改进的Berezin-Li-Yau不等式的几何形式
Geometrical Versions of improved Berezin-Li-Yau Inequalities
摘要：
本文研究了$\R^d$中任意有界开集上Dirichlet Laplace算子的特征值，$d \geq 2$ .特别是，我们推导出上界Riesz平均顺序$σ\geq 3/2$，改善了尖锐的Berezin不等式的负第二项。这个余项取决于集合边界的几何性质，并反映了半经典极限中增长的正确顺序。在一定的几何条件下，这些结果意味着特征值的新的下界，改进了Li-Yau不等式.
英文摘要：
We study the eigenvalues of the Dirichlet Laplace operator on an arbitrary bounded, open set in $\R^d$, $d \geq 2$. In particular, we derive upper bounds on Riesz means of order $σ\geq 3/2$, that improve the sharp Berezin inequality by a negative second term. This remainder term depends on geometric properties of the boundary of the set and reflects the correct order of growth in the semi-classical limit. Under certain geometric conditions these results imply new lower bounds on individual eigenvalues, which improve the Li-Yau inequality.
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