*Published Paper*

**Inserted:** 5 mar 2013

**Last Updated:** 30 oct 2017

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

**Volume:** 69

**Number:** 2

**Pages:** 199--231

**Year:** 2014

**Abstract:**

In this paper we prove that the shape optimisation problem'

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

\Omega

<+\infty\Big\},
\]
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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