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

Abstract

This article is about the ${\mathbb{Z}}^{\mathit{d}}$ -periodic Green function  ${\mathit{G}}_{\mathit{n}}\mathrm{(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{)}$ of the multiscale elliptic operator  $\mathit{L}\mathit{u}=-div\mathrm{(}\mathit{A}\mathrm{(}\mathit{n}·\mathrm{)}·\nabla \mathit{u}\mathrm{)}$ , where  $\mathit{A}\mathrm{(}\mathit{x}\mathrm{)}$ is a ${\mathbb{Z}}^{\mathit{d}}$ -periodic, coercive, and Hölder continuous matrix, and n is a large integer. We prove here pointwise estimates on  ${\mathit{G}}_{\mathit{n}}\mathrm{(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{)}$ , ${\nabla }_{\mathit{x}}{\mathit{G}}_{\mathit{n}}\mathrm{(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{)}$ , ${\nabla }_{\mathit{y}}{\mathit{G}}_{\mathit{n}}\mathrm{(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{)}$ and  ${\nabla }_{\mathit{x}}{\nabla }_{\mathit{y}}{\mathit{G}}_{\mathit{n}}\mathrm{(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{)}$ in dimensions  $\mathit{d}⩾2$ . Moreover, we derive an explicit decomposition of this Green function, which is of independent interest. These results also apply for systems.