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

TL;DR

This paper develops large-scale harmonic measure and nontangential maximal function estimates for elliptic operators with periodic, possibly irregular coefficients in homogenization, extending classical boundary behavior analysis to microscopic scales.

Contribution

It introduces large-scale nontangential maximal functions for homogenization problems with irregular coefficients and establishes uniform $L^p$ estimates across scales.

Findings

Established uniform $L^p$ estimates for large-scale nontangential maximal functions.

Extended classical boundary estimates to operators with irregular periodic coefficients.

Connected large-scale estimates with classical results under additional regularity assumptions.

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 $\Omega$ without any local smoothness assumption on $A = A(Y)$, where $\varepsilon \ll \text{diam}(\Omega)$ 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.