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Boundary estimates and Green function's expansion for elliptic systems with random coefficients

Li Wang, Qiang Xu

TL;DR

This work advances quantitative stochastic homogenization by providing sharp large-scale boundary regularity and Green-function expansion results for elliptic systems with stationary random coefficients. By integrating quenched boundary estimates, annealed Green-function decay, and boundary corrector analysis—including CLT-type scaling—the authors derive precise error bounds for the two-scale expansion of Green functions at the level of mixed derivatives. A key methodological contribution is a boundary-version extension of the Bella-Giunti-Otto lemma, coupled with a minimal-radius framework to leverage Gloria-Neukamm-Otto correctors in bounded domains. The results yield CLT-type fluctuations and robust, scale-invariant estimates applicable to boundary-layer problems, RVE/HMM analyses, and related homogenization contexts. Overall, the paper links boundary regularity, corrector theory, and Green-function expansions in a stochastic setting, offering practical implications for quantitative homogenization and numerical multiscale methods.

Abstract

We investigate boundary estimates for elliptic operators with stationary random coefficients exhibiting integrable correlations, arising from stochastic homogenization theory. As practical applications, we establish decay estimates for Green functions in both quenched and annealed senses. Furthermore, we derive notable annealed estimates for boundary correctors, including central limit theorem (CLT)-scaling type estimates. By extending the lemma of Bella, Giunti, and Otto [10] to accommodate boundary conditions, we ultimately obtain error estimates for the two-scale expansion of Green functions at the level of mixed derivatives, thereby establishing connections to other related fields.

Boundary estimates and Green function's expansion for elliptic systems with random coefficients

TL;DR

This work advances quantitative stochastic homogenization by providing sharp large-scale boundary regularity and Green-function expansion results for elliptic systems with stationary random coefficients. By integrating quenched boundary estimates, annealed Green-function decay, and boundary corrector analysis—including CLT-type scaling—the authors derive precise error bounds for the two-scale expansion of Green functions at the level of mixed derivatives. A key methodological contribution is a boundary-version extension of the Bella-Giunti-Otto lemma, coupled with a minimal-radius framework to leverage Gloria-Neukamm-Otto correctors in bounded domains. The results yield CLT-type fluctuations and robust, scale-invariant estimates applicable to boundary-layer problems, RVE/HMM analyses, and related homogenization contexts. Overall, the paper links boundary regularity, corrector theory, and Green-function expansions in a stochastic setting, offering practical implications for quantitative homogenization and numerical multiscale methods.

Abstract

We investigate boundary estimates for elliptic operators with stationary random coefficients exhibiting integrable correlations, arising from stochastic homogenization theory. As practical applications, we establish decay estimates for Green functions in both quenched and annealed senses. Furthermore, we derive notable annealed estimates for boundary correctors, including central limit theorem (CLT)-scaling type estimates. By extending the lemma of Bella, Giunti, and Otto [10] to accommodate boundary conditions, we ultimately obtain error estimates for the two-scale expansion of Green functions at the level of mixed derivatives, thereby establishing connections to other related fields.
Paper Structure (13 sections, 27 theorems, 262 equations)

This paper contains 13 sections, 27 theorems, 262 equations.

Table of Contents

  1. Introduction
  2. Motivation and assumptions
  3. Main results
  4. Organization of the paper
  5. Notations
  6. Quenched boundary estimates
  7. Hölder estimates
  8. Lipschitz estimates
  9. Green functions
  10. Boundary correctors
  11. Decay estimates of Green function's expansion
  12. A bootstrap argument
  13. Appendix

Key Result

Theorem 1

Let $\Omega\subset\mathbb{R}^d$ with $d\geq 2$ be a bounded domain, and $\varepsilon\in(0,1]$. Suppose that $\langle\cdot\rangle$ satisfies the spectral gap condition a:2 and the admissible coefficients satisfy the regularity condition a:3. Let $u_\varepsilon$ be a weak solution to $\mathcal{L}_\var

Theorems & Definitions (51)

  • Theorem 1: boundary regularity estimates at large scales
  • Theorem 2: boundary correctors
  • Theorem 3: CLT-scaling type estimates
  • Theorem 4: Green function's expansion
  • Remark 5
  • Remark 6
  • Remark 7
  • Remark 8
  • Proposition 9
  • lemma 10: extended correctors Gloria-Neukamm-Otto20
  • ...and 41 more