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Homogenization for the Poisson equation in domains perforated by random closed sets

Naoto Sato

Abstract

We study the homogenization of the Poisson equation in randomly perforated domains and obtain the strange term effect in the homogenized equation. The perforations are modeled by rescaled germ-grain processes, and the main assumption is stationarity of the capacities of the holes. We emphasize that the potential in the homogenized equation is constant, despite the possibly nonstationary spatial distribution of the holes. We also establish corrector results.

Homogenization for the Poisson equation in domains perforated by random closed sets

Abstract

We study the homogenization of the Poisson equation in randomly perforated domains and obtain the strange term effect in the homogenized equation. The perforations are modeled by rescaled germ-grain processes, and the main assumption is stationarity of the capacities of the holes. We emphasize that the potential in the homogenized equation is constant, despite the possibly nonstationary spatial distribution of the holes. We also establish corrector results.
Paper Structure (11 sections, 15 theorems, 164 equations)

This paper contains 11 sections, 15 theorems, 164 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Capacity
  5. Marked point processes
  6. Statement of the main result
  7. Proof of Theorem \ref{['thm: main thm']}
  8. Lemmas from
  9. Proof of Lemma \ref{['thm: construction of osc fct']}
  10. Corrector results
  11. Appendix

Key Result

Theorem 1.1

Let $\{(z_{i}, K_{i})\}_{i = 1}^{\infty}$ be a countable collection of random pairs $(z_{i}, K_{i})$, where $z_{i} \in \mathbb{R}^{d}$ and $K_{i} \subseteq \mathbb{R}^{d}$ is an arbitrary compact set centered at the origin. Suppose that the number of points $z_{i}$ in any cube has a finite first mom (the domain $W$ serves as an observation window of the processes $\{\varepsilon z_{i}\}_{i}$). Supp

Theorems & Definitions (35)

  • Theorem 1.1: a special case of Example \ref{['thm: c alpha']}
  • Definition 2.1
  • Lemma 2.2: MR817985
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Theorem 3.1
  • Corollary 3.2
  • proof
  • ...and 25 more