Free-Boundary Monotonicity for Almost-Minimizers of the Relative Perimeter

Let $E\subset \mathrm{\Omega }$ be a local almost-minimizer of the relative perimeter in the open set $\mathrm{\Omega }\subset {\mathbb{R}}^n$. We prove a free-boundary monotonicity inequality for $E$ at a point $x\in \mathrm{\partial }\mathrm{\Omega }$, under a geometric property called ``visibility’', that $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$ at $x$.