Steklov asymptotics of curvilinear polygons with reflex angles
Let $\Omega \subset \mathbb R^2$ be a curvilinear polygon: a simply connected domain whose boundary is a closed curve made of $n$ smooth arcs, of lengths $\ell_1, \dots, \ell_n$, meeting at corners with interior angles $\alpha_1, \dots, \alpha_n$. Levitin, Parnovski, Polterovich and Sher defined quasi-eigenvalues $\hat\sigma_k$, computed explicitly from the side lengths and the angles (when no angle is of the form $\pi/m$, as the values of $\sigma \ge 0$ at which a product of $2 \times 2$ transfer matrices has eigenvalue $1$), and proved that if every $\alpha_j$ lies in $(0, \pi)$, then the Steklov eigenvalues satisfy $$\sigma_k = \hat\sigma_k + O(k^{-\varepsilon_0})$$ for an explicit $\varepsilon_0 > 0$. Their proof depends strongly on the angles being less than $\pi$; numerical evidence suggests that the answer below is yes.
Problem. Can the condition that the angles $\alpha_j$ lie in $(0, \pi)$ be relaxed to $(0, 2\pi)$?
Source: Levitin, Parnovski, Polterovich and Sher, Sloshing, Steklov and corners, Proc. Lond. Math. Soc. 125 (2022), final section, where numerics suggest the extension. Listed as Open Question 8.11 in Colbois–Girouard–Gordon–Sher (2024).
The statement is adapted from doi.org/10.1007/s13163-023-00480-3, whose text is available under CC BY 4.0.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- AP
- Keywords
- Colbois–Girouard–Gordon–Sher Steklov problems, polygons, spectral asymptotics, Steklov eigenvalues
References
@article{levitin2022,
author = {Levitin, Michael and Parnovski, Leonid and Polterovich, Iosif and Sher, David A.},
title = {Sloshing, {S}teklov and corners: asymptotics of {S}teklov eigenvalues for curvilinear polygons},
journal = {Proceedings of the London Mathematical Society (3)},
volume = {125},
number = {3},
pages = {359--487},
year = {2022},
doi = {10.1112/plms.12461},
eprint = {1908.06455},
archivePrefix = {arXiv}
}
@misc{galkowski2026,
author = {Galkowski, Jeffrey and Wang, Ruoyu P. T.},
title = {Lower bounds on {S}teklov eigenfunctions for a class of curvilinear polygons},
year = {2026},
eprint = {2610.31387},
archivePrefix = {arXiv},
note = {One corner with angle in $(0, 2\pi)$; cites a forthcoming paper of Galkowski, Malagutti and Wang on curvilinear polygons}
}
@article{colbois2024,
author = {Colbois, Bruno and Girouard, Alexandre and Gordon, Carolyn and Sher, David},
title = {Some recent developments on the {S}teklov eigenvalue problem},
journal = {Revista Matem{\'a}tica Complutense},
volume = {37},
number = {1},
pages = {1--161},
year = {2024},
doi = {10.1007/s13163-023-00480-3},
eprint = {2212.12528},
archivePrefix = {arXiv},
note = {Open Question 8.11}
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01026, "Steklov asymptotics of curvilinear polygons with reflex angles", clopen.solutions.