Dirichlet spectral determination of smooth strictly convex planar domains
Let $\Omega \subset \mathbb{R}^2$ be a bounded open strictly convex domain with $C^\infty$ boundary, strict convexity meaning that the open segment between any two distinct points of $\overline{\Omega}$ lies in $\Omega$. Let $0 < \lambda_1(\Omega) \le \lambda_2(\Omega) \le \cdots$ be its Dirichlet eigenvalues of $-\Delta$, counted with multiplicity. Gordon, Webb and Wolpert constructed pairs of non-congruent planar domains with the same Dirichlet spectrum; all known such pairs are neither smooth nor convex. A disk is determined by its Dirichlet spectrum among bounded Lipschitz domains. Hezari and Zelditch proved that every ellipse of sufficiently small eccentricity is determined, up to isometry, by its Dirichlet spectrum among all smooth domains, and Koval proved that every ellipse outside a small exceptional set of eccentricities is so determined among smooth domains sufficiently close to it.
Problem. Let $\Omega_1$ and $\Omega_2$ be two such domains with $\lambda_j(\Omega_1) = \lambda_j(\Omega_2)$ for every $j \ge 1$. Must there exist $Q \in O(2)$ and $a \in \mathbb{R}^2$ such that $\Omega_2 = Q\Omega_1 + a$? Reflections are allowed.
Source: Kac (1966), for planar domains; Levitin, Mangoubi and Polterovich (2023), Open Problem 6.2.26, for smooth and for convex planar domains. Stated by AIM — Open Applied Problems for smooth strictly convex domains and the Dirichlet spectrum.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- AP, DG
- Keywords
- AIM Open Applied Problems, Dirichlet eigenvalues, inverse spectral problems, isospectral domains
References
@article{kac1966,
author = {Kac, Mark},
title = {Can one hear the shape of a drum?},
journal = {Amer. Math. Monthly},
volume = {73},
number = {4},
year = {1966},
pages = {1--23},
doi = {10.2307/2313748},
}
@book{levitin2023,
author = {Levitin, Michael and Mangoubi, Dan and Polterovich, Iosif},
title = {Topics in Spectral Geometry},
series = {Graduate Studies in Mathematics},
volume = {237},
publisher = {American Mathematical Society},
address = {Providence, RI},
year = {2023},
doi = {10.1090/gsm/237},
}
@article{gordon1992,
author = {Gordon, Carolyn and Webb, David L. and Wolpert, Scott},
title = {One cannot hear the shape of a drum},
journal = {Bull. Amer. Math. Soc. (N.S.)},
volume = {27},
number = {1},
year = {1992},
pages = {134--138},
doi = {10.1090/S0273-0979-1992-00289-6},
}
@article{hezari2022,
author = {Hezari, Hamid and Zelditch, Steve},
title = {One can hear the shape of ellipses of small eccentricity},
journal = {Ann. of Math. (2)},
volume = {196},
number = {3},
year = {2022},
pages = {1083--1134},
doi = {10.4007/annals.2022.196.3.4},
eprint = {1907.03882},
archivePrefix = {arXiv},
}
@article{koval2026,
author = {Koval, Illya},
title = {Local strong {B}irkhoff conjecture and local spectral uniqueness of almost every ellipse},
journal = {Invent. Math.},
volume = {244},
number = {1},
year = {2026},
pages = {221--298},
doi = {10.1007/s00222-025-01397-y},
eprint = {2111.12171},
archivePrefix = {arXiv},
}
@misc{aim-convex-dirichlet-spectrum,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Dirichlet spectral determination of smooth strictly convex planar domains}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/295-convex-dirichlet-spectrum.md},
note = {Entry 295 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-00285, "Dirichlet spectral determination of smooth strictly convex planar domains", clopen.solutions.