Quantitative regularity properties for the optimal design problem

TL;DR

This paper advances the regularity theory for the optimal design problem by establishing boundary rectifiability for a broader class of minimizers, improving bounds on component distances, and identifying conditions for full regularity in two dimensions.

Contribution

It extends regularity results for the optimal design problem, including boundary rectifiability and conditions for full regularity in 2D, partially answering existing open questions.

Findings

Boundary of optimal set is uniformly rectifiable for more minimizers.

Improved bounds on the distance between connected components in 2D.

Full regularity in 2D holds if the ratio of constants is not larger than 4.

Abstract

In this paper we slightly improve the regularity theory for the so called optimal design problem. We first establish the uniform rectifiability of the boundary of the optimal set, for a larger class of minimizers, in any dimension. As an application, we improve the bound obtained by Larsen in dimension~2 about the mutual distance between two connected components. Finally we also prove that the full regularity in dimension 2 holds true provided that the ratio between the two constants in front of the Dirichlet energy is not larger than 4, which partially answers to a question raised by Larsen.