发表机构
ETH Zürich（苏黎世联邦理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了非负质量Dirac算子在单连通区域上的严格Faber-Krahn不等式，即第一个正特征值不小于同面积圆盘的特征值，等号仅当区域为圆盘时成立；关键方法是利用Dirac旋量上分量的全局非零性构造度为一的球值映射，并结合平面等周不等式。
AI 中文摘要
本文证明了在具有无限质量边界的单连通区域上，非负质量Dirac算子的严格Faber-Krahn不等式。设相应Dirac算子质量$m\geq 0$的第一个正特征值为$\lambda_1^{+}(\Omega;m)$，我们的主要结果表明\begin{equation*} \lambda_1^{+}(\Omega;m)\geq \lambda_1^{+}(B_{|\Omega|};m), \end{equation*}其中$B_{|\Omega|}$是与区域$\Omega$面积相同的圆盘，且等号严格成立当且仅当$\Omega=B_{|\Omega|}$。证明的关键观察是，与$\lambda_1^{+}(\Omega;m)$相关的Dirac旋量的上分量在$\Omega$上非零，这通过利用其Dirac流和相关的流函数得以证明。基于这一全局非零性质，我们构造了一个度为一的球值旋量映射，从而利用平面等周不等式与径向圆盘轮廓进行尖锐比较。
英文摘要
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 $λ_1^{+}(Ω;m)$, our main result shows that \begin{equation*} λ_1^{+}(Ω;m)\geq λ_1^{+}(B_{|Ω|};m), \end{equation*} 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.
