*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.

**Keywords:**
shape optimization, spectral optimization, Dirichlet eigenvalues

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