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Homogenization of the stochastic double-porosity model

Elise Bonhomme, Mitia Duerinckx, Antoine Gloria

TL;DR

The paper tackles stochastic homogenization of elliptic equations in high-contrast double-porosity media in a resonant regime. It develops a rigorous framework under weak geometric assumptions, proving qualitative homogenization and a robust corrector theory via resonant cell problems, extended by an ergodic theorem for resonant equations. Under stronger regularity, it provides sharp quantitative two-scale expansion error estimates, introducing flux and inclusion correctors to achieve optimal rates, and it extends the analysis to oscillatory forcing data. The results substantially broaden the class of admissible random inclusions (beyond uniform bounds and separation) and supply practical error bounds for homogenized models of fractured media and metamaterials.

Abstract

This work is devoted to the homogenization of elliptic equations in high-contrast media in the so-called 'double-porosity' resonant regime, for which we solve two open problems of the literature. First, we prove qualitative stochastic homogenization under very weak conditions, which cover the case of inclusions that are not uniformly bounded or separated. Second, under stronger assumptions, we provide sharp error estimates for the two-scale expansion. The main difficulty is related to the loss of integrability of the control in the resonant zones.

Homogenization of the stochastic double-porosity model

TL;DR

The paper tackles stochastic homogenization of elliptic equations in high-contrast double-porosity media in a resonant regime. It develops a rigorous framework under weak geometric assumptions, proving qualitative homogenization and a robust corrector theory via resonant cell problems, extended by an ergodic theorem for resonant equations. Under stronger regularity, it provides sharp quantitative two-scale expansion error estimates, introducing flux and inclusion correctors to achieve optimal rates, and it extends the analysis to oscillatory forcing data. The results substantially broaden the class of admissible random inclusions (beyond uniform bounds and separation) and supply practical error bounds for homogenized models of fractured media and metamaterials.

Abstract

This work is devoted to the homogenization of elliptic equations in high-contrast media in the so-called 'double-porosity' resonant regime, for which we solve two open problems of the literature. First, we prove qualitative stochastic homogenization under very weak conditions, which cover the case of inclusions that are not uniformly bounded or separated. Second, under stronger assumptions, we provide sharp error estimates for the two-scale expansion. The main difficulty is related to the loss of integrability of the control in the resonant zones.

Paper Structure

This paper contains 20 sections, 10 theorems, 228 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Qualitative theory
  3. Quantitative error estimates
  4. Discussion of the assumptions
  5. Qualitative theory
  6. Compactness result
  7. Ergodic theorem for resonant equations
  8. Proof of Theorem \ref{['th:main']}
  9. Proof of Theorem \ref{['th:corr-qual']}
  10. Proof of quantitative error estimates
  11. Discussion of the assumptions
  12. Proof of Lemma \ref{['lem:ass1']}
  13. Proof of (i)
  14. Proof of (ii)
  15. Proof of (iii)
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let the random inclusion process $F=\cup_nI_n\subset \mathbb R^d$ satisfy Assumptions ass0, ass1, and ass2 for some $1<p \le 2$ and $q=p'$. Given a bounded Lipschitz domain $D\subset\mathbb R^d$, for all $\varepsilon>0$, let $F_\varepsilon(D):=\cup\{\varepsilon I_n:\varepsilon I_n\subset D\}$, let $ Then we have almost surely where:

Figures (8)

  • Figure 2:
  • Figure 3: This illustrates the construction of local neighborhoods at a boundary point, say $x_*=0\in\partial I_n'$.
  • Figure : (a) model $F_1$
  • Figure : (a) model $F_1$
  • Figure : (c.1) model $F_3$
  • ...and 3 more figures

Theorems & Definitions (14)

  • Theorem 1.1: Qualitative homogenization
  • Lemma 1.2: Correctors; see JKO94
  • Theorem 1.3: Corrector result
  • Remark 1.4: Extension to oscillatory data
  • Lemma 1.5: Flux and inclusion correctors; e.g. MR4103433bernou2023homogenization
  • Theorem 1.6
  • Lemma 1.8: Validity of \ref{['ass1']}
  • Lemma 1.9: Validity of \ref{['ass2']}
  • Lemma 2.1
  • Lemma 2.2
  • ...and 4 more