The sharp planar Pleijel constant
Let $\Omega \subset \mathbb{R}^2$ be a bounded connected domain with piecewise smooth Lipschitz boundary, and let $(u_k)$ be a real orthonormal basis of Dirichlet eigenfunctions of $-\Delta$ on $\Omega$, ordered by nondecreasing eigenvalue counted with multiplicity. Let $\nu(u_k)$ be the number of nodal domains of $u_k$, the connected components of $\Omega \setminus \{u_k = 0\}$. Courant's theorem gives $\nu(u_k) \le k$, and Pleijel proved $$\limsup_{k \to \infty} \frac{\nu(u_k)}{k} \le \frac{4}{j_{0,1}^2} \approx 0.692,$$ where $j_{0,1}$ is the first positive zero of the Bessel function $J_0$. Bourgain and Steinerberger improved this constant slightly. For a rectangle whose squared side ratio is irrational, the eigenvalues are simple and the limit superior equals $2/\pi \approx 0.637$ (Helffer and Hoffmann-Ostenhof).
Problem. Is $$\limsup_{k \to \infty} \frac{\nu(u_k)}{k} \le \frac{2}{\pi}$$ for every such domain $\Omega$ and every such eigenbasis $(u_k)$?
Source: Polterovich (2009), Remark 2.5, for regular bounded planar domains with Dirichlet or Neumann conditions; stated for the Dirichlet problem by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, Dirichlet eigenfunctions, nodal domains, Pleijel's theorem
References
@article{polterovich2009,
author = {Polterovich, Iosif},
title = {Pleijel's nodal domain theorem for free membranes},
journal = {Proc. Amer. Math. Soc.},
volume = {137},
number = {3},
year = {2009},
pages = {1021--1024},
doi = {10.1090/S0002-9939-08-09596-8},
eprint = {0805.1553},
archivePrefix = {arXiv},
}
@article{pleijel1956,
author = {Pleijel, {\AA}ke},
title = {Remarks on {C}ourant's nodal line theorem},
journal = {Comm. Pure Appl. Math.},
volume = {9},
number = {3},
year = {1956},
pages = {543--550},
doi = {10.1002/cpa.3160090324},
}
@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},
eprint = {1306.3103},
archivePrefix = {arXiv},
}
@incollection{helffer2015,
author = {Helffer, Bernard and Hoffmann-Ostenhof, Thomas},
title = {A review on large $k$ minimal spectral $k$-partitions and {P}leijel's theorem},
booktitle = {Spectral Theory and Partial Differential Equations},
series = {Contemp. Math.},
volume = {640},
publisher = {Amer. Math. Soc.},
address = {Providence, RI},
year = {2015},
pages = {39--57},
doi = {10.1090/conm/640/12841},
eprint = {1509.04501},
archivePrefix = {arXiv},
}
@misc{aim-sharp-pleijel-constant,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The sharp planar Pleijel constant}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/012-sharp-pleijel-constant.md},
note = {Entry 012 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-00307, "The sharp planar Pleijel constant", clopen.solutions.