Equilibrium solutions to generalized motion by mean curvature

In this paper we consider viscosity equilibria to the mean curvature level set flow with a Dirichlet condition. The main result shows that almost every level set of an equilibrium solution is analytic off of a singular set of Hausdorff dimension at most n − 8 and that these level sets are stationary and stable with respect to the area functional. A key tool developed is a maximum principle for solutions to obstacle problems where the obstacle consists of (viscosity) minimal surfaces. Convergence to equilibrium as t → ∞ is also established for the associated initial-boundary value problem.