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 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 \begin{align*} |λ_k(Ω)-λ_k(B)|\leq C (λ_1(Ω)-λ_1(B))^β\,, && λ_k(B)-λ_k(Ω)\leq Cd(Ω)^{β'}\,, \end{align*} 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.

13 pages