Bareket's conjecture for simply connected planar domains
For a bounded simply connected planar domain $\Omega$ with smooth boundary and $\alpha < 0$, let $$\lambda_1^\alpha(\Omega) = \inf_{0 \ne u \in H^1(\Omega)} \frac{\int_\Omega |\nabla u|^2 \, dx + \alpha \int_{\partial\Omega} |u|^2 \, ds}{\int_\Omega |u|^2 \, dx},$$ the first eigenvalue of the Laplacian with the attractive Robin condition $\partial_\nu u + \alpha u = 0$. Bareket conjectured that among planar domains of given area the disk maximises $\lambda_1^\alpha$ for every $\alpha < 0$, and proved it for a class of nearly circular domains. Freitas and Krejčiřík showed that every annulus of the same area as the disk has the larger first eigenvalue once $-\alpha$ is large enough, and that for bounded planar domains of class $C^2$ and fixed area the disk is the maximiser for all $\alpha$ in an interval $[\alpha_*, 0)$, where $\alpha_*$ depends only on the area. Antunes, Freitas and Krejčiřík proved that the disk maximises $\lambda_1^\alpha$ for every $\alpha < 0$ among bounded planar domains of class $C^2$ with the same perimeter. Krejčiřík, Lotoreichik and Vu proved that the equilateral triangle is a local maximiser among triangles of given area when $|\alpha|$ is small, and Gavitone, Krejčiřík and Paoli that the ball maximises $\lambda_1^\alpha$ among bounded convex domains of given volume in every dimension when $|\alpha|$ is small enough, depending only on the volume and the dimension.
Problem. If $B$ is a disk with $|B| = |\Omega|$, is $\lambda_1^\alpha(\Omega) \le \lambda_1^\alpha(B)$ for every $\alpha < 0$ and every such $\Omega$? The area, not the perimeter, is fixed.
Source: Bareket (1977), for planar domains of given area; restricted to simply connected domains by Antunes, Freitas and Krejčiřík (2017), Conjecture 2. Stated for smooth boundaries by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, isoperimetric inequalities, Robin eigenvalues
References
@article{bareket1977,
author = {Bareket, Miriam},
title = {On an isoperimetric inequality for the first eigenvalue of a boundary value problem},
journal = {SIAM J. Math. Anal.},
volume = {8},
number = {2},
year = {1977},
pages = {280--287},
doi = {10.1137/0508020},
}
@article{antunes2017,
author = {Antunes, Pedro R. S. and Freitas, Pedro and Krej{\v{c}}i{\v{r}}{\'\i}k, David},
title = {Bounds and extremal domains for {R}obin eigenvalues with negative boundary parameter},
journal = {Adv. Calc. Var.},
volume = {10},
number = {4},
year = {2017},
pages = {357--379},
doi = {10.1515/acv-2015-0045},
eprint = {1605.08161},
archivePrefix = {arXiv},
}
@article{freitas2015,
author = {Freitas, Pedro and Krej{\v{c}}i{\v{r}}{\'\i}k, David},
title = {The first {R}obin eigenvalue with negative boundary parameter},
journal = {Adv. Math.},
volume = {280},
year = {2015},
pages = {322--339},
doi = {10.1016/j.aim.2015.04.023},
eprint = {1403.6666},
archivePrefix = {arXiv},
}
@article{krejcirik2023,
author = {Krej{\v{c}}i{\v{r}}{\'\i}k, David and Lotoreichik, Vladimir and Vu, Tuyen},
title = {Reverse isoperimetric inequality for the lowest {R}obin eigenvalue of a triangle},
journal = {Appl. Math. Optim.},
volume = {88},
number = {2},
year = {2023},
pages = {63},
doi = {10.1007/s00245-023-10033-1},
eprint = {2204.03235},
archivePrefix = {arXiv},
}
@misc{gavitone2026,
author = {Gavitone, Nunzia and Krej{\v{c}}i{\v{r}}{\'\i}k, David and Paoli, Gloria},
title = {A reverse {F}aber--{K}rahn inequality for the {R}obin {L}aplacian with negative boundary parameter: small coupling in all dimensions},
year = {2026},
eprint = {2606.24341},
archivePrefix = {arXiv},
}
@misc{aim-bareket-simply-connected,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Bareket's conjecture for simply connected planar domains}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/020-bareket-simply-connected.md},
note = {Entry 020 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-00275, "Bareket's conjecture for simply connected planar domains", clopen.solutions.