Calculus of Variations and Geometric Measure Theory

D. Mazzoleni - A. Pratelli

Existence of minimizers for spectral problems

created by pratelli on 01 Dec 2011
modified on 16 Feb 2015


Published Paper

Inserted: 1 dec 2011
Last Updated: 16 feb 2015

Journal: J. Math. Pures Appl.
Year: 2013


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 $\Omega\in{\mathbb R}^N$.