Operator estimates for the crushed ice problem

Let $\Delta_{\Omega_\varepsilon}$ be the Dirichlet Laplacian in the domain $\Omega_\varepsilon:=\Omega\setminus\left(\cup_i D_{i \varepsilon}\right)$. Here $\Omega\subset\mathbb{R}^n$ and $\{D_{i \varepsilon}\}_{i}$ is a family of tiny identical holes ("ice pieces") distributed periodically in $\mathbb{R}^n$ with period $\varepsilon$. We denote by $\mathrm{cap}(D_{i \varepsilon})$ the capacity of a single hole. It was known for a long time that $-\Delta_{\Omega_\varepsilon}$ converges to the operator $-\Delta_{\Omega}+q$ in strong resolvent sense provided the limit $q:=\lim_{\varepsilon\to 0} \mathrm{cap}(D_{i\varepsilon}) \varepsilon^{-n}$ exists and is finite. In the current contribution we improve this result deriving estimates for the rate of convergence in terms of operator norms. As an application, we establish the uniform convergence of the corresponding semi-groups and (for bounded $\Omega$) an estimate for the difference of the $k$-th eigenvalue of $-\Delta_{\Omega_\varepsilon}$ and $-\Delta_{\Omega_\varepsilon}+q$. Our proofs relies on an abstract scheme for studying the convergence of operators in varying Hilbert spaces developed previously by the second author.