OP-01021 Open

A Pleijel theorem for Steklov eigenfunctions in dimension three and higher

Let $\Omega$ be a compact Riemannian manifold with boundary, let $u_k$ be a Steklov eigenfunction with eigenvalue $\sigma_k$, and let $N_k$ be the number of nodal domains of $u_k$ (with $\sigma_0 = 0$). Kuttler and Sigillito proved the analogue of Courant's theorem, $N_k \le k + 1$, so $\limsup_{k \to \infty} N_k / k \le 1$. Pleijel proved that for the Dirichlet Laplacian the corresponding $\limsup$ is at most an explicit constant $\gamma < 1$. In dimension two no such theorem holds for the Steklov problem: on the unit disk, $r^n \cos n\theta$ has $2n$ nodal domains and index about $2n$.

Problem. Is $\limsup_{k \to \infty} N_k / k < 1$ for the Steklov eigenfunctions of every compact Riemannian manifold $\Omega$ with boundary of dimension at least three?

Source: A. Hassannezhad and D. Sher, Nodal count for Dirichlet-to-Neumann operators with potential, Proc. Amer. Math. Soc., Conjecture 1.7 (the part on Steklov eigenfunctions). Listed, without a restriction on the dimension, as Open Question 10.13 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, nodal domains, Pleijel theorem, Steklov eigenfunctions

References

@misc{hassannezhad2021,
  author = {Hassannezhad, Asma and Sher, David},
  title = {Nodal count for {D}irichlet-to-{N}eumann operators with potential},
  year = {2021},
  doi = {10.1090/proc/16207},
  eprint = {2107.03370},
  archivePrefix = {arXiv},
  note = {Proc. Amer. Math. Soc.; Conjecture 1.7, verified there for cylinders and balls}
}

@article{kuttler1969,
  author = {Kuttler, J. R. and Sigillito, V. G.},
  title = {An inequality of a {S}tekloff eigenvalue by the method of defect},
  journal = {Proceedings of the American Mathematical Society},
  volume = {20},
  pages = {357--360},
  year = {1969},
  doi = {10.1090/S0002-9939-1969-0235323-5}
}

@article{pleijel1956,
  author = {Pleijel, {\AA}ke},
  title = {Remarks on {C}ourant's nodal line theorem},
  journal = {Communications on Pure and Applied Mathematics},
  volume = {9},
  pages = {543--550},
  year = {1956},
  doi = {10.1002/cpa.3160090324}
}

@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 10.13}
}

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-01021, "A Pleijel theorem for Steklov eigenfunctions in dimension three and higher", clopen.solutions.