MINIMAL GRAPHS AND GRAPHICAL MEAN CURVATURE FLOW IN Mn × R

Abstract. In this paper, we investigate the problem of finding minimal graphs in Mn × R with general boundary conditions using a variational approach. Follow-ing Giusti[3], we look at so called generalized solutions that minimize a functional adapted from the area functional. Generalizing results in Schulz-Williams [12] to Mn ×R, we show that for certain conditions on our boundary data φ the solutions obtain the boundary data φ(x). Following Oliker-Ural’tseva [10], [11] we also con-sider solutions u of a perturbed mean curvature flow (8.4) for > 0. We show that there are subsequences i where u i converges to a function u satisfying the mean curvature flow, and subsequences u(·, ti) converge to a generalized solution u ̄ of the Dirichlet problem. Furthermore, u ̄ depends only on the choice of sequence i