论文标题
有限域中狄拉克运算符的变性公式。应用于光谱几何不等式
A variational formulation for Dirac operators in bounded domains. Applications to spectral geometric inequalities
论文作者
论文摘要
我们在$ \ mathbb {r}^2 $的平滑界面域中研究了具有无限质量边界条件的狄拉克运算符的光谱特征。由光谱几何不平等的动机，我们证明了非线性变分的表述，以表征其主特征值。事实证明，这种表征非常强大，可以简单地证明Szegö型不平等以及该操作员的Faber-Krahn型不平等的新重新印象。该论文与支持Faber-Krahn类型不平等存在的强烈数值证据相辅相成。
We investigate spectral features of the Dirac operator with infinite mass boundary conditions in a smooth bounded domain of $\mathbb{R}^2$. Motivated by spectral geometric inequalities, we prove a non-linear variational formulation to characterize its principal eigenvalue. This characterization turns out to be very robust and allows for a simple proof of a Szegö type inequality as well as a new reformulation of a Faber-Krahn type inequality for this operator. The paper is complemented with strong numerical evidences supporting the existence of a Faber-Krahn type inequality.
