A LEVEL-SET METHOD FOR A MEAN CURVATURE FLOW WITH A PRESCRIBED BOUNDARY

We propose a level-set method for a mean curvature flow whose boundary is prescribed by interpreting the boundary as an obstacle. Since the corresponding obstacle problem is globally solvable, our method gives a global-in-time level-set mean curvature flow under a prescribed boundary with no restriction of the profile of an initial hyper-surface. We show that our solution agrees with a classical mean curvature flow under the Dirichlet condition. We moreover prove that our solution agrees with a level-set flow under the Dirichlet condition constructed by P. Sternberg and W. P. Ziemer (1994), where the initial hyper-surface is contained in a strictly mean-convex domain and the prescribed boundary is on the boundary of the domain.