The Mercedes-star partition of a disk
Let $B = \{x \in \mathbb{R}^2 : |x| < 1\}$, let $\lambda_1(D)$ be the first Dirichlet eigenvalue of $-\Delta$ on an open set $D$, and let $$\mathcal{L}_3(B) = \inf_{(D_1, D_2, D_3)} \max_{1 \le i \le 3} \lambda_1(D_i),$$ the infimum taken over all triples of pairwise disjoint nonempty connected open subsets of $B$, with no condition on their topology, symmetry or interfaces. The Mercedes star is the triple of open sectors cut out by three radii at angles $2\pi/3$; each has first Dirichlet eigenvalue $j_{3/2,1}^2$, where $j_{3/2,1}$ is the first positive zero of the Bessel function $J_{3/2}$. Helffer and Hoffmann-Ostenhof proved that a minimal 3-partition of $B$ whose cells are simply connected in $B \setminus \{0\}$ is the Mercedes star, and Bonnaillie-Noël and Helffer that the same holds if its boundary set has only one critical point. Berkolaiko, Canzani, Cox and Marzuola proved that the Mercedes star is minimal among the 3-partitions whose boundary set contains a curve from the origin to $\partial B$.
Problem. Do the three sectors of the Mercedes star attain the minimum, that is, is $\mathcal{L}_3(B) = j_{3/2,1}^2$?
Source: Helffer and Hoffmann-Ostenhof (2010), as a conjecture that the Mercedes star is the unique minimal 3-partition of the disk up to rotation (see Bonnaillie-Noël and Helffer (2013), §1.2); stated without uniqueness 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, spectral minimal partitions
References
@incollection{helffer2010,
author = {Helffer, Bernard and Hoffmann-Ostenhof, Thomas},
title = {On minimal partitions: new properties and applications to the disk},
booktitle = {Spectrum and Dynamics},
series = {CRM Proc. Lecture Notes},
volume = {52},
publisher = {Amer. Math. Soc.},
address = {Providence, RI},
year = {2010},
pages = {119--135},
doi = {10.1090/crmp/052/07},
}
@article{bonnaillienoel2013,
author = {Bonnaillie-No{\"e}l, Virginie and Helffer, Bernard},
title = {On spectral minimal partitions: the disk revisited},
journal = {Ann. Univ. Buchar. Math. Ser.},
volume = {4 (LXII)},
year = {2013},
pages = {321--342},
url = {https://www.math.ens.psl.eu/~bonnaillie/articles/BoHe13.pdf},
}
@article{berkolaiko2025,
author = {Berkolaiko, Gregory and Canzani, Yaiza and Cox, Graham and Marzuola, Jeremy L.},
title = {Homology of spectral minimal partitions},
journal = {J. Lond. Math. Soc. (2)},
volume = {111},
number = {1},
year = {2025},
pages = {e70065},
doi = {10.1112/jlms.70065},
eprint = {2406.04225},
archivePrefix = {arXiv},
}
@misc{aim-mercedes-disk-partition,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The Mercedes-star partition of a disk}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/010-mercedes-disk-partition.md},
note = {Entry 010 at commit a153f46, 2026-10-13},
}
Resolution claims
Among triples of pairwise disjoint nonempty connected open subsets of the unit disk, the three sectors of opening $2\pi/3$ minimize the largest first Dirichlet eigenvalue. The minimum is $j_{3/2,1}^2$, where $j_{3/2,1}$ is the first positive zero of the Bessel function $J_{3/2}$. The proof transfers any triple with smaller energy to a triple on the unit sphere with energy below its …
We prove that the Y-partition into three equal sectors is the minimal spectral three-partition both for the Dirichlet Laplacian on the unit disc and for the planar harmonic oscillator $-Δ+|x|^2$, with minimal energies $j_{3/2,1}^{2}$ and $5$; for the disc, this confirms a conjecture of Helffer and Hoffmann-Ostenhof. The minimizing regular strong partition is unique up to rotation, and every open …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: AIM @ a153f46, paper/mercedes_partition.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00759, "The Mercedes-star partition of a disk", clopen.solutions.