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, $ϕ(x)$, the solutions obtain the boundary data $ϕ(x)$. Following Oliker-Ural'tseva we also consider solutions $u^ε$ of a perturbed mean curvature flow for $ε> 0$. We show that there are subsequences $ε_i$ where $u^{ε_i}$ 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 $ε_i$.