Operator estimates for the Neumann sieve problem

Abstract

Let $\Omega$ be a domain in ${\mathbb {R}}^n$, $\Gamma$ be a hyperplane intersecting $\Omega$, $\varepsilon >0$ be a small parameter, and $D_{k,\varepsilon }$, $k=1,2,3\dots$ be a family of small “holes” in $\Gamma \cap \Omega$; when $\varepsilon \rightarrow 0$, the number of holes tends to infinity, while their diameters tends to zero. Let ${\mathscr {A}}_\varepsilon$ be the Neumann Laplacian in the perforated domain $\Omega _\varepsilon =\Omega \setminus \Gamma _\varepsilon$, where $\Gamma _\varepsilon =\Gamma \setminus (\cup _k D_{k,\varepsilon })$ (“sieve”). It is well-known that if the sizes of holes are carefully chosen, ${\mathscr {A}}_\varepsilon$ converges in the strong resolvent sense to the Laplacian on $\Omega \setminus \Gamma$ subject to the so-called $\delta '$-conditions on $\Gamma \cap \Omega$. 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 $L^2\rightarrow L^2$ and $L^2\rightarrow H^1$ operator norms; in the latter case a special corrector is required.