Inverse mean curvature flow with outer obstacle

We develop a new boundary condition for the weak inverse mean curvature flow, which gives canonical and nontrivial 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;˛r egularity 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.1. Introduction 3069 2. Preliminaries 3081 3. Weak formulations with outer obstacle 3095 4. Liouville theorems on the half space 3107 5. Parabolic estimates near smooth obstacles 3114 6.Initial value problems in smooth domains 3119 7. On maximal solutions 3133 Appendix A. Soliton examples in R 2 3137 Appendix B. Background in geometric measure theory 3140