Work

Sharp quantitative stability of the Dirichlet spectrum near the ball

Year: 2025

Type: article

Abstract: 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.

Source: Communications on Pure and Applied Mathematics

DOI: 10.1002/cpa.70021

Authors: Dorin Bucur, Jimmy Lamboley, Mickaël Nahon, Raphaël Prunier