# On some rescaled shape optimization problems

created by buttazzo on 14 Feb 2009

[BibTeX]

Submitted Paper

Inserted: 14 feb 2009

Year: 2009

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.