Operator estimates for Neumann sieve problem

Abstract

Let $Ω$ be a domain in $\mathbb{R}^n$, $Γ$ be a hyperplane intersecting $Ω$, $\varepsilon>0$ be a small parameter, and $D_{k,\varepsilon}$, $k=1,2,3\dots$ be a family of small "holes" in $Γ\capΩ$; when $\varepsilon \to 0$, 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 $Ω_\varepsilon=Ω\setminusΓ_\varepsilon$, where $Γ_\varepsilon=Γ\setminus (\cup_k D_{k,\varepsilon})$ ("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 $Ω\setminusΓ$ subject to the so-called $δ'$-conditions on $Γ$. 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.