Regularity of Free Boundary Minimal Surfaces in Locally Polyhedral Domains

Abstract

We prove an Allard‐type regularity theorem for free‐boundary minimal surfaces in Lipschitz domains locally modeled on convex polyhedra. We show that if such a minimal surface is sufficiently close to an appropriate free‐boundary plane, then the surface is C1,α graphical over this plane. We apply our theorem to prove partial regularity results for free‐boundary minimizing hypersurfaces, and relative isoperimetric regions. © 2022 Wiley Periodicals LLC.

Dihedral rigidity of parabolic polyhedrons in hyperbolic spaces

“…Remark 2.6. The condition that P = P 0 × R n−2 , P 0 is non-obtuse, is due to the boundary regularity theory developed by Edelen and the author [EL20].…”

Heintze-Karcher inequality and capillary hypersurfaces in a wedge

“…The latter was crucially used in a very recent work of Chao Li [21] mentioned at the beginning of the Introduction. See also his further work [12] with Edelen on surfaces in a polyhedral domain, which is also closely related to surfaces in a wedge domain.…”