Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantitative homogenization for log-normal coefficients

Nicolas Clozeau, Antoine Gloria, Siguang Qi

Abstract

We establish quantitative homogenization results for the popular log-normal coefficients. Since the coefficients are neither bounded nor uniformly elliptic, standard proofs do not apply directly. Instead, we take inspiration from the approach developed for the nonlinear setting by the first two authors and capitalize on large-scale regularity results by Bella, Fehrmann, and Otto for degenerate coefficients in order to leverage an optimal control (in terms of scaling and stochastic integrability) of oscillations and fluctuations.

Quantitative homogenization for log-normal coefficients

Abstract

We establish quantitative homogenization results for the popular log-normal coefficients. Since the coefficients are neither bounded nor uniformly elliptic, standard proofs do not apply directly. Instead, we take inspiration from the approach developed for the nonlinear setting by the first two authors and capitalize on large-scale regularity results by Bella, Fehrmann, and Otto for degenerate coefficients in order to leverage an optimal control (in terms of scaling and stochastic integrability) of oscillations and fluctuations.
Paper Structure (21 sections, 19 theorems, 137 equations)

This paper contains 21 sections, 19 theorems, 137 equations.

Table of Contents

  1. Main results
  2. Degenerate coefficients in stochastic homogenization
  3. Perturbative large-scale regularity
  4. Minimal radius and bounds on correctors
  5. Non-perturbative large-scale regularity
  6. Quantitative homogenization
  7. Large-scale Meyers estimates
  8. Proof of Theorem \ref{['th:Meyers']}
  9. Proof of Corollary \ref{['Lholefilling']}
  10. Proof of Theorem \ref{['th:annealedmeyers']}
  11. Control of correctors and of the minimal radius
  12. General strategy
  13. Proof of Proposition \ref{['prop:averages']}
  14. Proof of Proposition \ref{['prop:moment-en-dens']}
  15. Arguments for Theorems \ref{['th:bd-corr']} and \ref{['th:min-rad']}
  16. ...and 6 more sections

Key Result

Proposition 1.2

Let $a$ be as in Hypothesis hypo. Set $p_\diamond =d+ 1$. There exists a stationary $\frac{1}{8}$-Lipschitz field $r_\diamond$ such that for all $x\in\mathbb{R}^d$ and $r \ge r_\diamond(x)$ and which satisfies for some $C>0$. In what follows, for all $x\in \mathbb R^d$ we set $B_\diamond(x):=B_{r_\diamond(x)}(x)$. Let $0<\varepsilon \le 1$, we define for all $R\ge 1$ There exists $C>0$ (depending

Theorems & Definitions (24)

  • Proposition 1.2
  • Theorem 1.3: Quenched Meyers' estimates
  • Corollary 1.4: Hole-filling estimate in the large
  • Theorem 1.5: Annealed Meyers' estimate
  • Lemma 1.6
  • Definition 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Proposition 1.10: Large-scale Schauder estimates
  • Theorem 1.11: Quenched Calderón-Zygmund estimates
  • ...and 14 more