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