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

Abstract

Let $\epsilon >0$ be a small parameter. We consider the domain ${\Omega }_{\epsilon }:=\Omega \setminus D_{\epsilon }$, where $\Omega$ is an open domain in ${\mathbb{R}}^n$, and $D_{\epsilon }$ is a family of small balls of the radius $d_{\epsilon }=o(\epsilon )$ distributed periodically with period $\epsilon$. Let ${\Delta }_{\epsilon }$ be the Laplace operator in ${\Omega }_{\epsilon }$ subject to the Robin condition $\frac{\partial u}{\partial n}+{\gamma }_{\epsilon }u=0$ with ${\gamma }_{\epsilon }\ge 0$ 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 $d_{\epsilon }$ and ${\gamma }_{\epsilon }$, the operator ${\Delta }_{\epsilon }$ converges in the strong resolvent sense to the sum of the Dirichlet Laplacian in $\Omega$ and a constant potential. We improve this result deriving estimates on the rate of convergence in terms of $L^2\to L^2$ and $L^2\to H^1$ operator norms. As a byproduct we establish the estimate on the distance between the spectra of the associated operators.