Convergence rates for the homogenization of the Poisson problem in randomly perforated domains (2007.13386v1)

Abstract: In this paper we provide converge rates for the homogenization of the Poisson problem with Dirichlet boundary conditions in a randomly perforated domain of $\mathbb{R}^d$, $d \geq 3$. We assume that the holes that perforate the domain are spherical and are generated by a rescaled marked point process $(\Phi, \mathcal{R})$. The point process $\Phi$ generating the centres of the holes is either a Poisson point process or the lattice $\mathbb{Z}^d$; the marks $\mathcal{R}$ generating the radii are unbounded i.i.d random variables having finite $(d-2+\beta)$-moment, for $\beta > 0$. We study the rate of convergence to the homogenized solution in terms of the parameter $\beta$. We stress that, for certain values of $\beta$, the balls generating the holes may overlap with overwhelming probability.