A sharp universal inequality for clamped-plate eigenvalues
Let $d \ge 2$ and let $\Omega \subset \mathbb{R}^d$ be a bounded connected domain with smooth boundary. Let $0 < \Gamma_1 \le \Gamma_2 \le \cdots$ be the eigenvalues of the clamped plate problem $$\Delta^2 u = \Gamma u \ \text{ in } \Omega, \qquad u = \partial_\nu u = 0 \ \text{ on } \partial\Omega,$$ counted with multiplicity. As Chen, Cheng and Wei recall, the inequality below holds for every bounded domain with $8(d+2)/d^2$ in place of $8/d$; the constant $8/d$ is smaller.
Problem. Is $$\sum_{i=1}^{k} (\Gamma_{k+1} - \Gamma_i)^2 \le \frac{8}{d} \sum_{i=1}^{k} (\Gamma_{k+1} - \Gamma_i)\,\Gamma_i$$ for every $d \ge 2$, every such $\Omega$ and every integer $k \ge 1$?
Source: Q.-M. Cheng, open problem session of the 6th International Congress of Chinese Mathematicians (2013); Chen, Cheng and Wei (2016), Conjecture 1.1.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, biharmonic operators, clamped plates, universal inequalities
References
@misc{chen2016,
author = {Chen, Daguang and Cheng, Qing-Ming and Wei, Guoxin},
title = {A gap for eigenvalues of a clamped plate problem},
year = {2016},
eprint = {1610.05889},
archivePrefix = {arXiv},
}
@misc{aim-clamped-plate-universal-bound,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{A sharp universal inequality for clamped-plate eigenvalues}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/107-clamped-plate-universal-bound.md},
note = {Entry 107 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00334, "A sharp universal inequality for clamped-plate eigenvalues", clopen.solutions.