OP-00273 Open

The regular-polygon Steklov conjecture

Fix an integer $n \ge 3$ and $L > 0$. For a convex polygon $P \subset \mathbb{R}^2$ with at most $n$ sides and perimeter $L$, let $\sigma_1(P)$ be its first positive Steklov eigenvalue: the least $\sigma > 0$ for which $$\Delta u = 0 \ \text{ in } P, \qquad \partial_\nu u = \sigma u \ \text{ on } \partial P$$ has a nonzero weak solution. Let $R_n$ be the regular $n$-gon with perimeter $L$. By Weinstock's inequality, the disk of perimeter $L$ has the largest $\sigma_1$ among simply connected planar domains of perimeter $L$. Cheng, Gui, Hu, Li and Yao proved that $\sigma_1(R_n)$ increases strictly with $n$ at fixed perimeter, a consequence of the inequality below.

Problem. Is $\sigma_1(P) \le \sigma_1(R_n)$ for every $n \ge 3$, every $L > 0$ and every such $P$, with equality only when $P$ is congruent to $R_n$?

Source: Oudet, at the American Institute of Mathematics workshop on Steklov eigenproblems (2018), AimPL Problem 4.6, and Nigam (2020), Open Problem 8, for all n-gons; in this form, Cheng, Gui, Hu, Li and Yao (2026), Conjecture 1.1.

Imported from github.com/MColbrook/AIM.

Classification

Primary: SP — Spectral Theory
Secondary: AP
Keywords: AIM Open Applied Problems, isoperimetric inequalities, polygons, Steklov eigenvalues

References

@misc{aimpl2018steklov,
  title        = {{AimPL}: {S}teklov eigenproblems},
  howpublished = {American Institute of Mathematics problem list, from the workshop of 30 April to 4 May 2018},
  year         = {2018},
  url          = {http://aimpl.org/stekloveigen},
  note         = {Problem 4.6, proposed by {\'E}douard Oudet},
}
@misc{eigenvalues2020,
  title        = {Open problems from the miniconference on sharp eigenvalue estimates for partial differential operators},
  howpublished = {Miniconference, April 2020},
  year         = {2020},
  url          = {https://publish.illinois.edu/eigenvalues2020/files/2020/04/Open-problems.pdf},
  note         = {Open Problem 8, proposed by Nilima Nigam},
}
@misc{cheng2026,
  author        = {Cheng, Zhuo and Gui, Changfeng and Hu, Yeyao and Li, Qinfeng and Yao, Ruofei},
  title         = {Global strict monotonicity and asymptotics of the first {S}teklov eigenvalue for regular polygons},
  year          = {2026},
  eprint        = {2603.25116},
  archivePrefix = {arXiv},
}
@article{weinstock1954,
  author  = {Weinstock, Robert},
  title   = {Inequalities for a classical eigenvalue problem},
  journal = {J. Rational Mech. Anal.},
  volume  = {3},
  number  = {6},
  year    = {1954},
  pages   = {745--753},
  doi     = {10.1512/iumj.1954.3.53036},
}
@misc{aim-steklov-polygon-optimizer,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{The regular-polygon Steklov conjecture}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/c9929805cece044705abaaebe1cebddac1ba11a6/problems/022-steklov-polygon-optimizer.md},
  note         = {Entry 022 at commit c992980, 2026-10-05},
}

Citing this problem

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

OP-00273, "The regular-polygon Steklov conjecture", clopen.solutions.