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Regularity theorems for random elliptic operators on domains

Peter Bella, Julian Fischer, Marc Josien, Claudia Raithel

Abstract

Regularity theorems à la Avellaneda-Lin are an indispensable part of the modern quantitative theory of stochastic homogenization. While interior regularity results for random elliptic operators have been available for a while, on general smooth domains the existing theory has until recently remained limited to Lipschitz estimates. We establish $C^{1,α}$ regularity results for random elliptic operators on bounded sufficiently smooth domains, as well as for scalar problems on convex polytopes. We, furthermore, prove a number of auxiliary results typically employed in the derivation of fluctuation bounds, such as a weighted Meyers estimate.

Regularity theorems for random elliptic operators on domains

Abstract

Regularity theorems à la Avellaneda-Lin are an indispensable part of the modern quantitative theory of stochastic homogenization. While interior regularity results for random elliptic operators have been available for a while, on general smooth domains the existing theory has until recently remained limited to Lipschitz estimates. We establish regularity results for random elliptic operators on bounded sufficiently smooth domains, as well as for scalar problems on convex polytopes. We, furthermore, prove a number of auxiliary results typically employed in the derivation of fluctuation bounds, such as a weighted Meyers estimate.

Paper Structure

This paper contains 22 sections, 16 theorems, 157 equations.

Table of Contents

  1. Introduction
  2. Boundary layers in periodic and stochastic homogenization
  3. Discussion of main results
  4. Assumptions
  5. Assumptions on the domain.
  6. Assumptions on the ensemble.
  7. A suboptimal decay estimate for the boundary layer corrector
  8. Regularity for random $a^\varepsilon$-harmonic functions on regular domains
  9. A large-scale weighted Meyers estimate
  10. Regularity for $a^\varepsilon$-harmonic functions on convex polytopes
  11. Argument for Proposition \ref{['intermediate_lemma']}: A suboptimal decay estimate for the boundary layer
  12. Proof of Proposition \ref{['intermediate_lemma']}
  13. Proof of Lemma \ref{['Hardy']}: A weighted Hardy inequality
  14. Argument for Theorem \ref{['large_scale_reg_domain']}: Regularity for random $a$-harmonic functions on regular domains
  15. Proof of Theorem \ref{['large_scale_reg_domain']}
  16. ...and 7 more sections

Key Result

Proposition 3

Adopt the assumptions (A1)-(A4), let $0<\varepsilon\leq 1$, and let ${\mathcal{O}}\subseteq \mathbb{R}^d$ be either a bounded $C^{1,1}$-domain or $\mathbb{H}^d_+$. Let $\theta^\varepsilon\in H^1_{loc}({\mathcal{O}})$ denote the unique weak solution to defn_theta with $\lim_{r\rightarrow\infty} { \vc holds for any $x_0 \in {\mathcal{O}}$. In case of ${\mathcal{O}}=\mathbb{H}^d_+$, the random field

Theorems & Definitions (31)

  • Remark 1: Scalar PDEs versus systems
  • Remark 2: The case $d=2$
  • Proposition 3
  • Theorem 4
  • Corollary 5
  • Corollary 6
  • Proposition 7: Weighted Meyers Estimate
  • Theorem 8
  • Lemma 9
  • Lemma 10
  • ...and 21 more