A free boundary problem in thermal insulation with a prescribed heat source

TL;DR

This paper investigates a free boundary problem related to thermal insulation with a fixed heat source, establishing existence and regularity of optimal insulation configurations under convection boundary conditions.

Contribution

It introduces a new model for thermal insulation with a prescribed heat source and proves the existence and regularity of minimal insulating configurations.

Findings

Existence of a minimal insulation configuration.

Uniform density estimates for the optimal layer.

Model incorporating convection boundary conditions.

Abstract

We study the thermal insulation of a bounded body $\Omega\subset\mathbb{R}^n$, under a prescribed heat source $f>0$, via a bulk layer of insulating material. We consider a model of heat transfer between the insulated body and the environment determined by convection; this corresponds to Robin boundary conditions on the free boundary of the layer. We show that a minimal configuration exists and that it satisfies uniform density estimates.