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- Title
Minimal Graphs and Graphical Mean Curvature Flow in Mn×R.
- Authors
McGonagle, Matthew; Xiao, Ling
- Abstract
In this paper, we investigate the problem of finding minimal graphs in M n × R with general boundary conditions using a variational approach. Following Giusti (Minimal Surfaces and Functions of Bounded Variation. Monographs in Mathematics, vol. 80. Birkhäuser 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 φ 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 × 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 → ∞ , u (· , t) → u ¯ , where u ¯ is a generalized solution to the associated Dirichlet problem.
- Publication
Journal of Geometric Analysis, 2020, Vol 30, Issue 2, p2189
- ISSN
1050-6926
- Publication type
Article
- DOI
10.1007/s12220-019-00179-2