Boundary Properties for Almost-Minimizers of the Relative Perimeter

Let A be an Euclidean open Lipschitz set. This dissertation aims to discuss some results concerning the boundary regularity for almost-minimizers of the relative perimeter in A. An almost-minimizer of the relative perimeter in A is a measurable set E that minimizes the perimeter functional P(E;A), roughly speaking the (n-1)-area of the boundary of E in A, among local competitors for E. Important examples of almost-minimizers in A are given, for instance, by the solutions to relative isoperimetric problems like min { P(E;A) : E is contained in A and |E| = m}. While when A is smooth the theory of the boundary regularity for almost-minimizers is well-established, little is known when the boundary of A contains singular points such as edges, vertices, cusps, etc. In particular, we prove a boundary Monotonicity Formula, holding under a so-called visibility condition on A at a point x on the boundary of A, and a Vertex-skipping Theorem, valid when n = 3 and A is convex. This latter result establishes that the closure of the boundary of an almost-minimizer of the relative perimeter in a 3-dimensional open, convex set A cannot contain vertex-type singularities of the boundary of A. The optimality of the dimensional restriction n = 3 is also examined in the thesis.