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

摘要

This article is about the Z(d)-periodic Green function G(n)(x, y) of the multiscale elliptic operator Lu =-div(A(n.).del u), where A(x) is a Z(d)-periodic, coercive, and Holder continuous matrix, and n is a large integer. We prove here pointwise estimates on G(n)(x, y), del(x)G(n)(x, y), del(y)G(n)(x, y) and del(x)del(y)G(n)(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.