Infinity-harmonic functions on planar domains are C^{1,1/3} with isolated critical points and unique quasiradial blow-ups, via a p-to-infinity duality that produces inverse mean curvature flow clusters.
Inverse mean curvature flow with outer obstacle
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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.
years
2026 2representative citing papers
New mass invariants for positive cosmological constant initial data characterize de Sitter spacetime and yield positive mass and Penrose inequalities.
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Infinity-harmonic functions and inverse mean curvature flow clusters
Infinity-harmonic functions on planar domains are C^{1,1/3} with isolated critical points and unique quasiradial blow-ups, via a p-to-infinity duality that produces inverse mean curvature flow clusters.
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Mass-type invariants in the presence of a cosmological constant
New mass invariants for positive cosmological constant initial data characterize de Sitter spacetime and yield positive mass and Penrose inequalities.