On some rescaled shape optimization problems

TL;DR

This paper investigates rescaled Cheeger-like shape optimization problems, establishing existence, regularity, and optimality conditions for solutions involving compliance and Laplacian eigenvalues.

Contribution

It introduces a new class of rescaled shape optimization problems and provides existence, regularity, and optimality conditions for their solutions.

Findings

Existence of solutions for the rescaled shape optimization problems.

Optimal sets are proven to be open.

Necessary conditions of optimality are derived.

Abstract

We consider Cheeger-like shape optimization problems of the form $$\min\big\{|\Omega|^\alpha J(\Omega) : \Omega\subset D\big\}$$ where $D$ is a given bounded domain and $\alpha$ is above the natural scaling. We show the existence of a solution and analyze as $J(\Omega)$ the particular cases of the compliance functional $C(\Omega)$ and of the first eigenvalue $\lambda_1(\Omega)$ of the Dirichlet Laplacian. We prove that optimal sets are open and we obtain some necessary conditions of optimality.