Modeling and Analysis of an Optimal Insulation Problem on Non-Smooth Domains

In this paper, we study an insulation problem that seeks the optimal distribution of a fixed amount $m>0$ of insulating material coating an insulated boundary $\Gamma_I\subseteq \partial\Omega$ of a thermally conducting body $\Omega\subseteq \mathbb{R}^d$, $d\in \mathbb{N}$. The thickness of the thin insulating layer $\Sigma_{I}^{\varepsilon}$ is given locally via $\varepsilon \mathtt{d}$, where $\mathtt{d}\colon \Gamma_{I}\to [0,+\infty)$ specifies the (to be determined) distribution of the insulating material. We establish $\Gamma(L^2(\mathbb{R}^d))$-convergence of the problem (as $\varepsilon\to 0^+$). Different from the existing literature, which predominantly assumes that the thermally conducting body $\Omega$ has a $C^{1,1}$-boundary, we merely assume that $\Gamma_I$ is piece-wise flat. To overcome this lack of boundary regularity, we define the thin insulating boundary layer $\Sigma_{I}^{\varepsilon}$ using a Lipschitz continuous transversal vector field rather than the outward unit normal vector field. The piece-wise flatness condition on $\Gamma_I$ is only needed to prove the $\liminf$-estimate. In fact, for the $\limsup$-estimate is enough that the thermally conducting body $\Omega$ has a $C^{0,1}$-boundary.