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

Let $\varepsilon>0$ be a small parameter. We consider the domain $\Omega_\varepsilon:=\Omega\setminus D_\varepsilon$, where $\Omega$ is an open domain in $\mathbb{R}^n$, and $D_\varepsilon$ is a family of small balls of the radius $d_\varepsilon=o(\varepsilon)$ distributed periodically with period $\varepsilon$. Let $\Delta_\varepsilon$ be the Laplace operator in $\Omega_\varepsilon$ subject to the Robin condition ${\partial u\over \partial n}+\gamma_\varepsilon u = 0$ with $\gamma_\varepsilon\ge 0$ 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 $d_\varepsilon$ and $\gamma_\varepsilon$, the operator $\Delta_\varepsilon$ 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.