The polygonal Payne–Pólya–Weinberger ratio conjecture
For a bounded planar domain $D$ let $\lambda_1(D) \le \lambda_2(D) \le \cdots$ be the Dirichlet eigenvalues of $-\Delta$ on $D$, counted with multiplicity. The ratio $\lambda_2/\lambda_1$ is invariant under dilations. Payne, Pólya and Weinberger proved $\lambda_2/\lambda_1 \le 3$ for every bounded planar domain, and Ashbaugh and Benguria proved that the disk maximizes the ratio. Among triangles the equilateral triangle maximizes it: Siudeja proved this for acute triangles, and Arbon, Mannan, Psenka and Ragavan for all triangles. Arbon proved that the square is a strict local maximizer of the ratio among quadrilaterals.
Problem. Let $R_n$ be a regular $n$-gon. Is $$\frac{\lambda_2(P)}{\lambda_1(P)} \le \frac{\lambda_2(R_n)}{\lambda_1(R_n)}$$ for every integer $n \ge 4$ and every simple bounded planar $n$-gon $P$?
Source: Antunes and Freitas (2008), from a numerical study (see Arbon (2022)); stated as Conjecture 6.31 by Laugesen and Siudeja (2017), whose case n = 3 was proved by Arbon, Mannan, Psenka and Ragavan (2022). Stated for n ≥ 4 by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, Dirichlet eigenvalues, eigenvalue ratios, polygons
References
@article{antunes2008,
author = {Antunes, Pedro and Freitas, Pedro},
title = {A numerical study of the spectral gap},
journal = {J. Phys. A},
volume = {41},
number = {5},
year = {2008},
pages = {055201},
doi = {10.1088/1751-8113/41/5/055201},
}
@incollection{laugesen2017,
author = {Laugesen, Richard S. and Siudeja, Bart{\l}omiej A.},
title = {Triangles and other special domains},
booktitle = {Shape Optimization and Spectral Theory},
editor = {Henrot, Antoine},
publisher = {De Gruyter Open},
address = {Warsaw},
year = {2017},
pages = {149--200},
doi = {10.1515/9783110550887-006},
}
@article{arbon2022triangles,
author = {Arbon, Ryan and Mannan, Mohammed and Psenka, Michael and Ragavan, Seyoon},
title = {A proof of the triangular {A}shbaugh--{B}enguria--{P}ayne--{P}{\'o}lya--{W}einberger inequality},
journal = {J. Spectr. Theory},
volume = {12},
number = {2},
year = {2022},
pages = {515--533},
doi = {10.4171/JST/409},
eprint = {2009.00927},
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},
number = {1-4},
year = {1956},
pages = {289--298},
doi = {10.1002/sapm1956351289},
}
@article{ashbaugh1992,
author = {Ashbaugh, Mark S. and Benguria, Rafael D.},
title = {A sharp bound for the ratio of the first two eigenvalues of {D}irichlet {L}aplacians and extensions},
journal = {Ann. of Math. (2)},
volume = {135},
number = {3},
year = {1992},
pages = {601--628},
doi = {10.2307/2946578},
}
@article{siudeja2010,
author = {Siudeja, Bart{\l}omiej},
title = {Isoperimetric inequalities for eigenvalues of triangles},
journal = {Indiana Univ. Math. J.},
volume = {59},
number = {3},
year = {2010},
pages = {1097--1120},
doi = {10.1512/iumj.2010.59.3744},
eprint = {0707.3631},
archivePrefix = {arXiv},
}
@misc{arbon2022quadrilaterals,
author = {Arbon, Ryan},
title = {Global and local bounds on the fundamental ratio of triangles and quadrilaterals},
year = {2022},
eprint = {2207.05814},
archivePrefix = {arXiv},
}
@misc{aim-polygonal-ppw,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The polygonal Payne–Pólya–Weinberger ratio conjecture}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/019-polygonal-ppw.md},
note = {Entry 019 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-00297, "The polygonal Payne–Pólya–Weinberger ratio conjecture", clopen.solutions.