On the Dirac Faber-Krahn Conjecture

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 $m\ge 0$ as ${\lambda }_1^{+}(\mathrm{\Omega };m)$, our main result shows that \begin{equation*} λ_1^{+}(Ω;m)\geq λ_1^{+}(B_{|Ω|};m), \end{equation*} where $B_{\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }}$ is the disk with the same area as the domain $\mathrm{\Omega }$, and the equality is strictly valid if and only if $\mathrm{\Omega }=B_{\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }}$. The key observation leading to the proof is that the upper component of the Dirac spinor associated with ${\lambda }_1^{+}(\mathrm{\Omega };m)$ is non-vanishing over $\mathrm{\Omega }$, 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.