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[J]. Networks and Heterogeneous Media, 2021, 16(3): 341-375. doi: 10.3934/nhm.2021009
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.
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Arianna Giunti. Convergence rates for the homogenization of the Poisson problem in randomly perforated domains[J]. Networks and Heterogeneous Media, 2021, 16(3): 341-375. doi: 10.3934/nhm.2021009
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