Homogenisation of nonlinear Dirichlet problems in randomly perforated domains under minimal assumptions on the size of perforations

IF 1.5 1区 数学 Q2 STATISTICS & PROBABILITY
Lucia Scardia, Konstantinos Zemas, Caterina Ida Zeppieri
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Abstract

In this paper we study the convergence of nonlinear Dirichlet problems for systems of variational elliptic PDEs defined on randomly perforated domains of \(\mathbb {R}^n\). Under the assumption that the perforations are small balls whose centres and radii are generated by a stationary short-range marked point process, we obtain in the critical-scaling limit an averaged nonlinear analogue of the extra term obtained in the classical work of Cioranescu and Murat (Res Notes Math III, 1982). In analogy to the random setting recently introduced by Giunti, Höfer and Velázquez (Commun Part Differ Equ 43(9):1377–1412, 2018) to study the Poisson equation, we only require that the random radii have finite \((n-q)\)-moment, where \(1<q<n\) is the growth-exponent of the associated energy functionals. This assumption on the one hand ensures that the expectation of the nonlinear q-capacity of the spherical holes is finite, and hence that the limit problem is well defined. On the other hand, it does not exclude the presence of balls with large radii, that can cluster up. We show however that the critical rescaling of the perforations is sufficient to ensure that no percolating-like structures appear in the limit.

Abstract Image

随机穿孔域中非线性德里赫特问题的均质化(关于穿孔大小的最小假设条件
在本文中,我们研究了定义在 \(\mathbb {R}^n\) 随机穿孔域上的变分椭圆 PDEs 系统的非线性 Dirichlet 问题的收敛性。假设穿孔是小球,其中心和半径由静止的短程标记点过程产生,我们在临界规模极限中得到了西奥拉内斯库和缪拉的经典著作(Res Notes Math III, 1982)中得到的额外项的平均非线性类似物。与 Giunti、Höfer 和 Velázquez (Commun Part Differ Equ 43(9):1377-1412, 2018) 最近为研究泊松方程而引入的随机设置类似,我们只要求随机半径具有有限的 \((n-q)\)-动量,其中 \(1<q<n\) 是相关能量函数的增长指数。这一假设一方面确保了球洞非线性q容量的期望值是有限的,从而确保了极限问题的定义。另一方面,它并不排除大半径球的存在,这些球可能会聚集在一起。然而,我们证明了穿孔的临界重缩足以确保在极限中不会出现类似于渗滤的结构。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Probability Theory and Related Fields
Probability Theory and Related Fields 数学-统计学与概率论
CiteScore
3.70
自引率
5.00%
发文量
71
审稿时长
6-12 weeks
期刊介绍: Probability Theory and Related Fields publishes research papers in modern probability theory and its various fields of application. Thus, subjects of interest include: mathematical statistical physics, mathematical statistics, mathematical biology, theoretical computer science, and applications of probability theory to other areas of mathematics such as combinatorics, analysis, ergodic theory and geometry. Survey papers on emerging areas of importance may be considered for publication. The main languages of publication are English, French and German.
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