Propagation of smallness in elliptic periodic homogenization

TL;DR

This paper establishes the first approximate three-ball inequality for solutions in elliptic periodic homogenization, enabling quantitative propagation of smallness and advancing the understanding of solution behavior in homogenized media.

Contribution

It introduces a novel approximate three-ball inequality for elliptic homogenization, utilizing Poisson kernel representation and interpolation techniques.

Findings

First such inequality in homogenization theory

Quantitative propagation of smallness demonstrated

Additional propagation results provided

Abstract

The paper is mainly concerned with an approximate three-ball inequality for solutions in elliptic periodic homogenization. We consider a family of second order operators $\mathcal{L}_\epsilon$ in divergence form with rapidly oscillating and periodic coefficients. It is the first time such an approximate three-ball inequality for homogenization theory is obtained. It implies an approximate quantitative propagation of smallness. The proof relies on a representation of the solution by the Poisson kernel and the Lagrange interpolation technique. Another full propagation of smallness result is also shown.