On the spectral counting function for the Dirichlet Laplacian (Q1193914)

We obtain upper and lower bounds for the spectral counting function of the Dirichlet Laplacian associated to general open sets satisfying the following conditions: \((H_ 1)\) There exists a constant \(c>0\), such that \(-\Delta_ D \geq c/d^ 2(x)\) in the sense of quadratic forms, where \(d(x)=\inf_{y \in \mathbb{R}^ m \backslash D} | y-x |\). \((H_ 2)\) For all \(\varepsilon>0\), \(\mu(\varepsilon)=\int_{\{x \in D:d(x)> \varepsilon\}} <\infty\).