Norm-resolvent convergence in perforated domains

For several different boundary conditions (Dirichlet, Neumann, Robin), we prove norm-resolvent convergence for the operator Δ in the perforated domain Ω\∪ _i∈2ϵℤd B _aϵ (i), a _ϵ ϵ, to the limit operator -Δ+ μ _i on L ²(Ω), where μ _i ∈ ℂ is a constant depending on the choice of boundary conditions. This is an improvement of previous results [Progress in Nonlinear Differential Equations and Their Applications 31 (1997), 45-93; in: Proc. Japan Acad., 1985], which show strong resolvent convergence. In particular, our result implies Hausdorff convergence of the spectrum of the resolvent for the perforated domain problem.