $\theta(x)$. Under certain assumptions such as $\Omega$ is strictly convex and $\|\cos\theta\|_{C^2}$ is small, or $\Omega$ is not necessarily convex but $|A|$ is sufficiently large, we derive some {\it uniform-in-time gradient bounds} for the solutions to initial boundary value problems. Then, we present a trichotomy result as well as its criterion for the asymptotic behavior of the solutions, that is, when $I:= A|\Omega|+\int_{\partial \Omega} \cos\theta(x) d\sigma>0$ (resp. $=0$, $<0$), the solution $u$ converges as $t\to \infty$ to a translating solution with positive speed (resp. stationary solution, a translating solution with negative speed).