Quantitative stability control of the full spectrum of the Dirichlet Laplacian by the second eigenvalue

2025/06/06 by Alexis de Villeroché, de Villeroché, Alexis
Engineering · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2506.05870

openalex publication_date 2025/06/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Let Ω⊂ ℝ^d be an open set of finite measure and let Θ be a disjoint union of two balls of half measure. We study the stability of the full Dirichlet spectrum of Ω when its second eigenvalue is close to the second eigenvalue of Θ. Precisely, for every integer k ≥ 1, we provide a quantitative control of the difference |λ_k(Ω)-λ_k(Θ)| by the variation of the second eigenvalue C(d,k)(λ₂(Ω)-λ₂(Θ))^α, for a suitable exponent α and a positive constant C(d,k) depending only on the dimension of the space and the index k. We are able to find such an estimate for general k and arbitrary Ω with α=α_d/(d+1)² where α₂ = 1/2 and 0<α_d<1 in higher dimensions. In the particular case where λ_k(Ω)≥ λ_k(Θ), we can improve the inequality and find an estimate with the sharp exponent α= 1/2.

Related

• Sharp Quantitative Stability of the Dirichlet spectrum near the ball
• Some estimates for the higher eigenvalues of sets close to the ball
• Regularity of the optimal sets for the second Dirichlet eigenvalue
• Uniform stability of the ball with respect to the first Dirichlet and Neumann ∞-eigenvalues
• Spectral stability estimates for the Dirichlet and Neumann Laplacian in rough domains