关于凸区域上Berezin不等式的余项
On the remainder term of the Berezin inequality on a convex domain
摘要：
利用$n\geq 2$研究了$\mathbb{R}^n$中凸区域上Laplacian算子的Dirichlet特征值.特别地，我们推广和改进了Geisinger，Laptev和Weidl在一篇文章中建立的$σ\geq 3/2$阶Riesz平均的上界.这是通过改进Berezin不等式中负第二项的估计来实现的。所得到的余项反映了半经典极限中增长的正确顺序，并且仅依赖于域的边界的测量。我们强调，这样的改进是一般的$Ω\subset\mathbb{R}^n$不可能的，以前已知的，以满足一定的几何条件的平面凸区域举行。作为推论，我们得到了各个特征值$λ_k$的下界，在一定范围内的$k$改进了凸区域的Li-Yau不等式.然而，对于凸区域，可以使用不同的方法来获得更强的这种下界。
英文摘要：
We study the Dirichlet eigenvalues of the Laplacian on a convex domain in $\mathbb{R}^n$, with $n\geq 2$. In particular, we generalize and improve upper bounds for the Riesz means of order $σ\geq 3/2$ established in an article by Geisinger, Laptev and Weidl. This is achieved by refining estimates for a negative second term in the Berezin inequality. The obtained remainder term reflects the correct order of growth in the semi-classical limit and depends only on the measure of the boundary of the domain. We emphasize that such an improvement is for general $Ω\subset\mathbb{R}^n$ not possible and was previously known to hold only for planar convex domains satisfying certain geometric conditions. As a corollary we obtain lower bounds for the individual eigenvalues $λ_k$, which for a certain range of $k$ improves the Li--Yau inequality for convex domains. However, for convex domains one can use different methods to obtain even stronger such lower bounds.
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