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On the Dirac Faber-Krahn Conjecture

by H. Ammari and J. Qiu

(Report number 2026-37)

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 \(m\geq 0\) as \(\lambda_1^{+}(\Omega;m)\), our main result shows that \begin{equation*} \lambda_1^{+}(\Omega;m)\geq \lambda_1^{+}(B_{|\Omega|};m), \end{equation*} where \(B_{|\Omega|}\) is the disk with the same area as the domain \(\Omega\), and the equality is strictly valid if and only if \(\Omega=B_{|\Omega|}\). The key observation leading to the proof is that the upper component of the Dirac spinor associated with \(\lambda_1^{+}(\Omega;m)\) is non-vanishing over \(\Omega\), 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.

Keywords: Dirac operator; infinite-mass boundary condition; Faber-Krahn inequality.

@Techreport{AQ26_1192,
  author = {H. Ammari and J. Qiu},
  title = {On the Dirac Faber-Krahn Conjecture},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2026-37},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2026/2026-37.pdf },
  year = {2026}
}

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