Does the disk minimize the third Dirichlet eigenvalue?
For a bounded open set $\Omega \subset \mathbb{R}^2$ let $\lambda_1(\Omega) \le \lambda_2(\Omega) \le \lambda_3(\Omega) \le \cdots$ be the Dirichlet eigenvalues of $-\Delta$ on $\Omega$, counted with multiplicity, and let $j_{1,1}$ be the first positive zero of the Bessel function $J_1$. For the disk $B$ of area $A$, $\lambda_2(B) = \lambda_3(B)$. Faber and Krahn proved that among open sets of given area the disk minimizes $\lambda_1$, and Krahn proved that $\lambda_2$ is minimized by the union of two disjoint disks of equal area, a result sometimes attributed to Szegő. Berger proved that in the plane, apart from the first four, no Dirichlet eigenvalue can be minimized by disks or disjoint unions of disks. Oudet's numerical minimization of $\lambda_3$ (2004) produced the disk.
Problem. Let $A > 0$ and let $B$ be the disk of area $A$. Is $$\lambda_3(\Omega) \ge \lambda_3(B) = \frac{\pi j_{1,1}^2}{A}$$ for every bounded open set $\Omega \subset \mathbb{R}^2$ with $|\Omega| = A$? Disconnected $\Omega$ are allowed.
Source: Attributed to Szegő by Oudet (2004), citing Wolf and Keller (1994); see Henrot (2006, p. 82) and Laugesen (2020), Open Problem 1, which asks, for bounded domains in R^n, whether the ball of the same volume minimizes λ_{n+1}. Stated by AIM — Open Applied Problems for bounded open sets in the plane.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, Dirichlet eigenvalues, spectral shape optimization
References
@misc{laugesen2020,
author = {Laugesen, Richard S.},
title = {Open problems from the miniconference on sharp eigenvalue estimates for partial differential operators},
howpublished = {Problem list, miniconference ``Sharp eigenvalue estimates for partial differential operators'', University of Illinois},
month = apr,
year = {2020},
url = {https://publish.illinois.edu/eigenvalues2020/files/2020/04/Open-problems.pdf},
note = {Open Problem 1, proposed by R. S. Laugesen},
}
@article{oudet2004,
author = {Oudet, {\'E}douard},
title = {Numerical minimization of eigenmodes of a membrane with respect to the domain},
journal = {ESAIM Control Optim. Calc. Var.},
volume = {10},
number = {3},
year = {2004},
pages = {315--330},
doi = {10.1051/cocv:2004011},
}
@article{wolf1994,
author = {Wolf, Sven Andreas and Keller, Joseph B.},
title = {Range of the first two eigenvalues of the {L}aplacian},
journal = {Proc. R. Soc. Lond. A},
volume = {447},
number = {1930},
year = {1994},
pages = {397--412},
doi = {10.1098/rspa.1994.0147},
}
@article{faber1923,
author = {Faber, Georg},
title = {Beweis, dass unter allen homogenen {M}embranen von gleicher {F}l{\"a}che und gleicher {S}pannung die kreisf{\"o}rmige den tiefsten {G}rundton gibt},
journal = {Sitzungsber. Bayer. Akad. Wiss. M{\"u}nchen, Math.-Phys. Kl.},
year = {1923},
pages = {169--172},
}
@article{krahn1925,
author = {Krahn, Edgar},
title = {{\"U}ber eine von {R}ayleigh formulierte {M}inimaleigenschaft des {K}reises},
journal = {Math. Ann.},
volume = {94},
number = {1},
year = {1925},
pages = {97--100},
doi = {10.1007/BF01208645},
}
@article{krahn1926,
author = {Krahn, Edgar},
title = {{\"U}ber {M}inimaleigenschaften der {K}ugel in drei und mehr {D}imensionen},
journal = {Acta Comm. Univ. Tartu (Dorpat) A},
volume = {9},
year = {1926},
pages = {1--44},
}
@article{berger2015,
author = {Berger, Amandine},
title = {The eigenvalues of the {L}aplacian with {D}irichlet boundary condition in $\mathbb{R}^2$ are almost never minimized by disks},
journal = {Ann. Global Anal. Geom.},
volume = {47},
number = {3},
year = {2015},
pages = {285--304},
doi = {10.1007/s10455-014-9446-9},
}
@misc{aim-third-dirichlet-disk,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Does the disk minimize the third Dirichlet eigenvalue?}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/005-third-dirichlet-disk.md},
note = {Entry 005 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-00309, "Does the disk minimize the third Dirichlet eigenvalue?", clopen.solutions.