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

Let ε > 0 be a small parameter. We consider the domain Ω ε : = Ω ∖ D ε , where Ω is an open domain in R n , and D ε is a family of small balls of the radius d ε = o (ε) distributed periodically with period ε. Let Δ ε be the Laplace operator in Ω ε subject to the Robin condition ∂ u ∂ n + γ ε u = 0 with γ ε ≥ 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 d ε and γ ε , the operator Δ 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 L 2 → L 2 and L 2 → H 1 operator norms. As a byproduct we establish the estimate on the distance between the spectra of the associated operators. [ABSTRACT FROM AUTHOR]