A survey on the optimal partition problem

Abstract

This survey synthesizes the current state of the art on the regularity theory for solutions to the optimal partition problem. Namely, we consider non-negative, vector-valued Sobolev functions whose components have mutually disjoint support, and which are either local minimizers of the Dirichlet energy or, more generally, critical points satisfying a system of variational inequalities. This is particularly meaningful as the problem has emerged on several occasions and in diverse contexts: our aim is then to provide a coherent point of view and an up-to-date account of the progress concerning regularity of the solutions and their free boundaries, both in the interior and up to a fixed boundary.