Sharp remainder estimates in the Weyl formula for the Neumann Laplacian on a class of planar regions
Let \(\Omega\) be a bounded domain in \(\mathbb{R}^2\). In contradistinction to the Dirichlet Laplacian on \(\Omega\), the Neumann Laplacian may have essential spectrum or/and the Weyl formula for the counting function may fail. On the other hand, the counting function \(N_N(\Omega,\lambda)\) satisfies the Weyl formula \[ N_N(\Omega,\lambda)= {\mu_2(\Omega)\over 4\pi}+ o(\lambda),\quad\lambda\to \infty, \] if \(\Omega\) has a Lipschitz boundary. Here \(\mu_2\) is the two-dimensional Lebesgue measure. In the paper the author treats a larger class of domains \(\Omega\) bounded by the graph of a lower-semicontinuous \(L^1\)-function. Via a new variational method order sharp estimates for the remainder term \[ R_N(\Omega,\lambda)= N_N(\Omega, \lambda)- {\mu_2(\Omega)\over 4\pi}\lambda \] are obtained. These estimates provide, in particular, a criterion in order that the classical Weyl formula holds for the domain \(\Omega\).
scientific article; zbMATH DE number 5190794
scientific article