*Preprint*

**Inserted:** 10 may 2017

**Last Updated:** 10 may 2017

**Year:** 2017

**Abstract:**

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 we prove that, for all $k\in\N$, there is a positive constant $C=C(k,N)$ such that for every open set $\Omega\subseteq \R^N$ with unit measure and with $\lambda_1(\Omega)$ not excessively large one has
\[

\lambda_k(\Omega)-\lambda_k(B)

\leq C (\lambda_1(\Omega)-\lambda_1(B))^\beta\,, \qquad \lambda_k(B)-\lambda_k(\Omega)\leq Cd(\Omega)^{\beta'}\,,
\]
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.

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