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.