Minimal Graphs and Graphical Mean Curvature Flow in $$M^n\times \mathbb {R}$$Mn×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. Following Giusti (Minimal Surfaces and Functions of Bounded Variation. Monographs in Mathematics, vol. 80. Birkhauser Verlag, Basel, 1984), we study the so-called generalized solutions that minimize the adapted area functional (AAF) (1.2). We also show that when the boundary data $$\varphi $$ satisfy certain conditions, the generalized solution is actually the classical solution. This generalizes the results in Schulz–Williams (Analysis 7(3–4):359–374, 1987) to $$M^n\times \mathbb {R}.$$ Finally, following the idea of Oliker–Ural’tseva (Commun Pure Appl Math 46(1):97–135, 1993 and Topol Methods Nonlinear Anal 9(1):17–28, 1997), we consider the long time existence and convergence of the graphical mean curvature flow (1.3). We show that as $$t\rightarrow \infty ,$$ $$u(\cdot , t)\rightarrow \bar{u},$$ where $$\bar{u}$$ is a generalized solution to the associated Dirichlet problem.