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

In this paper we investigate the behavior of the eigenvalues of the Dirichlet Laplacian on sets in $R^N$ whose first eigenvalue is close to the one of the ball with the same volume. In particular in our main Theorem~ ef{finalmain} we prove that, for all $kinN$, there is a positive constant $C=C(k,N)$ such that for every open set $Omegasubseteq R^N$ with unit measure and with $lambda_1(Omega)$ not excessively large one has egin{align*} |lambda_k(Omega)-lambda_k(B)|leq C (lambda_1(Omega)-lambda_1(B))^eta,, && lambda_k(B)-lambda_k(Omega)leq Cd(Omega)^{eta'},, end{align*} where $d(Omega)$ is the Fraenkel asymmetry of $Omega$, and where $eta$ and $eta'$ are explicit exponents, not depending on $k$ nor on $N$; for the special case $N=2$, a better estimate holds.