On some rescaled shape optimization problems

We consider Cheeger-like shape optimization problems of the form $$\min \{\mathrm{\mid }\mathrm{\Omega }{\mathrm{\mid }}^{\alpha }J(\mathrm{\Omega }):\mathrm{\Omega }\subset D\}$$ where $D$ is a given bounded domain and $\alpha$ is above the natural scaling. We show the existence of a solution and analyze as $J(\mathrm{\Omega })$ the particular cases of the compliance functional $C(\mathrm{\Omega })$ and of the first eigenvalue ${\lambda }_1(\mathrm{\Omega })$ of the Dirichlet Laplacian. We prove that optimal sets are open and we obtain some necessary conditions of optimality.