2023/04/21 by Dorin Bucur, Bucur, Dorin, Jimmy Lamboley +5 · 2 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Spectral Theory in Mathematical Physics #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2304.10916
openalex publication_date 2023/04/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Ω⊂ℝn be an open set with the same volume as the unit ball B and let λk(Ω) be the k-th eigenvalue of the Laplace operator of Ω with Dirichlet boundary conditions on ∂Ω. In this work, we answer the following question: if λ1(Ω)-λ1(B) is small, how large can |λk(Ω)-λk(B)| be ? We establish quantitative bounds of the form |λk(Ω)-λk(B)|≤ C (λ1(Ω)-λ1(B))α with sharp exponents α depending on the multiplicity of λk(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 λk(B) is simple. We also obtain a similar result for the whole cluster of eigenvalues when λk(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.