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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.05870 |
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Table of Contents:
- Let $Ω\subset \mathbb{R}^d$ 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 $k \ge 1$, we provide a quantitative control of the difference $|λ_k(Ω)-λ_k(Θ)|$ by the variation of the second eigenvalue $C(d,k)(λ_2(Ω)-λ_2(Θ))^α$, 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 $α_2 = 1/2$ and $0<α_d<1$ in higher dimensions. In the particular case where $λ_k(Ω)\ge λ_k(Θ)$, we can improve the inequality and find an estimate with the sharp exponent $α= 1/2$.