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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.13053 |
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| _version_ | 1866915469552779264 |
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| author | Allen, Mark Kriventsov, Dennis Neumayer, Robin |
| author_facet | Allen, Mark Kriventsov, Dennis Neumayer, Robin |
| contents | For a bounded open set $Ω\subset \mathbb{R}^n$ with the same volume as the unit ball, the classical Faber-Krahn inequality says that the first Dirichlet eigenvalue $λ_1(Ω)$ of the Laplacian is at least that of the unit ball $B$. We prove that the deficit $λ_1(Ω)- λ_1(B)$ in the Faber-Krahn inequality controls the square of the distance between the resolvent operator $(-Δ_Ω)^{-1}$ for the Dirichlet Laplacian on $Ω$ and the resolvent operator on the nearest unit ball $B(x_Ω)$. The distance is measured by the operator norm from $L^{\infty}$ to $L^2$. As a main application, we show that the Faber-Krahn deficit $λ_1(Ω)- λ_1(B)$ controls the squared $L^2$ norm between $k$th eigenfunctions on $Ω$ and $B(x_Ω)$ for every $k \in \mathbb{N}.$ In both of these main theorems, the quadratic power is optimal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_13053 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantitative Resolvent and Eigenfunction Stability for the Faber-Krahn Inequality Allen, Mark Kriventsov, Dennis Neumayer, Robin Analysis of PDEs For a bounded open set $Ω\subset \mathbb{R}^n$ with the same volume as the unit ball, the classical Faber-Krahn inequality says that the first Dirichlet eigenvalue $λ_1(Ω)$ of the Laplacian is at least that of the unit ball $B$. We prove that the deficit $λ_1(Ω)- λ_1(B)$ in the Faber-Krahn inequality controls the square of the distance between the resolvent operator $(-Δ_Ω)^{-1}$ for the Dirichlet Laplacian on $Ω$ and the resolvent operator on the nearest unit ball $B(x_Ω)$. The distance is measured by the operator norm from $L^{\infty}$ to $L^2$. As a main application, we show that the Faber-Krahn deficit $λ_1(Ω)- λ_1(B)$ controls the squared $L^2$ norm between $k$th eigenfunctions on $Ω$ and $B(x_Ω)$ for every $k \in \mathbb{N}.$ In both of these main theorems, the quadratic power is optimal. |
| title | Quantitative Resolvent and Eigenfunction Stability for the Faber-Krahn Inequality |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2504.13053 |