Sharp quantitative stability of the Dirichlet spectrum near the ball

Let be an open set with the same volume as the unit ball and let be the ‐th eigenvalue of the Laplace operator of with Dirichlet boundary conditions on . In this work, we answer the following question: If is small, how large can be? We establish quantitative bounds of the form with sharp exponents depending on the multiplicity of . We first show that such an inequality is valid with for any , 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 if is simple. We also obtain a similar result for the whole cluster of eigenvalues when 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.