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

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.

Figures (6)

• Figure 1: Eigenvalues $\lambda_{m,\ell_{\min}}(B_1(0))$ with $B_1(0) \subset \mathbb{R}^3$ for $\widehat{m} = \{2, \dots, 5\}$ and $m = \widehat{m} - \ell_{\min} \vert B_1(0)\vert$ (left to right, top to bottom) as functions of $\ell_{\min} = (\widehat{m}/ \vert \partial B_1(0) \vert) q$, for $q \in [0,1]$, evaluated for $q = i/20, i = 0, \dots, 20$, approximated on a triangulation with mesh size $h = 2^{-3}$.
• Figure 2: Optimal distribution to the eigenvalue $\lambda_{m,\ell_{\min}}(B_1(0))$ with $B_1(0) \subset \mathbb{R}^3$ for $\widehat{m}\in\{2,3,5\}$ and $m=\widehat{m}-\ell_{\min}|B_1(0)|$ (left to right), and $\ell_{\min}=(\widehat{m}/ \vert \partial B_1(0) \vert) q$, for $q = i/5$, for $i = 0, \dots ,4$ (top to bottom), the optimal distribution $\widehat{\ell} = \ell_{\min} + h_{u}$ is shown on the boundary; approximated on a triangulation with mesh size $h = 2^{-3}$. The optimal distributions appear to be non-radial but rotationally symmetric for all $q<5$, i.e. where the free mass $m = \widehat{m} - \vert \partial B_1(0) \vert \ell_{\min} >0$.
• Figure 3: Eigenvalues $\widehat{\lambda}_{\widehat{m},\ell_{\min}}$ of the initial domain $B_1(0) \subset \mathbb{R}^3$ and the approximated optimal domains $\Omega^\star$ with $\ell_{\min}=(\widehat{m}/|\partial B_1(0)|)q$, for $q\in[0,1]$, evaluated for $q=i/5$, $i=0,\dots,5$, for $\widehat{m}\in\{2,3,4,5\}$ (left to right, top to bottom).
• Figure 4: Eigenvalues $\widehat{\lambda}_{\widehat{m},\ell_{\min}}$ of the approximated optimal domains $\Omega^\star$ with $\ell_{\min}=(\widehat{m}/|\partial B_1(0)|)q$, for $q\in[0,1]$, evaluated for $q=i/5$, $i=0,\dots,5$, for $\widehat{m}\in\{6,\dots,9\}$ (left to right, top to bottom).
• Figure 5: Approximated optimal domains for $\widehat{\lambda}_{\widehat{m},\ell_{\min}}$ for $\widehat{m}\in\{2,\dots,5\}$ (left to right) and $\ell_{\min} = (\widehat{m}/|\partial B_1(0)|)q$, for $q\in[0,1)$, evaluated for $q = i/5$, $i=0,\dots,4$ (top to bottom), with initial domain approximated on a mesh with maximal mesh size $h = 2^{-3}$. With increasing $\ell_{\min}$, the optimal domains transform from the asymmetric domain to the ball, and the observed kink in the boundary smoothens. The optimal domains appear to be rotationally symmetric, and symmetry breaking in the optimal insulation can be observed for all values of $\widehat{m}$ and $\ell_{\min} < (\widehat{m}/|\partial B_1(0)|)$. For the value $i = 5$ the optimal domains are balls with constant distributions and are not shown here.

Theorems & Definitions (13)

• Theorem 2.1: BBN17
• Proposition 2.2: keller
• Proposition 3.1: della2021optimization
• Theorem 3.2: della2021optimization
• Remark 3.3
• Proposition 4.1
• Proposition 4.2
• Remark 4.3: Non-existence
• Proposition 5.2
• Proposition 5.3