Minimal Graphs and Graphical Mean Curvature Flow in $M\times \mathbb{R}$

Abstract

In this paper, we investigate the problem of finding minimal graphs in $M^n\times \mathbb{R}$ with general boundary conditions using a variational approach. We look at so called generalized solutions of the Dirichlet Problem that minimize a functional adapted from the area functional. We construct barriers to show that for certain conditions on our boundary data, $\varphi (x)$, the solutions obtain the boundary data $\varphi (x)$. Following Oliker-Ural'tseva we also consider solutions $u^{\epsilon }$ of a perturbed mean curvature flow for $\epsilon >0$. We show that there are subsequences ${\epsilon }_i$ where $u^{{\epsilon }_i}$ converges to a function $u$ satisfying the mean curvature flow, and subsequences $u(\cdot ,t_i)$ converge to a generalized solution $\bar{u}$ of the Dirichlet problem. Furthermore, $\bar{u}$ depends only on the choice of sequence ${\epsilon }_i$.

37 pages, 2 figures