The Neumann sieve problem revisited

Let Ω be a domain in Rⁿ, Γ be a hyperplane intersecting Ω, ε>0 be a small parameter, and Ω_ε=ΩΣ_ε̄, where the set Σ_ε has a geometry of a thin "sieve" - a layer of thickness 2ε centered on Γ with a lot of drilled passages in it; when ε → 0, the number of passages (per finite volume) tends to infinity, while the diameters of their cross-sections tend to zero. For the case of identical straight periodically distributed passages T. Del Vecchio [Ann. Mat. Pura Appl., 1987] proved that the Neumann Laplacian on Ω_ε converges in a kind of strong resolvent sense to the Laplacian on ΩΓ subject to the so-called δ'-conditions on Γ provided the passages are appropriately scaled. In the current work we refine this result deriving estimates on the rate of convergence in terms of L²→ L² and L²→ H¹ operator norms; also we provide the estimate for the distance between the spectra of these operators in the weighted Hausdorff metrics. The assumptions we impose on the geometry and distribution of the passages are rather general; several examples obeying these assumptions are presented. For $n=2$ the results of T. Del Vecchio are not complete and some cases remain as open problems; we fill these gaps in the current work.