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Calderon-Zygmund estimates for stochastic elliptic systems on bounded Lipschitz domains

Li Wang, Qiang Xu

Abstract

Concerned with elliptic operators with stationary random coefficients of integrable correlations and bounded Lipschitz domains, arising from stochastic homogenization theory, this paper is mainly devoted to studying Calderón-Zygmund estimates. As an application, we obtain the homogenization error in the sense of oscillation and fluctuation, respectively. These results are optimal up to a quantity $O(\ln(1/\varepsilon))$, which is caused by the quantified sublinearity of correctors in dimension two and the less smoothness of the boundary. In this paper, we find a novel form of \emph{minimal radius}, which is proved to be a suitable tool for quantitative stochastic homogenization on boundary value problems, when we adopt Gloria-Neukamm-Otto's strategy originally inspired by the pioneering work of Naddaf and Spencer.

Calderon-Zygmund estimates for stochastic elliptic systems on bounded Lipschitz domains

Abstract

Concerned with elliptic operators with stationary random coefficients of integrable correlations and bounded Lipschitz domains, arising from stochastic homogenization theory, this paper is mainly devoted to studying Calderón-Zygmund estimates. As an application, we obtain the homogenization error in the sense of oscillation and fluctuation, respectively. These results are optimal up to a quantity , which is caused by the quantified sublinearity of correctors in dimension two and the less smoothness of the boundary. In this paper, we find a novel form of \emph{minimal radius}, which is proved to be a suitable tool for quantitative stochastic homogenization on boundary value problems, when we adopt Gloria-Neukamm-Otto's strategy originally inspired by the pioneering work of Naddaf and Spencer.
Paper Structure (14 sections, 20 theorems, 312 equations, 1 figure)

This paper contains 14 sections, 20 theorems, 312 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation and basic ideas
  3. Main results and relation to previous works
  4. Organization of the paper
  5. Notations
  6. Preliminaries
  7. Calderón-Zygmund estimates
  8. Outline some basic ingredients
  9. Quenched estimates
  10. Annealed estimates
  11. Homogenization errors
  12. Strong norm estimates
  13. Weak norm estimates
  14. Appendix

Key Result

Theorem 3

Let $\Omega\subset\mathbb{R}^d$ with $d\geq 2$ be a bounded Lipschitz domain and $\varepsilon\in(0,1]$. Suppose that $\langle\cdot\rangle$ is stationary and satisfies spectral gap condition a:2, and the (admissible) coefficient additionally satisfies a:3 with the symmetry condition $a=a^*$. Let $u_\ holds for any $p>1$ satisfying $|\frac{1}{p}-\frac{1}{2}|\leq \frac{1}{2d}+\theta$ with $0<\theta\l

Figures (1)

  • Figure 1: A covering of the domain $\Omega$

Theorems & Definitions (53)

  • Definition 1: Lipschitz, $C^1$ domains Kenig94
  • Definition 2: $A_q$-weights
  • Theorem 3: Calderón-Zygmund estimates
  • Remark 4
  • Theorem 5: homogenization errors
  • lemma 6: extended correctors Gloria-Neukamm-Otto20
  • proof
  • lemma 7: qualitative theory
  • Remark 8
  • Remark 9
  • ...and 43 more