Boundedness of minimizers for spectral problems in ${\mathbb{R}}^N$

Abstract

In [8] it was proved 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 this paper we show that every minimizer is uniformly bounded by a constant depending only on $k,N$.