  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 hypersurface. 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 hypersurface is contained in a strictly mean-convex domain and the
prescribed boundary is on the boundary of the domain.
