Convection Effects and Optimal Insulation: Modelling and Analysis

In this paper, we study an insulation problem that seeks to determine the optimal distribution of a given amount $m>0$ of insulating material coating an insulated boundary part $\Gamma_I\subseteq \partial\Omega$ of a thermally conducting body $\Omega\subseteq \mathbb{R}^d$, $d\in \mathbb{N}$, subject to convective heat transfer. The `$\textit{thickness}$' of the insulating layer $\Sigma_{I}^{\varepsilon}\subseteq \mathbb{R}^d$ is given locally via $\varepsilon \mathtt{d}$, where $\varepsilon>0$ denotes the (arbitrarily small) conductivity and $\mathtt{d}\colon \Gamma_{I}\to [0,+\infty)$ the (to be determined) distribution of the insulating material. Then, the physical process is modelled by the stationary heat equation in the insulated thermally conducting body $\Omega_{I}^{\varepsilon}:= \Omega\cup\Sigma_{I}^{\varepsilon}$ with Robin-type boundary conditions on the interacting insulation boundary $\Gamma_I^{\varepsilon}\subseteq \partial\Omega_{I}^{\varepsilon}$ (reflecting convective heat transfer between the thermally conducting body $\Omega$ and its surrounding medium) as well as Dirichlet and Neumann boundary conditions at the remaining boundary parts, $\textit{i.e.}$, $\partial\Omega_{I}^{\varepsilon}\setminus \Gamma_I^{\varepsilon}$. More precisely, we establish $\Gamma(L^2(\mathbb{R}^d))$-convergence of the heat loss formulation (as ${\varepsilon \to 0^+}$), in the case that the thermally conducting body $\Omega$ is a bounded Lipschitz domain having a $C^{1,1}$-regular or piece-wise flat insulated boundary $\Gamma_I$.