A survey on the optimal partition problem

Roberto Ognibene; Bozhidar Velichkov

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arXiv · 2510.08241

A survey on the optimal partition problem

Roberto Ognibene·Bozhidar Velichkov

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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.

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Roberto Ognibene, Bozhidar Velichkov. 2025-10-09. A survey on the optimal partition problem. https://arxiv.org/abs/2510.08241

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