The Neumann sieve problem revisited

Abstract

Let $\Omega\subset\mathbb{R}^n$ be a domain, $\Gamma$ be a hyperplane intersecting it. Let $\varepsilon>0$, and $\Omega_\varepsilon=\Omega\setminus\overline{\Sigma_\varepsilon}$, where $\Sigma_\varepsilon$ ("sieve") is an $\varepsilon$-neighbourhood of $\Gamma$ punctured by many narrow passages. When $\varepsilon\to0$, the number of passages tends to infinity, while the diameters of their cross-sections tend to zero. For the case of identical straight periodically distributed and appropriately scaled passages T. Del Vecchio (1987) proved that the Neumann Laplacian on $\Omega_\varepsilon$ converges in a strong resolvent sense to the Laplacian on $\Omega\setminus\Gamma$ subject to the so-called $\delta'$-conditions on $\Gamma$. We will refine this result by deriving estimates on the rate of convergence in terms of various operator norms, and providing the estimate for the distance between the spectra. The assumptions we impose on the passages are rather general. For $n=2$ the results of T. Del Vecchio are not complete, some cases remain as open problems, and in this work we will fill these gaps.