Quantitative Kröger inequalities for Neumann eigenvalues of convex domains
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| Format: | Preprint |
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2026
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| _version_ | 1866918447131131904 |
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| author | Bucur, Dorin Gentile, Andrea Henrot, Antoine |
| author_facet | Bucur, Dorin Gentile, Andrea Henrot, Antoine |
| contents | Refining the sharp upper bounds $μ_{k,d}^* $ obtained by Kröger (1999) for the $k$-th Neumann eigenvalue of a convex domain $Ω\subset \mathbb{R}^d$, we prove the following inequalities: for any $k\in \mathbb{N}$ there exists a constant $C(k,d) >0$ such that
$$D_Ω^2 μ_k(Ω) \leq μ_{k,d}^* - C(k,d) a_2(Ω)^2/D_Ω^2$$
where $D_Ω$ is the diameter of $Ω$ and $a_2(Ω)$ is the second largest semiaxis of the John ellipsoid of $Ω$. In the planar case, for $k=1$ we also give an explicit value of the constant $C(1,2)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_13246 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quantitative Kröger inequalities for Neumann eigenvalues of convex domains Bucur, Dorin Gentile, Andrea Henrot, Antoine Analysis of PDEs Spectral Theory 35P15, 52A20 Refining the sharp upper bounds $μ_{k,d}^* $ obtained by Kröger (1999) for the $k$-th Neumann eigenvalue of a convex domain $Ω\subset \mathbb{R}^d$, we prove the following inequalities: for any $k\in \mathbb{N}$ there exists a constant $C(k,d) >0$ such that $$D_Ω^2 μ_k(Ω) \leq μ_{k,d}^* - C(k,d) a_2(Ω)^2/D_Ω^2$$ where $D_Ω$ is the diameter of $Ω$ and $a_2(Ω)$ is the second largest semiaxis of the John ellipsoid of $Ω$. In the planar case, for $k=1$ we also give an explicit value of the constant $C(1,2)$. |
| title | Quantitative Kröger inequalities for Neumann eigenvalues of convex domains |
| topic | Analysis of PDEs Spectral Theory 35P15, 52A20 |
| url | https://arxiv.org/abs/2604.13246 |