Operator estimates for homogenization of the Robin Laplacian in a perforated domain

Let ε>0 be a small parameter. We consider the domain Ωε:=Ω∖Dε, where Ω is an open in Rn, and Dε family of balls radius dε=o(ε) distributed periodically with period ε. Δε Laplace operator Ωε subject to Robin condition ∂u∂n+γεu=0 γε≥0 on boundary holes Dirichlet exterior boundary. Kaizu (1985, 1989) Brillard (1988) have shown that, under appropriate assumptions dε γε, converges strong resolvent sense sum Laplacian constant potential. improve this result deriving estimates rate convergence terms L2→L2 L2→H1 norms. As byproduct we establish estimate distance between spectra associated operators.