*preprint*

**Inserted:** 28 apr 2023

**Year:** 2023

**Abstract:**

Let $\Omega\subset\mathbb{R}^n$ be an open set with same volume as the unit
ball $B$ and let $\lambda_k(\Omega)$ be the $k$-th eigenvalue of the Laplace
operator of $\Omega$ with Dirichlet boundary conditions in $\partial\Omega$. In
this work, we answer the following question: if
$\lambda_1(\Omega)-\lambda_1(B)$ is small, how large can
$

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

$ be ?
We establish quantitative bounds of the form
$

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

\le C (\lambda_1(\Omega)-\lambda_1(B))^\alpha$
with sharp exponents $\alpha$ depending on the multiplicity of $\lambda_k(B)$.
We first show that such an inequality is valid with $\alpha=1/2$ for any $k$,
improving previous known results and providing the sharpest possible exponent.
Then, through the study of a vectorial free boundary problem, we show that one
can achieve the better exponent $\alpha=1$ if $\lambda_{k}(B)$ is simple. We
also obtain a similar result for the whole cluster of eigenvalues when
$\lambda_{k}(B)$ is multiple, thus providing a complete answer to the question
above. As a consequence of these results, we obtain the persistence of the ball
as minimizer for a large class of spectral functionals which are small
perturbations of the fundamental eigenvalue on the one hand, and a full reverse
Kohler-Jobin inequality on the other hand, solving an open problem formulated
by M. Van Den Berg, G. Buttazzo and A. Pratelli.