Abstract:
In $\Rn$, we present the quasilinear degenerate parabolic second order PDE which arises as the natural parabolic generalization of the level set formulation of the inverse mean curvature flow. We develop a notion of weak solution that renders precisely the weak solutions of the inverse mean
curvature flow introduced by Huisken and Ilmanen (2001) in case the situation in view is stationary.
Using, as Huisken and Ilmanen did, the approximation scheme of epsilon regularization, we prove existence of a Lipschitz continuous weak solution for appropriate initial and boundary value problems and prove their uniqueness for a wide class of initial values.
Within this class, we identify initial values for which the solutions approximate a stationary solution as time tends to infinity. For the boundaries of the sets $\{ x \vert u(x,t) \le z \}$ of these solutions, we show that in dimension $n \le 7$, they converge towards the corresponding sets of the stationary solution locally uniformly with respect to the $C^{1,\alpha}-$norm $(0<\alpha< 1/2)$ in the sense of sinlge-layered convergence. In arbitrary dimension, we obtain $C^{1,\alpha}-$ regularity of the respective sets up to a singular set of Hausdorff dimension $k \le n-8$. The convergence then holds up to this set and under an additional assumption on the stationary solution.