Shape optimization for a nonlinear elliptic problem related to thermal insulation

Abstract

In this paper, we consider a minimization problem of a nonlinear functional $I_{\beta ,p}(D,\Omega )$ related to a thermal insulation problem with a convection term, where $\Omega$ is a bounded connected open set in ${\mathbb{R}}^n$ and $D\subset \overline{\Omega }$ is a compact set. The Euler–Lagrange equation relative to $I_{\beta ,p}$ is a $p$-Laplace equation, $1<p<\infty$, with a Robin boundary condition with parameter $\beta >0$. The main aim is to study extremum problems for $I_{\beta ,p}(D,\Omega )$, among domains $D$ with given geometrical constraints and $\Omega \setminus D$ of fixed thickness. In the planar case, we show that under perimeter constraint the disk maximizes $I_{\beta ,p}$. 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.