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

Arianna Giunti

doi:10.3934/nhm.2021009

Article Contents

2021, Volume 16, Issue 3: 341-375. Doi: 10.3934/nhm.2021009

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

Arianna Giunti

Imperial College London, Department of Mathematics, London, UK

Received: November 30, 2020

Revised: February 28, 2021

Early access: April 2021

Published: September 2021

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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 \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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Mathematics Subject Classification: 35B27, 35J25, 60F15, 60F99, 60G55.

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Figure 1. 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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