Decomposition and pointwise estimates of periodic Green functions of some elliptic equations with periodic oscillatory coefficients

TL;DR

This paper derives pointwise estimates and an explicit decomposition for the periodic Green function of multiscale elliptic operators with oscillatory coefficients, applicable in multiple dimensions and systems.

Contribution

It provides new pointwise bounds and a decomposition of the Green function for elliptic operators with periodic oscillatory coefficients, extending understanding of their behavior.

Findings

Established pointwise estimates for Green functions and their derivatives.

Derived an explicit decomposition of the Green function.

Results applicable to systems and higher dimensions.

Abstract

This article is about the $\mathbb{Z}^d$-periodic Green function $G_n(x,y)$ of the multiscale elliptic operator $Lu=-{\rm div}\left( A(n\cdot) \cdot \nabla u \right)$, where $A(x)$ is a $\mathbb{Z}^d$-periodic, coercive, and H\"older continuous matrix, and $n$ is a large integer. We prove here pointwise estimates on $G_n(x,y)$, $\nabla_x G_n(x,y)$, $\nabla_y G_n(x,y)$ and $\nabla_x \nabla_y G_n(x,y)$ in dimensions $d \geq 2$. Moreover, we derive an explicit decomposition of this Green function, which is of independent interest. These results also apply for systems.