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Large-scale harmonic measures and nontangential maximal functions in periodic homogenization

Zhongwei Shen, Jinping Zhuge

Abstract

In this paper, we consider the elliptic operators $\mathcal{L}_\varepsilon = -\nabla\cdot (A(X/\varepsilon) \nabla )$ with periodic coefficients in a bounded domain $Ω$ without any local smoothness assumption on $A = A(Y)$, where $\varepsilon \ll \text{diam}(Ω)$ is a microscopic scale. Due to the irregularity of the coefficients at $\varepsilon$ scale, we introduce the correct forms of the large-scale nontangential maximal functions for the Dirichlet, Neumann and regularity problems that measure the behaviors of solutions at an $\varepsilon$ distance away from the boundary. The $L^p$ estimates uniform in $\varepsilon$ are established for these nontangential maximal functions for the same and optimal ranges of $p$ as the Laplace operator in the Lipschitz or $C^1$ domains. With some additional regularity assumption on the coefficients, the large-scale estimates combined with the small-scale estimates recover the classical full-scale estimates of the nontangential maximal functions. Our proofs are based on the notion of large-scale $\mathcal{L}_\varepsilon$-harmonic measures, the periodic structure of operators in the transversal direction to the boundaries, and the homogenization tools, including convergence rates and large-scale regularity.

Large-scale harmonic measures and nontangential maximal functions in periodic homogenization

Abstract

In this paper, we consider the elliptic operators with periodic coefficients in a bounded domain without any local smoothness assumption on , where is a microscopic scale. Due to the irregularity of the coefficients at scale, we introduce the correct forms of the large-scale nontangential maximal functions for the Dirichlet, Neumann and regularity problems that measure the behaviors of solutions at an distance away from the boundary. The estimates uniform in are established for these nontangential maximal functions for the same and optimal ranges of as the Laplace operator in the Lipschitz or domains. With some additional regularity assumption on the coefficients, the large-scale estimates combined with the small-scale estimates recover the classical full-scale estimates of the nontangential maximal functions. Our proofs are based on the notion of large-scale -harmonic measures, the periodic structure of operators in the transversal direction to the boundaries, and the homogenization tools, including convergence rates and large-scale regularity.
Paper Structure (26 sections, 42 theorems, 281 equations, 2 figures)

This paper contains 26 sections, 42 theorems, 281 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivations
  3. Statement of main results
  4. Proof sketch
  5. Notations
  6. Large-scale $\mathcal{L}_\varepsilon$-harmonic measure and Dirichlet problem
  7. $\mathcal{L}_\varepsilon$-harmonic measure
  8. Localization
  9. $L^2$ Rellich estimates
  10. Large-scale $\mathcal{L}_\varepsilon$-harmonic measure
  11. Dirichlet problem
  12. Regularity and Neumann problems
  13. Difference operator
  14. $L^2$ estimates
  15. $L^p$ estimates for $1<p<2$
  16. ...and 11 more sections

Key Result

Theorem 1.1

Assume that $A$ is real, symmetric and satisfies ellipticity and periodicity. Let $\Omega$ be a bounded Lipschitz domain and $f\in H^{1/2}(\partial\Omega)\cap C(\partial\Omega)$. Let $u_\varepsilon\in H^1(\Omega)$ be a weak solution of $\mathcal{L}_\varepsilon (u_\varepsilon)=0$ in $\Omega$ with the where $C$ depends on $d$, $p$, $\Lambda$ and $\Omega$. Moreover, if $\Omega$ is a bounded $C^1$ dom

Figures (2)

  • Figure 1: The flowchart of proofs
  • Figure 2: A graph domain

Theorems & Definitions (83)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Proposition 2.1
  • Proposition 2.2: KS11
  • Proposition 2.3: KS11
  • Remark 2.4
  • Proposition 2.5
  • Lemma 2.6
  • ...and 73 more