Shape optimization for a nonlinear elliptic problem related to thermal insulation

TL;DR

This paper investigates shape optimization problems for a nonlinear elliptic PDE related to thermal insulation, demonstrating that under certain constraints, the optimal shape is a disk in 2D and a ball in higher dimensions.

Contribution

It establishes the optimal shapes for a nonlinear elliptic problem under perimeter and convexity constraints, extending known results to more general settings.

Findings

In 2D, the disk maximizes the functional under perimeter constraints.

In higher dimensions, the ball maximizes the functional among convex sets.

The results connect geometric constraints with optimal thermal insulation configurations.

Abstract

In this paper we consider a minimization problem of the type $$ I_{\beta,p}(D;\Omega)=\inf\biggl\{\int_\Omega \lvert{D\phi}\rvert^pdx+\beta \int_{\partial^* \Omega}\lvert{\phi}\rvert^pd\mathcal{H}^{n-1},\; \phi \in W^{1,p}(\Omega),\;\phi \geq 1 \;\textrm{in}\;D\biggl\}, $$ 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.