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

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 R d , d 3. We assume that the holes that perforate the domain are spherical and are generated by a rescaled marked point process (Φ, R). The point process Φ generating the centres of the holes is either a Poisson point process or the lattice Z d ; the marks R generating the radii are unbounded i.i.d random variables having finite (d − 2 + β)-moment, for β > 0. We study the rate of convergence to the homogenized solution in terms of the parameter β. We stress that, for low values of β, the balls generating the holes may overlap with overwhelming probability.

Derivation of Darcy's law in randomly punctured domains

“…We divide the proof into steps. The strategy of this proof is similar to the one for [11][Theorem 2.1, (b)].…”

“…In the remaining part of the proof we tackle inequalities (3.25). We follow the same lines of [11][Theorem 1.1, (b)]. and thus only sketch the main steps for the argument.…”

“…The explicit formulation of the harmonic functions {w ε,z } z∈n ε (D) defined in (3.16) (c.f. also [11][(2.24)]) implies that for every z ∈ n ε (D)…”

“…It thus remains to pass from (3.27) to (3.26): We do this by taking the expectation and arguing as for [11][Inequality (4.22)]. We rely on the stationarity of (φ, R), the properties of the Poisson point process and the fact that z ∈ n ε implies that ε α ρ z ε 1+γ and R ε,z ε 1+ 1 2 γ .…”

“…These allow to pass to the limit in the equation and identify the effective problem. We stress that a crucial ingredient in these arguments is given by the quantitative bounds obtained in [11] in the case α = 3. These bounds may indeed also be extended to the current setting sot that the rate of convergence of the measures −σ −2 ε ∆w ε ∈ H −1 (D) is quantified.…”