On the dimension of the singular set in optimization problems with measure constraint

TL;DR

This paper establishes bounds on the dimension of the singular set in shape optimization problems with measure constraints, focusing on heat conduction and Bernoulli problems, using a novel stability formulation.

Contribution

It introduces a new stability notion for the one-phase problem under volume-preserving variations and applies existing methods to estimate the singular set's Hausdorff dimension.

Findings

Bounds on the Hausdorff dimension of the singular set are derived.

A new stability formulation for the one-phase problem is proposed.

The approach extends previous frameworks to measure-constrained optimization problems.

Abstract

In this paper, we prove estimates on the dimension of the singular part of the free boundary for solutions to shape optimization problems with measure constraints. The focus is on the heat conduction problem studied by Aguilera, Caffarelli, and Spruck and the one-phase Bernoulli problem with measure constraint introduced by Aguilera, Alt and Caffarelli. To estimate the Hausdorff dimension of the singular set, we introduce a new formulation of the notion of stability for the one-phase problem along volume-preserving variations, which is preserved under blow-up limits. Finally, the result follows by applying the program developed in [Buttazzo et al. 2022] to this class of domain variation.