论文标题
随机穿孔域中泊松问题均质化的收敛速率
Convergence rates for the homogenization of the Poisson problem in randomly perforated domains
论文作者
论文摘要
在本文中，我们在$ \ mathbb {r}^d $，$ d \ geq 3 $的随机穿孔域中提供了泊松问题的均质汇率。我们假设穿孔域的孔是球形的，并且由重新标记的点过程$（φ，\ Mathcal {r}）$生成。点过程$φ$生成孔的中心是Poisson点过程，也可以是晶格$ \ Mathbb {z}^d $;标记$ \ MATHCAL {r} $生成半径是无界的I.I.D随机变量，其有限$（D-2+β）$ - 时刻，对于$β> 0 $。我们根据参数$β$研究了与均质溶液的收敛速率。我们强调的是，对于$β$的某些值，产生孔的球可能与压倒性的概率重叠。
In this paper we provide converge rates for the homogenization of the Poisson problem with Dirichlet boundary conditions in a randomly perforated domain of $\mathbb{R}^d$, $d \geq 3$. We assume that the holes that perforate the domain are spherical and are generated by a rescaled marked point process $(Φ, \mathcal{R})$. The point process $Φ$ 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 $(d-2+β)$-moment, for $β> 0$. We study the rate of convergence to the homogenized solution in terms of the parameter $β$. We stress that, for certain values of $β$, the balls generating the holes may overlap with overwhelming probability.
