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