The spectral honeycomb conjecture
Let $\Omega \subset \mathbb{R}^2$ be a bounded connected Lipschitz domain of area $A$, and let $\lambda_1$ denote the first Dirichlet eigenvalue of $-\Delta$. For $k \ge 1$ let $$\mathcal{L}_k(\Omega) = \inf_{(D_1, \dots, D_k)} \max_{1 \le i \le k} \lambda_1(D_i),$$ the infimum taken over all $k$-tuples of pairwise disjoint nonempty connected open subsets of $\Omega$, which need not be convex. Let $H$ be a regular hexagon of area one. Keeping the cells of a hexagonal tiling that lie in $\Omega$ gives $\limsup_{k \to \infty} A \mathcal{L}_k(\Omega)/k \le \lambda_1(H)$, and the Faber–Krahn inequality gives $A \mathcal{L}_k(\Omega)/k \ge \lambda_1(B)$ for the disk $B$ of area one, a lower bound improved asymptotically by Bourgain and by Steinerberger. Bucur, Fragalà, Velichkov and Verzini proved honeycomb asymptotics of this kind for partitions into convex cells for functionals including the Cheeger constant, and reduced the case of $\lambda_1$ with convex cells to properties of $\lambda_1$ on convex pentagons, hexagons and heptagons.
Problem. Does $A \mathcal{L}_k(\Omega)/k$ converge to $\lambda_1(H)$ as $k \to \infty$?
Source: Van den Berg (2006, unpublished; see Bonnaillie-Noël and Helffer (2017), §9.1, the hexagonal conjecture); the analogue for the mean of the eigenvalues is mentioned by Caffarelli and Lin (2007).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, Dirichlet eigenvalues, honeycomb conjecture, spectral minimal partitions
References
@incollection{bonnaillienoel2017,
author = {Bonnaillie-No{\"e}l, Virginie and Helffer, Bernard},
title = {Nodal and spectral minimal partitions -- the state of the art in 2016},
booktitle = {Shape Optimization and Spectral Theory},
editor = {Henrot, Antoine},
publisher = {De Gruyter Open},
address = {Warsaw},
year = {2017},
pages = {353--397},
doi = {10.1515/9783110550887-010},
eprint = {1506.07249},
archivePrefix = {arXiv},
}
@article{caffarelli2007,
author = {Caffarelli, Luis A. and Lin, Fang Hua},
title = {An optimal partition problem for eigenvalues},
journal = {J. Sci. Comput.},
volume = {31},
number = {1-2},
year = {2007},
pages = {5--18},
doi = {10.1007/s10915-006-9114-8},
}
@article{bourgain2015,
author = {Bourgain, Jean},
title = {On {P}leijel's nodal domain theorem},
journal = {Int. Math. Res. Not. IMRN},
volume = {2015},
number = {6},
year = {2015},
pages = {1601--1612},
doi = {10.1093/imrn/rnt241},
eprint = {1308.4422},
archivePrefix = {arXiv},
}
@article{steinerberger2014,
author = {Steinerberger, Stefan},
title = {A geometric uncertainty principle with an application to {P}leijel's estimate},
journal = {Ann. Henri Poincar{\'e}},
volume = {15},
number = {12},
year = {2014},
pages = {2299--2319},
doi = {10.1007/s00023-013-0310-4},
}
@article{bucur2018,
author = {Bucur, Dorin and Fragal{\`a}, Ilaria and Velichkov, Bozhidar and Verzini, Gianmaria},
title = {On the honeycomb conjecture for a class of minimal convex partitions},
journal = {Trans. Amer. Math. Soc.},
volume = {370},
number = {10},
year = {2018},
pages = {7149--7179},
doi = {10.1090/tran/7326},
eprint = {1703.05383},
archivePrefix = {arXiv},
}
@misc{aim-spectral-honeycomb,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The spectral honeycomb conjecture}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/c9929805cece044705abaaebe1cebddac1ba11a6/problems/009-spectral-honeycomb.md},
note = {Entry 009 at commit c992980, 2026-10-05},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00272, "The spectral honeycomb conjecture", clopen.solutions.