Sharpness of the logarithmic factor in the higher-order Cheeger–Steklov inequality
Let $\Omega$ be a compact Riemannian manifold with boundary, with Steklov eigenvalues $0 = \sigma_0 < \sigma_1 \le \sigma_2 \le \dots$, and let $i_k(\Omega)$ be the $k$-th Cheeger–Steklov constant of Hassannezhad and Miclo, an infimum over $k$-tuples of disjoint domains. They proved that there are universal constants $C, C' > 0$ such that, for every $k \ge 1$, $$\sigma_k(\Omega) \ge \frac{C}{k^6}\, i_k(\Omega) \qquad \text{and} \qquad \sigma_{2k+1}(\Omega) \ge \frac{C'}{\log^2(k+2)}\, i_{k+1}(\Omega).$$ In comparable settings, for the combinatorial Laplacian (Lee, Oveis Gharan and Trevisan) and for Markov operators (Miclo), the corresponding logarithmic factor is known to be sharp.
Problem. Is the coefficient $C'/\log^2(k+2)$ of $i_{k+1}(\Omega)$ in the second inequality sharp?
Source: A. Hassannezhad and L. Miclo, Higher order Cheeger inequalities for Steklov eigenvalues, Ann. Sci. Éc. Norm. Supér. 53 (2020), Remark 1. Listed as Open Question 4.7 in Colbois–Girouard–Gordon–Sher (2024).
The statement is adapted from doi.org/10.1007/s13163-023-00480-3, whose text is available under CC BY 4.0.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- DG
- Keywords
- Cheeger inequality, Colbois–Girouard–Gordon–Sher Steklov problems, Steklov eigenvalues
References
@article{hassannezhad2020,
author = {Hassannezhad, Asma and Miclo, Laurent},
title = {Higher order {C}heeger inequalities for {S}teklov eigenvalues},
journal = {Annales Scientifiques de l'{\'E}cole Normale Sup{\'e}rieure (4)},
volume = {53},
number = {1},
pages = {43--88},
year = {2020},
doi = {10.24033/asens.2417},
eprint = {1705.08643},
archivePrefix = {arXiv}
}
@article{colbois2024,
author = {Colbois, Bruno and Girouard, Alexandre and Gordon, Carolyn and Sher, David},
title = {Some recent developments on the {S}teklov eigenvalue problem},
journal = {Revista Matem{\'a}tica Complutense},
volume = {37},
number = {1},
pages = {1--161},
year = {2024},
doi = {10.1007/s13163-023-00480-3},
eprint = {2212.12528},
archivePrefix = {arXiv},
note = {Open Question 4.7}
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01008, "Sharpness of the logarithmic factor in the higher-order Cheeger–Steklov inequality", clopen.solutions.