Sharp Quantitative Stability of the Dirichlet spectrum near the ball

Abstract

Let $\Omega \subset {\mathbb{R}}^n$ be an open set with the 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 on $\partial \Omega$. 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 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.