Shape optimization of a thermal insulation problem

TL;DR

This paper investigates the optimal shape of insulating layers around a heated object to minimize energy, providing full solutions for convection cases and partial results for general heat transfer conditions.

Contribution

It offers a comprehensive analysis of shape optimization for thermal insulation, including existence, regularity, and characterization of minimizers under various heat transfer laws.

Findings

Full characterization of minimizers in convection cases

Existence and regularity results for general heat transfer laws

Partial description of optimal shapes in non-convection scenarios

Abstract

We study a shape optimization problem involving a solid $K\subset\mathbb{R}^n$ that is maintained at constant temperature and is enveloped by a layer of insulating material $\Omega$ which obeys a generalized boundary heat transfer law. We minimize the energy of such configurations among all $(K,\Omega)$ with prescribed measure for $K$ and $\Omega$, and no topological or geometrical constraints. In the convection case (corresponding to Robin boundary conditions on $\partial\Omega$) we obtain a full description of minimizers, while for general heat transfer conditions, we prove the existence and regularity of solutions and give a partial description of minimizers.