Regularity of free boundary minimal surfaces in locally polyhedral domains

TL;DR

This paper establishes a regularity theorem for free-boundary minimal surfaces in polyhedral domains, showing they are smooth when close to a plane, with applications to hypersurfaces and isoperimetric regions.

Contribution

It introduces an Allard-type regularity result for free-boundary minimal surfaces in polyhedral domains, extending regularity theory to Lipschitz convex polyhedral settings.

Findings

Minimal surfaces are $C^{1,eta}$ regular near free-boundary planes.

Partial regularity results for free-boundary minimizing hypersurfaces.

Regularity results for relative isoperimetric regions.

Abstract

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 $C^{1,\alpha}$ graphical over this plane. We apply our theorem to prove partial regularity results for free-boundary minimizing hypersurfaces, and relative isoperimetric regions.