Inverse mean curvature flow with outer obstacle

TL;DR

This paper introduces a new boundary condition for inverse mean curvature flow in bounded domains, ensuring existence, uniqueness, and regularity of solutions with an outer obstacle, advancing the mathematical understanding of geometric flows.

Contribution

It develops a novel boundary condition for weak inverse mean curvature flow that guarantees well-posedness and regularity in bounded domains with obstacles.

Findings

Established existence and uniqueness of solutions.

Proved $C^{1,eta}$ regularity of level sets.

Addressed well-posedness for initial value problems.

Abstract

We develop a new boundary condition for the weak inverse mean curvature flow, which gives canonical and non-trivial solutions in bounded domains. Roughly speaking, the boundary of the domain serves as an outer obstacle, and the evolving hypersurfaces are assumed to stick tangentially to the boundary upon contact. In smooth bounded domains, we prove an existence and uniqueness theorem for weak solutions, and establish $C^{1,\alpha}$ regularity of the level sets up to the obstacle. The proof combines various techniques, including elliptic regularization, blow-up analysis, and certain parabolic estimates. As an analytic application, we address the well-posedness problem for the usual weak inverse mean curvature flow, showing that the initial value problem always admits a unique maximal (or innermost) weak solution.