An optimal insulation problem

In this paper we consider a minimization problem which arises from thermal insulation. A compact connected set $K$, which represents a conductor of constant temperature, say $1$, is thermally insulated by surrounding it with a layer of thermal insulator, the open set $\Omega\setminus K$ with $K\subset\bar\Omega$. The heat dispersion is then obtained as $$\inf\left\{ \int_{\Omega}|\nabla \varphi|^{2}dx +\beta\int_{\partial^{*}\Omega}\varphi^{2}d\mathcal H^{n-1} ,\;\varphi\in H^{1}(\mathbb R^{n}), \, \varphi\ge 1\text{ in } K\right\},$$ for some positive constant $\beta$. We mostly restrict our analysis to the case of an insulating layer of constant thickness. We let the set $K$ vary, under prescribed geometrical constraints, and we look for the best (or worst) geometry in terms of heat dispersion. We show that under perimeter constraint the disk in two dimensions is the worst one. The same is true for the ball in higher dimension but under different constraints. We finally discuss few open problems.