Higher integrability of the gradient for the Thermal Insulation problem

TL;DR

This paper proves that the gradient of solutions to the thermal insulation problem exhibits higher integrability, leading to insights about the structure and size of the free boundary's singular set.

Contribution

It establishes higher integrability of the gradient for the thermal insulation problem, analogous to De Giorgi's conjecture for the Mumford-Shah functional.

Findings

Higher integrability of the gradient for minimizers

Singular part of the free boundary has Hausdorff dimension less than n-1

Advances understanding of free boundary regularity

Abstract

We prove the higher integrability of the gradient for minimizers of the thermal insulation problem, an analogue of De Giorgi's conjecture for the Mumford-Shah functional. We deduce that the singular part of the free boundary has Hausdorff dimension strictly less than $n-1$.