Faber–Krahn inequality for graphene quantum dots
Let $\Omega \subset \mathbb{R}^2$ be a bounded connected $C^2$ domain with outward unit normal $n = (n_1, n_2)$. On $L^2(\Omega; \mathbb{C}^2)$ consider the massless Dirac operator with infinite-mass boundary condition $$D_\Omega = \begin{pmatrix} 0 & -i\partial_1 - \partial_2 \\ -i\partial_1 + \partial_2 & 0 \end{pmatrix}, \qquad \operatorname{dom} D_\Omega = \{u \in H^1(\Omega; \mathbb{C}^2) : u_2 = i(n_1 + i n_2) u_1 \text{ on } \partial\Omega\}.$$ Write $\lambda_+(\Omega)$ for its smallest positive eigenvalue, and let $B$ be a disk with $|B| = |\Omega|$. Benguria, Fournais, Stockmeyer and Van Den Bosch proved $\lambda_+(\Omega)^2 \ge 2\pi/|\Omega|$ for simply connected $\Omega$, a bound attained by no domain. Antunes, Benguria, Lotoreichik and Ourmières-Bonafos gave a variational characterisation of $\lambda_+(\Omega)$, and their computations support the inequality below. Duran, Mas and Sanz-Perela showed that the Faber–Krahn inequalities for the family of quantum dot Dirac operators with non-negative mass are equivalent to Faber–Krahn inequalities for $\overline\partial$-Robin Laplacians, and proved them for boundary conditions close enough, depending on the domain, to one of the two zigzag conditions, and, for simply connected domains, also close enough to the other.
Problem. Is $\lambda_+(\Omega) \ge \lambda_+(B)$ for every such $\Omega$, with equality only when $\Omega$ is a disk?
Source: Ourmières-Bonafos, in AimPL: Shape optimization with surface interactions (2019), Problem 5.1; also Duran, Mas and Sanz-Perela (2026), Conjecture 1.1, the case of zero mass and infinite-mass boundary condition.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- AP, MP
- Keywords
- AIM Open Applied Problems, Dirac operators, Faber–Krahn inequalities
References
@misc{aimpl2019,
editor = {Krej{\v{c}}i{\v{r}}{\'\i}k, David and Larson, Simon and Lotoreichik, Vladimir},
title = {{AimPL}: Shape optimization with surface interactions},
howpublished = {American Institute of Mathematics problem list},
year = {2019},
url = {http://aimpl.org/shapesurface},
note = {Problem 5.1, posed by T. Ourmi{\`e}res-Bonafos},
}
@article{duran2026,
author = {Duran, Joaquim and Mas, Albert and Sanz-Perela, Tom{\'a}s},
title = {A connection between quantum dot {D}irac operators and $\overline\partial$-{R}obin {L}aplacians in the context of shape optimization problems},
journal = {J. Funct. Anal.},
volume = {290},
number = {10},
year = {2026},
pages = {Paper No. 111398},
doi = {10.1016/j.jfa.2026.111398},
eprint = {2507.18698},
archivePrefix = {arXiv},
}
@article{benguria2017,
author = {Benguria, Rafael D. and Fournais, S{\o}ren and Stockmeyer, Edgardo and Van Den Bosch, Hanne},
title = {Spectral gaps of {D}irac operators describing graphene quantum dots},
journal = {Math. Phys. Anal. Geom.},
volume = {20},
number = {2},
year = {2017},
pages = {Paper No. 11},
doi = {10.1007/s11040-017-9242-4},
eprint = {1601.06607},
archivePrefix = {arXiv},
}
@article{antunes2021,
author = {Antunes, Pedro R. S. and Benguria, Rafael D. and Lotoreichik, Vladimir and Ourmi{\`e}res-Bonafos, Thomas},
title = {A variational formulation for {D}irac operators in bounded domains. {A}pplications to spectral geometric inequalities},
journal = {Comm. Math. Phys.},
volume = {386},
number = {2},
year = {2021},
pages = {781--818},
doi = {10.1007/s00220-021-03959-6},
eprint = {2003.04061},
archivePrefix = {arXiv},
}
@misc{aim-quantum-dot-dirac-faber-krahn,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Faber–Krahn inequality for graphene quantum dots}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/102-quantum-dot-dirac-faber-krahn.md},
note = {Entry 102 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-00374, "Faber–Krahn inequality for graphene quantum dots", clopen.solutions.