Abstract

Davies’ version of the Hardy inequality gives a lower bound for the Dirichlet integral of a function vanishing on the boundary of a domain in terms of the integral of the squared function with a weight containing the averaged distance to the boundary. This inequality is applied to easily derive two classical results of spectral theory, E. Lieb’s inequality for the first eigenvalue of the Dirichlet Laplacian and G. Rozenblum’s estimate for the spectral counting function of the Laplacian in an unbounded domain in terms of the number of disjoint balls of preset size whose intersection with the domain is large enough.

中文翻译：

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戴维斯哈代不等式的两个后果

摘要

哈代不等式的戴维斯版本给出了在域边界上消失的函数的狄利克雷积分的下限，根据平方函数的积分，权重包含到边界的平均距离。应用此不等式很容易推导出谱理论的两个经典结果，即狄利克雷拉普拉斯算子的第一特征值的 E. Lieb 不等式和无界域中拉普拉斯算子的谱计数函数的估计，以不相交的数量表示与域的交点足够大的预设大小的球。