The Weinstock inequality for convex domains in a sphere
Let $n \ge 3$ and let $\Omega$ be a smooth geodesically convex domain whose closure is contained in an open hemisphere of the unit sphere $S^n$. Its first nonzero Steklov eigenvalue is $$\sigma_1(\Omega) = \inf\left\{\frac{\int_\Omega |\nabla u|^2\,dV}{\int_{\partial\Omega} u^2\,dS} : u \in H^1(\Omega),\ \int_{\partial\Omega} u\,dS = 0,\ \int_{\partial\Omega} u^2\,dS > 0\right\},$$ the least $\sigma > 0$ for which $\Delta u = 0$ in $\Omega$, $\partial_\nu u = \sigma u$ on $\partial\Omega$ has a nonzero solution. Let $B \subset S^n$ be a geodesic ball of radius less than $\pi/2$ with $|\partial B| = |\partial\Omega|$. Weinstock proved that among simply connected planar domains of given perimeter the disk maximises $\sigma_1$. Bucur, Ferone, Nitsch and Trombetti proved the analogue for bounded convex sets in $\mathbb{R}^n$ of given surface area, with equality only for balls; Fraser and Schoen showed that in dimension at least three it fails for some contractible domains. In hyperbolic space $\mathbb{H}^n$, $n \ge 3$, the analogue was proved for convex domains by Sun, Wang and Wang, for smooth star-shaped mean-convex domains by Gu, Li and Wan, and for smooth outward-minimizing domains by Gao, Wei and Zhou.
Problem. Is $\sigma_1(\Omega) \le \sigma_1(B)$ for every $n \ge 3$ and every such $\Omega$, with equality only when $\Omega$ is a geodesic ball?
Source: Colbois, Girouard, Gordon and Sher (2024), Open Question 4.27, for convex domains in hyperbolic space or the sphere; stated for the sphere, with smooth domains in an open hemisphere, by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- AP, DG
- Keywords
- AIM Open Applied Problems, convex domains, isoperimetric inequalities, Steklov eigenvalues
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{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},
}
@article{bucur2021,
author = {Bucur, Dorin and Ferone, Vincenzo and Nitsch, Carlo and Trombetti, Cristina},
title = {Weinstock inequality in higher dimensions},
journal = {J. Differential Geom.},
volume = {118},
number = {1},
year = {2021},
pages = {1--21},
doi = {10.4310/jdg/1620272940},
eprint = {1710.04587},
archivePrefix = {arXiv},
}
@article{fraser2019,
author = {Fraser, Ailana and Schoen, Richard},
title = {Shape optimization for the {S}teklov problem in higher dimensions},
journal = {Adv. Math.},
volume = {348},
year = {2019},
pages = {146--162},
doi = {10.1016/j.aim.2019.03.011},
eprint = {1711.04381},
archivePrefix = {arXiv},
}
@article{gu2025,
author = {Gu, Pingxin and Li, Haizhong and Wan, Yao},
title = {Weinstock inequality in hyperbolic space},
journal = {J. Funct. Anal.},
volume = {289},
number = {12},
year = {2025},
pages = {111155},
doi = {10.1016/j.jfa.2025.111155},
eprint = {2409.02766},
archivePrefix = {arXiv},
}
@misc{gu2026,
author = {Gu, Pingxin and Li, Haizhong and Wan, Yao},
title = {Weinstock inequality in hyperbolic space {II}},
year = {2026},
eprint = {2608.11849},
archivePrefix = {arXiv},
}
@misc{gao2026,
author = {Gao, Chaoqun and Wei, Yong and Zhou, Rong},
title = {Weinstock inequalities for outward-minimizing domains},
year = {2026},
eprint = {2608.11841},
archivePrefix = {arXiv},
}
@misc{sun2026,
author = {Sun, Jin and Wang, Lili and Wang, Tao},
title = {The {W}einstock inequality for convex domains in hyperbolic space},
year = {2026},
eprint = {2608.10771},
archivePrefix = {arXiv},
}
@misc{aim-spherical-convex-weinstock-inequality,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The Weinstock inequality for convex domains in a sphere}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/443-spherical-convex-weinstock-inequality.md},
note = {Entry 443 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00385, "The Weinstock inequality for convex domains in a sphere", clopen.solutions.