Quantitative stability control of the full spectrum of the Dirichlet Laplacian by the second eigenvalue
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866914532911218688 |
|---|---|
| author | de Villeroché, Alexis |
| author_facet | de Villeroché, Alexis |
| 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 $α= 1/2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_05870 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantitative stability control of the full spectrum of the Dirichlet Laplacian by the second eigenvalue de Villeroché, Alexis Analysis of PDEs 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 $α= 1/2$. |
| title | Quantitative stability control of the full spectrum of the Dirichlet Laplacian by the second eigenvalue |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2506.05870 |