Asymptotic behaviour of Green functions of divergence form operators with periodic coefficients

TL;DR

This paper investigates the asymptotic behavior of Green functions for divergence form operators with periodic, coercive, and bounded coefficients, focusing on their behavior at infinity and at the origin.

Contribution

It provides new insights into the asymptotic properties of Green functions for periodic divergence form operators, enhancing understanding of their behavior in different regimes.

Findings

Green functions exhibit specific asymptotic decay at infinity.

Green functions have characteristic singularities at the origin.

Results apply to a broad class of periodic elliptic operators.

Abstract

This article is concerned with the asymptotic behaviour, at infinity and at the origin, of Green functions of operators of the form $Lu = -\text{div} (A \nabla u),$ where $A$ is a periodic, coercive and bounded matrix.