preprint

Inserted: 28 apr 2023
Last Updated: 22 dec 2025

Year: 2023

Abstract:

Let $Ω\subset\mathbb{R}^n$ be an open set with the same volume as the unit ball $B$ and let $λ_k(Ω)$ be the $k$-th eigenvalue of the Laplace operator of $Ω$ with Dirichlet boundary conditions on $\partialΩ$. In this work, we answer the following question: if $λ_1(Ω)-λ_1(B)$ is small, how large can $
λ_k(Ω)-λ_k(B)
$ be ? We establish quantitative bounds of the form $
λ_k(Ω)-λ_k(B)
\le C (λ_1(Ω)-λ_1(B))^α$ with sharp exponents $α$ depending on the multiplicity of $λ_k(B)$. We first show that such an inequality is valid with $α=1/2$ 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 $α=1$ if $λ_{k}(B)$ is simple. We also obtain a similar result for the whole cluster of eigenvalues when $λ_{k}(B)$ 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 the 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.