American Institute of Mathematical Sciences

doi: 10.3934/nhm.2021009

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

 Imperial College London, Department of Mathematics, London, UK

Received  December 2020 Revised  March 2021 Published  April 2021

In this paper we provide converge rates for the homogenization of the Poisson problem with Dirichlet boundary conditions in a randomly perforated domain of $\mathbb{R}^d$, $d \geqslant 3$. 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 low values of $\beta$, the balls generating the holes may overlap with overwhelming probability.

Citation: Arianna Giunti. Convergence rates for the homogenization of the Poisson problem in randomly perforated domains. Networks & Heterogeneous Media, doi: 10.3934/nhm.2021009
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The construction of $K_{ \varepsilon, z}$ from the cube $Q_{k, \varepsilon}$. The dashed grey area corresponds to the set $K_{ \varepsilon, z}$, while $Q_{ \varepsilon, z}$ is the square bounded by the thick black line. The green dots are the points of $\Phi^ \varepsilon_\delta$ that fall inside the set $Q_{k-1, z}$ (here bounded by the dashed blue line). The red dots are the points that are outside of $Q_{k, z}$ but whose associated cube intersects $\partial Q_{k, z}$. The black dots are the points that are in $Q_{k, z} \backslash Q_{k-1, z}$. Note that the cubes associated to the black and red dots are typically smaller than the ones associated to the green dots due to the cut-off $\tilde R_{ \varepsilon, z}$
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