  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, $\phi(x)$,
the solutions obtain the boundary data $\phi(x)$. 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 $\epsilon_i$.
