Existence of minimizers for spectral problems

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 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 R^N.