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

Abstract

This article is about the ${\mathbb{Z}}^d$-periodic Green function $G_n(x,y)$ of the multiscale elliptic operator $Lu=-\mathrm{div}\left(A(n\cdot )\cdot \nabla u\right)$, where $A(x)$ is a ${\mathbb{Z}}^d$-periodic, coercive, and Hölder 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 $d\ge 2$. Moreover, we derive an explicit decomposition of this Green function, which is of independent interest. These results also apply for systems.

20 pages, 1 figure