High-order Regularity Theory for High-contrast Elliptic Homogenization

TL;DR

The paper develops a global, high-order regularity theory for high-contrast elliptic homogenization, focusing on the equation $-\nabla\cdot \mathbf{a}(x)\nabla u=0$ with a random, stationary coefficient field. It introduces a coarse-grained ellipticity framework through $\mathbf{A}(U)$ and its homogenized limit $\overline{\mathbf{A}}$, using an adapted geometry defined by $\mathbf{q}_0$ and scale $\mathcal{X}_{\mathcal{H}}$, under axioms (P1)-(P3). The authors establish a non-iterative Caccioppoli inequality tailored to high-contrast media and prove a Liouville-type high-order regularity theorem linking $\mathcal{A}_k$ to $\overline{\mathcal{A}}_k$ with explicit decay rates depending on the random scale. They also show a dimension equality for the polynomial spaces, and develop an induction framework to obtain a global, quantitative regularity theory in this setting, enabling sharper homogenization error control in complex random media.

Abstract

The purpose of this article is to formulate and prove a global high-order regularity result within the high-contrast framework of elliptic homogenization. In order to achieve this, we also present a version of the high-contrast Caccioppoli inequality.

Theorems & Definitions (19)

• Proposition 1.2
• Theorem 1.3
• Proposition 2.1
• proof
• Proposition 2.2: Controlling the homogenization error
• proof
• Lemma 2.3: Controlling the Sobolev seminorms
• proof
• Proposition 2.4: Harmonic approximation in high contrast
• proof