Uniform Calderón-Zygmund estimates in multiscale elliptic homogenization

TL;DR

This work addresses the problem of obtaining uniform regularity for multiscale elliptic equations with coefficients $A_ε(x)=A(x/ε_1,...,x/ε_n)$ that are 1-periodic in each fast variable. It introduces a novel combination of Dirichlet's simultaneous Diophantine approximation, a reperiodization technique, iterative (reiterated) homogenization, and a large-scale real-variable framework to obtain uniform gradient bounds for all $p∈(1,∞)$ that are independent of the scales, and extends to quasiperiodic coefficients without Diophantine assumptions. A quantitative scale-reduction mechanism separates scales one by one, enabling reduced problems with fewer scales and explicit error terms, which underpins interior $W^{1,p}$ estimates and large-scale (mesoscopic) Lipschitz bounds. These results provide robust, scale-free regularity in general multiscale and quasiperiodic media, with implications for diffusion and transport in complex composite materials.

Abstract

This paper is concerned with the elliptic equation $-\text{div} (A_\varepsilon \nabla u_\varepsilon) = \text{div} f$ in a bounded $C^1$ domain, where $A_\varepsilon$ takes a form of $A_\varepsilon(x) = A(x/\varepsilon_1, x/\varepsilon_2,\cdots, x/\varepsilon_n)$, with $A(y_1,y_2,\cdots,y_n)$ being 1-periodic in each $y_i$. We prove the uniform Calderón-Zygmund estimate, namely, the uniform $L^p$ boundedness of the linear map $f\mapsto \nabla u_\varepsilon$ for any $p\in (1,\infty)$ with a constant independent of small parameters $(\varepsilon_1,\varepsilon_2,\cdots, \varepsilon_n) \in (0,1]^n$. Our result includes the uniform Calderón-Zygmund estimate in quasiperiodic elliptic homogenization (even without the Diophantine condition), which was previously unknown. The proof novelly combines the Dirichlet's theorem on the simultaneous Diophantine approximation from number theory, a technique of reperiodization, reiterated periodic homogenization and a large-scale real-variable argument. Using the idea of reperiodization, we also obtain some large-scale or mesoscopic-scale Lipschitz estimates.

Theorems & Definitions (20)

• Theorem 1.1
• Corollary 1.2
• Theorem 2.1
• Theorem 3.1: Dirichlet (1842)
• Theorem 3.2
• Remark 3.3
• Remark 3.4
• Theorem 3.5
• proof : Proof of Theorem \ref{['th-app']}
• proof : Proof of Theorem \ref{['th-app-n']}