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

Article type: Research Article

Authors: Josien, Marc^{; *}

Affiliations: CERMICS, École des Ponts, 6 et 8 avenue Blaise Pascal, 77455 Marne-La-Vallée Cedex 2, France. E-mail: marc.josien@mis.mpg.de

Correspondence: [*] Corresponding author. E-mail: marc.josien@mis.mpg.de.

Abstract: This article is about the Zd-periodic Green function Gn(x,y) of the multiscale elliptic operator Lu=−div(A(n·)·∇u), where A(x) is a Zd-periodic, coercive, and Hölder continuous matrix, and n is a large integer. We prove here pointwise estimates on Gn(x,y), ∇xGn(x,y), ∇yGn(x,y) and ∇x∇yGn(x,y) in dimensions d⩾2. Moreover, we derive an explicit decomposition of this Green function, which is of independent interest. These results also apply for systems.

Keywords: Green function, periodic homogenization, multiscale problems

DOI: 10.3233/ASY-181504

Journal: Asymptotic Analysis, vol. 112, no. 3-4, pp. 227-246, 2019