Optimization of an eigenvalue arising in optimal insulation with a lower bound

TL;DR

This paper studies an eigenvalue problem in optimal insulation, proving existence of optimal convex shapes, analyzing symmetry breaking, and proposing a numerical scheme to approximate solutions considering a lower bound constraint.

Contribution

It introduces a new shape optimization problem with a lower bound constraint, proves existence and stability of optimal shapes, and investigates symmetry breaking through numerical experiments.

Findings

Optimal convex shapes exist for the insulation problem.

Symmetry breaking occurs unless mass is near a critical value or the lower bound is large.

Numerical results align with previous symmetric shape optimization when the lower bound is zero.

Abstract

An eigenvalue problem arising in optimal insulation related to the minimization of the heat decay rate of an insulated body is adapted to enforce a positive lower bound imposed on the distribution of insulating material. We prove the existence of optimal domains among a class of convex shapes and propose a numerical scheme to approximate the eigenvalue. The stability of the shape optimization among convex, bounded domains in $\mathbb{R}^3$ is proven for an approximation with polyhedral domains under a non-conformal convexity constraint. We prove that on the ball, symmetry breaking of the optimal insulation can be expected in general. To observe how the lower bound affects the breaking of symmetry in the optimal insulation and the shape optimization, the eigenvalue and optimal domains are approximated for several values of mass $m$ and lower bounds $\ell_{\min}\ge0$. The numerical experiments suggest, that in general symmetry breaking still arises, unless $m$ is close to a critical value $m_0$, and $\ell_{\min}$ large enough such that almost all of the mass $m$ is fixed through the lower bound. For $\ell_{\min}=0$, the numerical results are consistent with previous numerical experiments on shape optimization restricted to rotationally symmetric, convex domains.