Homogenization and operator estimates for Steklov problems in perforated domains

Let the set ${\mathrm{\Omega }}_{\epsilon }$ be obtained from the bounded domain $\mathrm{\Omega }$ by removing a family of $\epsilon$-periodically distributed identical balls. In ${\mathrm{\Omega }}_{\epsilon }$ one considers the Steklov spectral problem. It is known from [Girouard-Henrot-Lagacé, ARMA (2021)] that, if the radii of the holes shrink at a critical rate such that the surface area of a single hole is comparable to the volume of a periodicity cell, then, in the limit $\epsilon \to 0$, the Steklov spectrum converges to the spectrum of the problem $-\mathrm{\Delta }u=\lambda Qu$ on $\mathrm{\Omega }$ with some weight $Q>0$. In the present work, we extend this result by proving, under fairly general assumptions on the locations and shapes of the holes, convergence of the associated resolvent operators in the operator norm topology, together with quantitative estimates for the Hausdorff distance between the spectra. The underlying domain $\mathrm{\Omega }$ is not assumed to be bounded.