Convergence rates for the homogenization of the Poisson problem in randomly perforated domains

Abstract

<p style='text-indent:20px;'>In this paper we provide converge rates for the homogenization of the Poisson problem with Dirichlet boundary conditions in a randomly perforated domain of <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^d $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ d \geqslant 3 $\end{document}</tex-math></inline-formula>. We assume that the holes that perforate the domain are spherical and are generated by a rescaled marked point process <inline-formula><tex-math id="M3">\begin{document}$ (\Phi, \mathcal{R}) $\end{document}</tex-math></inline-formula>. The point process <inline-formula><tex-math id="M4">\begin{document}$ \Phi $\end{document}</tex-math></inline-formula> generating the centres of the holes is either a Poisson point process or the lattice <inline-formula><tex-math id="M5">\begin{document}$ \mathbb{Z}^d $\end{document}</tex-math></inline-formula>; the marks <inline-formula><tex-math id="M6">\begin{document}$ \mathcal{R} $\end{document}</tex-math></inline-formula> generating the radii are unbounded i.i.d random variables having finite <inline-formula><tex-math id="M7">\begin{document}$ (d-2+\beta) $\end{document}</tex-math></inline-formula>-moment, for <inline-formula><tex-math id="M8">\begin{document}$ \beta &gt; 0 $\end{document}</tex-math></inline-formula>. We study the rate of convergence to the homogenized solution in terms of the parameter <inline-formula><tex-math id="M9">\begin{document}$ \beta $\end{document}</tex-math></inline-formula>. We stress that, for low values of <inline-formula><tex-math id="M10">\begin{document}$ \beta $\end{document}</tex-math></inline-formula>, the balls generating the holes may overlap with overwhelming probability.</p>