A homotopy approach to solving the inverse mean curvature flow

Abstract

In R n , n < 7, we treat the quasilinear, degenerate parabolic initial and boundary value problem which is the natural parabolic extension of Huisken and Ilmanen's weak inverse mean curvature flow (IMCF). We prove long time existence and partial uniqueness of Lipschitz continuous weak solutions u(x, t) and show C 1,α -regularity for the sets ∂{x|u(x, t) < z}. Our approach offers a new approximation for weak solutions of the IMCF starting from a class of interesting and easily obtainable initial values; for these, the above sets are shown to converge against corresponding surfaces of the IMCF as t −→ ∞ globally in Hausdorff distance and locally uniformly with respect to the C 1,α -norm.

Weak solutions of inverse mean curvature flow for hypersurfaces with boundary

“…Brought to you by | University of California Authenticated Download Date | 6/11/15 9:18 PM (6) Another related problem was considered by Hein [10] who proved the existence of weak solutions similar to those of Huisken-Ilmanen for a parabolic version of the level-set equation. This approach is also interesting with regard to a numerical treatment of IMCF.…”

The Dirichlet problem for a singular elliptic equation arising in the level set formulation of the inverse mean curvature flow

“…This differential operator appears in the level set formulation of the inverse mean curvature flow ( [12], see also [11], [17] and [18]). …”

Gradient estimates for the p-Laplace heat equation under the Ricci flow

“…We also use p to denote the p-Laplace operator determined by g(t). We should point out that this regularization procedure has been used in [9,12]. The main aim of this paper is to get the space-only gradient estimates for positive continuous weak solutions to the system (1.3) and (1.4) in the continuous weak sense.…”