Symmetry breaking for a problem in optimal insulation

TL;DR

This paper investigates optimal insulation problems in Euclidean domains, revealing symmetry breaking phenomena in eigenvalue-based optimization and exploring shape optimization when the domain varies.

Contribution

It introduces a novel analysis of symmetry breaking in eigenvalue optimization for insulation and extends the study to shape optimization of the domain.

Findings

Symmetry breaking occurs for small insulator amounts in eigenvalue optimization.

Optimal insulation thickness can be non-symmetric even for symmetric domains like a ball.

Shape optimization results are discussed when the domain itself varies.

Abstract

We consider the problem of optimally insulating a given domain $\Omega$ of ${\mathbb{R}}^d$; this amounts to solve a nonlinear variational problem, where the optimal thickness of the insulator is obtained as the boundary trace of the solution. We deal with two different criteria of optimization: the first one consists in the minimization of the total energy of the system, while the second one involves the first eigenvalue of the related differential operator. Surprisingly, the second optimization problem presents a symmetry breaking in the sense that for a ball the optimal thickness is nonsymmetric when the total amount of insulator is small enough. In the last section we discuss the shape optimization problem which is obtained letting $\Omega$ to vary too.