The sharp ratio of the first two buckling eigenvalues
Let $d \ge 2$ and let $\Omega \subset \mathbb{R}^d$ be a bounded connected domain with smooth boundary. Let $0 < \Lambda_1 \le \Lambda_2 \le \cdots$ be the eigenvalues, counted with multiplicity, of the clamped buckling problem $$\Delta^2 u = -\Lambda \Delta u \ \text{ in } \Omega, \qquad u = \partial_\nu u = 0 \ \text{ on } \partial\Omega,$$ and let $j_{a,1}$ be the first positive zero of the Bessel function $J_a$. For a ball, $\Lambda_2 / \Lambda_1 = j_{d/2+1,1}^2 / j_{d/2,1}^2$, about $1.796$ for the disk. Payne, Pólya and Weinberger proved $\Lambda_2 / \Lambda_1 < 3$ for planar domains, and Hile and Yeh improved this to $\Lambda_2 / \Lambda_1 \le (d^2 + 8d + 20)/(d + 2)^2$ in $\mathbb{R}^d$, which is $2.5$ in the plane.
Problem. Is $$\frac{\Lambda_2(\Omega)}{\Lambda_1(\Omega)} \le \frac{j_{d/2+1,1}^2}{j_{d/2,1}^2}$$ for every $d \ge 2$ and every such $\Omega$?
Source: Ashbaugh (1999), §4, (4.8), by analogy with the Payne–Pólya–Weinberger conjecture for membranes; see Ashbaugh, Gesztesy, Mitrea, Shterenberg and Teschl (2013), Remark 8.10. Stated for bounded connected 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, buckling eigenvalues, clamped plates, eigenvalue ratios
References
@incollection{ashbaugh1999,
author = {Ashbaugh, Mark S.},
title = {Isoperimetric and universal inequalities for eigenvalues},
booktitle = {Spectral Theory and Geometry},
editor = {Davies, E. B. and Safarov, Yu.},
series = {London Math. Soc. Lecture Note Ser.},
volume = {273},
publisher = {Cambridge University Press},
address = {Cambridge},
year = {1999},
pages = {95--139},
doi = {10.1017/CBO9780511566165.007},
eprint = {math/0008087},
archivePrefix = {arXiv},
}
@incollection{ashbaugh2013,
author = {Ashbaugh, Mark S. and Gesztesy, Fritz and Mitrea, Marius and Shterenberg, Roman and Teschl, Gerald},
title = {A survey on the {K}rein--von {N}eumann extension, the corresponding abstract buckling problem, and {W}eyl-type spectral asymptotics for perturbed {K}rein {L}aplacians in nonsmooth domains},
booktitle = {Mathematical Physics, Spectral Theory and Stochastic Analysis},
editor = {Demuth, Michael and Kirsch, Werner},
series = {Oper. Theory Adv. Appl.},
volume = {232},
publisher = {Birkh{\"a}user},
address = {Basel},
year = {2013},
pages = {1--106},
doi = {10.1007/978-3-0348-0591-9_1},
eprint = {1203.5713},
archivePrefix = {arXiv},
}
@article{payne1956,
author = {Payne, L. E. and P{\'o}lya, G. and Weinberger, H. F.},
title = {On the ratio of consecutive eigenvalues},
journal = {J. Math. and Phys.},
volume = {35},
year = {1956},
pages = {289--298},
doi = {10.1002/sapm1956351289},
}
@article{hile1984,
author = {Hile, G. N. and Yeh, R. Z.},
title = {Inequalities for eigenvalues of the biharmonic operator},
journal = {Pacific J. Math.},
volume = {112},
number = {1},
year = {1984},
pages = {115--133},
doi = {10.2140/pjm.1984.112.115},
}
@misc{aim-buckling-eigenvalue-ratio,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The sharp ratio of the first two buckling eigenvalues}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/008-buckling-eigenvalue-ratio.md},
note = {Entry 008 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-00280, "The sharp ratio of the first two buckling eigenvalues", clopen.solutions.