Some estimates for the higher eigenvalues of sets close to the ball

Abstract

In this paper we investigate the behavior of the eigenvalues of the Dirichlet Laplacian on sets in ${\mathbb{R}}^N$ whose first eigenvalue is close to the one of the ball with the same volume. In particular in our main Theorem 1.1 we prove that, for all $k\in \mathbb{N}$, there is a positive constant $C=C(k,N)$ such that for every open set $\Omega \subseteq {\mathbb{R}}^N$ with unit measure and with ${\lambda }_1(\Omega )$ not excessively large one has

where $d(\Omega )$ is the Fraenkel asymmetry of $\Omega$, and where $\beta$ and ${\beta }^{\prime }$ are explicit exponents, not depending on $k$ nor on $N$; for the special case $N=2$, a better estimate holds.