Fluctuations in the homogenization of the Poisson and Stokes equations in perforated domains

Abstract

We study the homogenization problem of the Poisson and Stokes equations in \(\mathbb{R}^3\) perforated by \(m\) spherical holes, identically and independently distributed. In the critical regime when the radii of the holes are of order \(m^{-1}\), we consider the fluctuations of the solutions \(u_m\) around the homogenization limit \(u\). In the central limit scaling, we show that these fluctuations converge to a Gaussian field, locally in \(L^2(\mathbb{R}^3)\), with an explicit covariance.