Operator estimates for Neumann sieve problem

Abstract

Let $\Omega$ be a domain in ${\mathbb{R}}^n$, $\Gamma$ be a hyperplane intersecting $\Omega$, $\epsilon >0$ be a small parameter, and $D_{k,\epsilon }$, $k=1,2,3\dots$ be a family of small "holes" in $\Gamma \cap \Omega$; when $\epsilon \to 0$, the number of holes tends to infinity, while their diameters tends to zero. Let ${\mathcal{A}}_{\epsilon }$ be the Neumann Laplacian in the perforated domain ${\Omega }_{\epsilon }=\Omega \setminus {\Gamma }_{\epsilon }$, where ${\Gamma }_{\epsilon }=\Gamma \setminus ({\cup }_k{D}_{k,\epsilon })$ ("sieve"). It is well-known that if the sizes of holes are carefully chosen, ${\mathcal{A}}_{\epsilon }$ converges in the strong resolvent sense to the Laplacian on $\Omega \setminus \Gamma$ subject to the so-called ${\delta }^{\prime }$-conditions on $\Gamma$. In the current work we improve this result: under rather general assumptions on the shapes and locations of the holes we derive estimates on the rate of convergence in terms of $L^2\to L^2$ and $L^2\to H^1$ operator norms; in the latter case a special corrector is required.

33 pages, 3 figures