Regularity of Free Boundary Minimal Surfaces in Locally Polyhedral Domains

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 graphical over this plane. We apply our theorem to prove partial regularity results for free‐boundary minimizing hypersurfaces, and relative isoperimetric regions.