Exponential interior decay for smooth Steklov domains
Let $\Omega \subset \mathbb{R}^2$ be a bounded connected domain with $C^\infty$ boundary. A Steklov eigenfunction with eigenvalue $\sigma$ is a function $u$ with $\Delta u = 0$ in $\Omega$ and $\partial_\nu u = \sigma u$ on $\partial\Omega$, where $\partial_\nu$ is the outward normal derivative; normalize $\|u\|_{L^2(\partial\Omega)} = 1$.
Hislop and Lutzer proved that on domains with smooth boundary the Steklov eigenfunctions decay into the interior faster than any power of the eigenvalue. Galkowski and Toth proved, for real-analytic boundary, that they decay exponentially in the eigenvalue on every compact subset of the interior.
Problem. Is it true that for every such $\Omega$, every sequence of normalized Steklov eigenfunctions $u_j$ on $\Omega$ with eigenvalues $\sigma_j \to \infty$, and every compact $K \subset \Omega$, there are constants $C_K, c_K > 0$, independent of $j$ and of the choice of normalized eigenfunctions, such that $$\sup_{x \in K} |u_j(x)| \le C_K e^{-c_K \sigma_j}?$$
Source: Colbois, Girouard, Gordon and Sher (2024), Open Question 10.4, for smooth rather than real-analytic boundary in any dimension. Stated by AIM — Open Applied Problems for smooth planar domains.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, interior decay, Steklov eigenfunctions
References
@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 = {Rev. Mat. Complut.},
volume = {37},
number = {1},
year = {2024},
pages = {1--161},
doi = {10.1007/s13163-023-00480-3},
eprint = {2212.12528},
archivePrefix = {arXiv},
}
@article{hislop2001,
author = {Hislop, P. D. and Lutzer, C. V.},
title = {Spectral asymptotics of the {D}irichlet-to-{N}eumann map on multiply connected domains in {$\mathbb{R}^d$}},
journal = {Inverse Problems},
volume = {17},
number = {6},
year = {2001},
pages = {1717--1741},
doi = {10.1088/0266-5611/17/6/313},
}
@article{galkowski2019,
author = {Galkowski, Jeffrey and Toth, John A.},
title = {Pointwise bounds for {S}teklov eigenfunctions},
journal = {J. Geom. Anal.},
volume = {29},
number = {1},
year = {2019},
pages = {142--193},
doi = {10.1007/s12220-018-9984-7},
eprint = {1611.05363},
archivePrefix = {arXiv},
}
@misc{aim-steklov-smooth-interior-decay,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Exponential interior decay for smooth Steklov domains}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/research/resolved/653-steklov-smooth-interior-decay.md},
note = {Entry 653 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
We give an explicit bounded, simply connected planar domain with smooth strictly convex boundary on which uniform exponential interior decay of boundary-normalized Steklov eigenfunctions fails. At a fixed interior point, the squared eigenfunction values have no finite positive exponential spectral moment. The construction uses a conformal map with reciprocal boundary speed having nonnegative, subexponentially decaying Fourier coefficients. Positivity supplies lower …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: AIM @ a153f46, siavash-sadeghi-steklov-024/steklov_counterexample.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00724, "Exponential interior decay for smooth Steklov domains", clopen.solutions.