*Published Paper*

**Inserted:** 5 mar 2013

**Last Updated:** 23 apr 2014

**Journal:** Appl. Math. Optim.

**Volume:** 69

**Number:** 2

**Pages:** 199--231

**Year:** 2014

**Abstract:**

In this paper we prove that the shape optimization problem

$\min\left\{\lambda_k(\Omega):\ \Omega\subset\mathbb{R}^d,\ \Omega\ \hbox{open},\ P(\Omega)=1,\

\Omega

<+\infty\right \},$

has a solution for any $k\in\mathbb{N}$ and dimension $d$. Moreover, every solution is a bounded connected open set with boundary which is $C^{1,\alpha}$ outside a closed set of Hausdorff dimension $d-8$. Our results are more general and apply to spectral functionals of the form $f(\lambda_{k_1}(\Omega),\dots,\lambda_{k_p}(\Omega))$, for increasing functions $f$ satisfying some suitable bi-Lipschitz type condition.

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