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 $\mathbb{R}^N$ 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*} |\lambda_k(\Omega)-\lambda_k(B)|\leq C (\lambda_1(\Omega)-\lambda_1(B))^\beta\,, && \lambda_k(B)-\lambda_k(\Omega)\leq Cd(\Omega)^{\beta'}\,, \end{align*} where $d(\Omega)$ is the Fraenkel asymmetry of $\Omega$, and where $\beta$ and $\beta'$ are explicit exponents, not depending on $k$ nor on $N$; for the special case $N=2$, a better estimate holds.