Free-boundary monotonicity for almost-minimizers of the relative perimeter

Abstract

Let $E\subset \Omega$ be a local almost-minimizer of the relative perimeter in the open set $\Omega \subset {\mathbb{R}}^n$. We prove a free-boundary monotonicity inequality for $E$ at a point $x\in \delta \Omega$, under a geometric property called “visibility”, that $\Omega$ is required to satisfy in a neighborhood of $x$. Incidentally, the visibility property is satisfied by a considerably large class of Lipschitz and possibly non-smooth domains. Then, we prove the existence of the density of the relative perimeter of $E$ at $x$, as well as the fact that any blow-up of $E$ at $x$ is necessarily a perimeter-minimizing cone within the tangent cone to $\Omega$ at $x$.