A Free Boundary Problem Related to Thermal Insulation

TL;DR

This paper investigates a free boundary problem in thermal insulation theory, focusing on the existence, regularity, and geometric properties of minimal sets where harmonic functions meet Robin boundary conditions.

Contribution

It introduces a novel free boundary problem with Robin conditions on the boundary, establishing existence, density estimates, and regularity results for minimal sets.

Findings

Minimal sets exist for the problem.

Minimal sets satisfy uniform density estimates.

Under certain conditions, boundaries are locally flat.

Abstract

We study a free boundary problem arising from the theory of thermal insulation. The outstanding feature of this set optimization problem is that the boundary of the set being optimized is not a level surface of a harmonic function, but rather a hypersurface along which a harmonic function satisfies a Robin condition. We show that minimal sets exist, satisfy uniform density estimates, and, under some geometric conditions, have "locally flat" boundaries.