On the Dirac Faber-Krahn Conjecture
Abstract
In this paper, we prove the strict Faber-Krahn inequality for non-negative-mass Dirac operators on simply connected domains with infinite-mass boundary condition. Denoting the first positive eigenvalue of the corresponding Dirac operator with mass m≥ 0 as λ1+(Ω;m), our main result shows that λ1+(Ω;m)≥ λ1+(B|Ω|;m), where B|Ω| is the disk with the same area as the domain Ω, and the equality is strictly valid if and only if Ω=B|Ω|. The key observation leading to the proof is that the upper component of the Dirac spinor associated with λ1+(Ω;m) is non-vanishing over Ω, which is proved by exploiting its Dirac current and the associated stream function. Based on this global non-vanishing property, we construct a sphere-valued spinor map of degree one, which allows a sharp comparison with the radial disk profile using the planar isoperimetry.
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BibTeX
@misc{ammari2026dirac,
title = {{On the Dirac Faber-Krahn Conjecture}},
author = {Habib Ammari and Jiayu Qiu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08665},
eprint = {2610.08665},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08665},
}
RIS
TY - UNPB
TI - On the Dirac Faber-Krahn Conjecture
AU - Habib Ammari
AU - Jiayu Qiu
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08665
UR - https://arxiv.org/abs/2610.08665
AB - In this paper, we prove the strict Faber-Krahn inequality for non-negative-mass Dirac operators on simply connected domains with infinite-mass boundary condition. Denoting the first positive eigenvalue of the corresponding Dirac operator with mass m≥ 0 as λ_1^+(Ω;m), our main result shows that λ_1^+(Ω;m)≥ λ_1^+(B_|Ω|;m), where B_|Ω| is the disk with the same area as the domain Ω, and the equality is strictly valid if and only if Ω=B_|Ω|. The key observation leading to the proof is that the upper component of the Dirac spinor associated with λ_1^+(Ω;m) is non-vanishing over Ω, which is proved by exploiting its Dirac current and the associated stream function. Based on this global non-vanishing property, we construct a sphere-valued spinor map of degree one, which allows a sharp comparison with the radial disk profile using the planar isoperimetry.
ER -