Nonparametric mean curvature type flows of graphs with contact angle conditions

TL;DR

This paper investigates nonparametric mean curvature flows of graphs with contact angle conditions on Riemannian manifolds, establishing long-term existence and convergence results under specific conditions, with applications to boundary value problems.

Contribution

It provides new results on the long-time existence and convergence of mean curvature type flows with contact angle conditions in Riemannian manifolds, including applications to boundary problems.

Findings

Long time existence of solutions under certain conditions.

Uniform convergence of the flow as time approaches infinity.

Applications to mean curvature equations with boundary conditions.

Abstract

In this paper we study nonparametric mean curvature type flows in $M\times\mathbb{R}$ which are represented as graphs $(x,u(x,t))$ over a domain in a Riemannian manifold $M$ with prescribed contact angle. The speed of $u$ is the mean curvature speed minus an admissible function $\psi(x,u,Du)$. Long time existence and uniformly convergence are established if $\psi(x,u, Du)\equiv 0$ with vertical contact angle and $\psi(x,u,Du)=h(x,u)\omega$ with $h_u(x,u)\geq h_0>0$ and $\omega=\sqrt{1+|Du|^2}$. Their applications include mean curvature type equations with prescribed contact angle boundary condition and the asymptotic behavior of nonparametric mean curvature flows of graphs over a convex domain in $M^2$ which is a surface with nonnegative Ricci curvature.