Shape optimization for a nonlinear elliptic problem related to thermal insulation

Abstract

In this paper we consider a minimization problem of the type $$I_{\beta ,p}(D;\Omega )=\inf \{{\int }_{\Omega }|D\varphi {|}^{p}dx+\beta {\int }_{{\partial }^{\ast }\Omega }|\varphi {|}^{p}d{\mathcal{H}}^{n-1},\varphi \in W^{1,p}(\Omega ),\varphi \ge 1\text{in}D\},$$ where $\Omega$ is a bounded connected open set in ${\mathbb{R}}^n$, $D\subset \bar{\Omega }$ is a compact set and $\beta$ is a positive constant. We let the set $D$ vary under prescribed geometrical constraints and $\Omega \setminus D$ 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.

14 pages. arXiv admin note: text overlap with arXiv:2005.11934 by other authors

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