Rayleigh's clamped-plate conjecture in higher dimensions
For a bounded smooth domain $\Omega \subset \mathbb{R}^d$ let $$\Gamma_1(\Omega) = \inf_{0 \ne u \in H^2_0(\Omega)} \frac{\int_\Omega |\Delta u|^2 \, dx}{\int_\Omega |u|^2 \, dx},$$ the first eigenvalue of the clamped plate, $\Delta^2 u = \Gamma u$ in $\Omega$ with $u = \partial_\nu u = 0$ on $\partial\Omega$. Rayleigh conjectured that among plane plates of given area the disk has the least $\Gamma_1$. Following an approach of Talenti, Nadirashvili proved this in the plane, and Ashbaugh and Benguria proved it in dimensions two and three, both in 1995. Ashbaugh and Laugesen showed that this approach does not settle dimensions $d \ge 4$, and proved there that $\Gamma_1(\Omega) \ge c_d\, \Gamma_1(B)$ for the ball $B$ of the same volume, with explicit constants $c_d$ ($c_4 \approx 0.95$) tending to $1$ as $d \to \infty$. Leylekian proved that a minimizer of $\Gamma_1$ at fixed volume, if one exists, is a ball under additional hypotheses: if it has $C^4$ boundary and its first eigenfunction has no critical values in $\Omega$ other than its minimum and maximum (for $4 \le d \le 9$), or if its boundary is $C^4$ and connected and its first eigenfunction has minimal mean in a suitable sense or its Laplacian has constant normal derivative on the boundary.
Problem. For every integer $d \ge 4$ and every bounded smooth domain $\Omega \subset \mathbb{R}^d$, is $\Gamma_1(\Omega) \ge \Gamma_1(B)$ for a ball $B$ with $|B| = |\Omega|$?
Source: Lord Rayleigh (The Theory of Sound, 1877), for plane plates of given area; in d dimensions as stated by Ashbaugh and Benguria (1995) and Leylekian (2024). Stated for d ≥ 4 and smooth domains by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, biharmonic eigenvalues, clamped plates, isoperimetric inequalities
References
@book{rayleigh1877,
author = {{Lord Rayleigh}},
title = {The Theory of Sound},
volume = {1},
publisher = {Macmillan},
address = {London},
year = {1877},
note = {Second edition 1894},
}
@article{ashbaugh1995,
author = {Ashbaugh, Mark S. and Benguria, Rafael D.},
title = {On {R}ayleigh's conjecture for the clamped plate and its generalization to three dimensions},
journal = {Duke Math. J.},
volume = {78},
number = {1},
year = {1995},
pages = {1--17},
doi = {10.1215/S0012-7094-95-07801-6},
}
@article{leylekian2024arma,
author = {Leylekian, Rom{\'e}o},
title = {Towards the optimality of the ball for the {R}ayleigh conjecture concerning the clamped plate},
journal = {Arch. Ration. Mech. Anal.},
volume = {248},
number = {2},
year = {2024},
pages = {28},
doi = {10.1007/s00205-024-01972-2},
eprint = {2306.15523},
archivePrefix = {arXiv},
}
@article{leylekian2024annali,
author = {Leylekian, Rom{\'e}o},
title = {Sufficient conditions yielding the {R}ayleigh conjecture for the clamped plate},
journal = {Ann. Mat. Pura Appl. (4)},
volume = {203},
number = {6},
year = {2024},
pages = {2529--2547},
doi = {10.1007/s10231-024-01454-y},
eprint = {2302.06313},
archivePrefix = {arXiv},
}
@article{talenti1981,
author = {Talenti, Giorgio},
title = {On the first eigenvalue of the clamped plate},
journal = {Ann. Mat. Pura Appl. (4)},
volume = {129},
year = {1981},
pages = {265--280},
doi = {10.1007/BF01762146},
}
@article{nadirashvili1995,
author = {Nadirashvili, Nikolai S.},
title = {Rayleigh's conjecture on the principal frequency of the clamped plate},
journal = {Arch. Rational Mech. Anal.},
volume = {129},
number = {1},
year = {1995},
pages = {1--10},
doi = {10.1007/BF00375124},
}
@article{ashbaugh1996,
author = {Ashbaugh, Mark S. and Laugesen, Richard S.},
title = {Fundamental tones and buckling loads of clamped plates},
journal = {Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)},
volume = {23},
number = {2},
year = {1996},
pages = {383--402},
url = {http://www.numdam.org/item/ASNSP_1996_4_23_2_383_0/},
}
@misc{aim-rayleigh-clamped-plate,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Rayleigh's clamped-plate conjecture in higher dimensions}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/006-rayleigh-clamped-plate.md},
note = {Entry 006 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-00303, "Rayleigh's clamped-plate conjecture in higher dimensions", clopen.solutions.