An optimization problem in thermal insulation with Robin boundary conditions

TL;DR

This paper analyzes an optimization problem in thermal insulation involving Robin boundary conditions, deriving a limit functional, and proving that a uniform insulating layer on a ball maximizes heat retention.

Contribution

It introduces a Γ-limit analysis for the energy functional with small insulating layers and proves the optimality of uniform insulation on a spherical body.

Findings

Γ-limit of the energy functional computed

Minimizers satisfy a PDE with non-uniform Robin boundary conditions

Uniform insulation on a sphere maximizes heat content

Abstract

We study thermal insulating of a bounded body $\Omega\subset \mathbb{R}^n$. Under a prescribed heat source $f\geq 0$, we consider a model of heat transfer between $\Omega$ and the environment determined by convection; this corresponds, before insulation, to Robin boundary conditions. The body is then surrounded by a layer of insulating material of thickness of size $\varepsilon>0$, and whose conductivity is also proportional to $\varepsilon$. This corresponds to the case of a small amount of insulating material, with excellent insulating properties. We then compute the $\Gamma$-limit of the energy functional $F_\varepsilon$ and prove that this is a functional $F$ whose minimizers still satisfy an elliptic PDEs system with a non uniform Robin boundary condition depending on the distribution of insulating layer around $\Omega$. In a second step we study the maximization of heat content (which measures the goodness of the insulation) among all the possible distributions of insulating material with fixed mass, and prove an optimal upper bound in terms of geometric properties. Eventually we prove a conjecture which states that the ball surrounded by a uniform distribution of insulating material maximizes the heat content.