*Published Paper*

**Inserted:** 1 dec 2011

**Last Updated:** 16 feb 2015

**Journal:** J. Math. Pures Appl.

**Year:** 2013

**Abstract:**

In this paper we show that any increasing functional of the first $k$ eigenvalues of the Dirichlet Laplacian admits a (quasi-)open minimizer among the subsets of ${\mathbb R}^N$ of unit measure. In particular, there exists such a minimizer which is bounded, where the bound depends on $k$ and $N$, but not on the functional. In the meantime, we show that the ratio $\lambda_k(\Omega)/\lambda_1(\Omega)$ is uniformly bounded for sets $\Omega\in{\mathbb R}^N$.

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