Shape optimization for a nonlinear elliptic problem related to thermal insulation

In this paper we consider a minimization problem of the type $$I_{\beta ,p}(D;\mathrm{\Omega })=\inf \{{\int }_{\mathrm{\Omega }}\mid D\phi {\mid }^{p}dx+\beta {\int }_{{\mathrm{\partial }}^{\ast }\mathrm{\Omega }}\mid \phi {\mid }^{p}d{\mathcal{H}}^{n-1},\phi \in W^{1,p}(\mathrm{\Omega }),\phi \ge 1\text{in}D\},$$ where $\mathrm{\Omega }$ is a bounded connected open set in ${\mathbb{R}}^n$, $D\subset \bar{\mathrm{\Omega }}$ is a compact set and $\beta$ is a positive constant. We let the set $D$ vary under prescribed geometrical constraints and $\mathrm{\Omega }\setminus D$ of fixed thickness, in order to look for the best (or worst) geometry in terms of minimization (or maximization) of $I_{\beta ,p}$. In the planar case, we show that under perimeter constraint the disk maximize $I_{\beta ,p}$. In the $n$-dimensional case we restrict our analysis to convex sets showing that the same is true for the ball but under different geometrical constraints.