Pólya–Szegő conjecture for the first buckling load
For a bounded planar domain $\Omega$ with smooth boundary, let $$\Lambda_1(\Omega) = \inf_{0 \ne u \in H^2_0(\Omega)} \frac{\int_\Omega |\Delta u|^2 \, dx}{\int_\Omega |\nabla u|^2 \, dx}$$ be its first clamped buckling eigenvalue, the least $\Lambda$ for which $\Delta^2 u = -\Lambda \Delta u$ in $\Omega$, $u = \partial_\nu u = 0$ on $\partial\Omega$, has a nonzero solution. Szegő proved the inequality below under the assumption that some first eigenfunction does not change sign, and Antunes exhibited an analytic convex domain whose first eigenfunction changes sign, by a computation with the method of fundamental solutions. Bramble and Payne proved $\Lambda_1(\Omega) > 2\pi j_{0,1}^2 / |\Omega|$, where $j_{0,1}$ is the first positive zero of $J_0$, and Ashbaugh and Laugesen extended this to a bound in $\mathbb{R}^d$ whose ratio to the conjectured one tends to $1$ as $d \to \infty$. Ashbaugh and Bucur proved that a minimiser exists among simply connected open sets of given area, and Stollenwerk and Wagner that a smooth domain minimising $\Lambda_1$ among smooth domains of given volume with connected boundary must be a ball.
Problem. If $B$ is a disk of the same area as $\Omega$, is $\Lambda_1(\Omega) \ge \Lambda_1(B)$ for every such $\Omega$? No sign condition is imposed on a first eigenfunction.
Source: Pólya and Szegő (1951); see Ashbaugh (1999), §2.5, and Antunes (2011), §1. Stated for 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, isoperimetric inequalities
References
@book{polya1951,
author = {P{\'o}lya, G. and Szeg{\H{o}}, G.},
title = {Isoperimetric Inequalities in Mathematical Physics},
series = {Annals of Mathematics Studies},
number = {27},
publisher = {Princeton University Press},
address = {Princeton, NJ},
year = {1951},
doi = {10.1515/9781400882663},
}
@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},
}
@article{antunes2011,
author = {Antunes, Pedro R. S.},
title = {On the buckling eigenvalue problem},
journal = {J. Phys. A},
volume = {44},
number = {21},
year = {2011},
pages = {215205},
doi = {10.1088/1751-8113/44/21/215205},
}
@article{szego1950,
author = {Szeg{\H{o}}, G.},
title = {On membranes and plates},
journal = {Proc. Nat. Acad. Sci. U.S.A.},
volume = {36},
number = {3},
year = {1950},
pages = {210--216},
doi = {10.1073/pnas.36.3.210},
}
@article{bramble1963,
author = {Bramble, J. H. and Payne, L. E.},
title = {Pointwise bounds in the first biharmonic boundary value problem},
journal = {J. Math. and Phys.},
volume = {42},
year = {1963},
pages = {278--286},
doi = {10.1002/sapm1963421278},
}
@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/},
}
@article{ashbaugh2003,
author = {Ashbaugh, Mark S. and Bucur, Dorin},
title = {On the isoperimetric inequality for the buckling of a clamped plate},
journal = {Z. Angew. Math. Phys.},
volume = {54},
number = {5},
year = {2003},
pages = {756--770},
doi = {10.1007/s00033-003-3204-3},
}
@article{stollenwerk2015,
author = {Stollenwerk, Kathrin and Wagner, Alfred},
title = {Optimality conditions for the buckling of a clamped plate},
journal = {J. Math. Anal. Appl.},
volume = {432},
number = {1},
year = {2015},
pages = {254--273},
doi = {10.1016/j.jmaa.2015.06.035},
eprint = {1408.6982},
archivePrefix = {arXiv},
}
@misc{aim-buckling-load-ball,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Pólya–Szegő conjecture for the first buckling load}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/007-buckling-load-ball.md},
note = {Entry 007 at commit a153f46, 2026-10-13},
}
Resolution claims
We prove the planar Pólya–Szegő conjecture for the first clamped buckling eigenvalue. Our main result establishes the sharp comparison Λ₁(Ω) ≥ (j₁,₁²/j₀,₁²) λ₁ᴰ(Ω) for every planar open set Ω of finite positive area, where Λ₁ denotes the first buckling eigenvalue and λ₁ᴰ the first Dirichlet eigenvalue. The comparison constant is optimal and is attained on disks. Combined with the Faber–Krahn …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Open → Solution claimed | Quanyu Tang | Proof of the Planar Pólya–Szegő Buckling Conjecture v1 announced. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00281, "Pólya–Szegő conjecture for the first buckling load", clopen.solutions.