Regularity of free boundary minimal surfaces in locally polyhedral domains

Chao Li, Princeton University

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We prove an Allard-type regularity theorem for free- boundary minimal surfaces in Lipschitz domains locally modelled on convex polyhedra. We show that if such a minimal surface is sufficiently close to an appropriate free-boundary plane, then the surface is graphical over this plane. We apply our theorem to prove partial regularity results for free-boundary minimizing hypersurfaces, and isoperimetric regions. This is based on a joint work with Nick Edelen.