Inverse mean curvature flow with outer obstacle

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.