Can a mixed Steklov–Neumann problem be isospectral to a pure Steklov problem?
In a mixed Steklov–Neumann problem on a connected compact Riemannian manifold with boundary, a harmonic function satisfies the Steklov condition $\partial_\nu u = \sigma u$ on a nonempty part of the boundary and the Neumann condition $\partial_\nu u = 0$ on the rest, also nonempty. The Steklov problem on an orbifold whose only singularities are reflectors is such a mixed problem on the underlying manifold, with the Neumann condition on the reflectors. Whether the Steklov spectrum of a two-dimensional orbifold detects reflectors that do not meet its boundary is a special case of the question below.
Problem. Can a mixed Steklov–Neumann problem on a connected compact Riemannian manifold $\Omega_1$ and the Steklov problem on a connected compact Riemannian manifold $\Omega_2$ have the same spectrum?
Source: Posed by Colbois, Girouard, Gordon and Sher (2024), Open Question 8.21.
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
- DG
- Keywords
- Colbois–Girouard–Gordon–Sher Steklov problems, isospectral manifolds, orbifolds, Steklov eigenvalues
References
@article{ariasmarco2019,
author = {Arias-Marco, Teresa and Dryden, Emily B. and Gordon, Carolyn S. and Hassannezhad, Asma and Ray, Allie and Stanhope, Elizabeth},
title = {Spectral geometry of the {S}teklov problem on orbifolds},
journal = {International Mathematics Research Notices},
number = {1},
pages = {90--139},
year = {2019},
doi = {10.1093/imrn/rnx117},
eprint = {1609.05142},
archivePrefix = {arXiv}
}
@article{gordon2021,
author = {Gordon, Carolyn and Herbrich, Peter and Webb, David},
title = {Steklov and {R}obin isospectral manifolds},
journal = {Journal of Spectral Theory},
volume = {11},
number = {1},
pages = {39--61},
year = {2021},
doi = {10.4171/JST/335},
eprint = {1808.10741},
archivePrefix = {arXiv},
note = {Isospectral pairs for mixed problems}
}
@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 8.21}
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01015, "Can a mixed Steklov–Neumann problem be isospectral to a pure Steklov problem?", clopen.solutions.