Lipschitz estimate for elliptic equations with oscillatory coefficients

Regularity for elliptic equations with oscillatory coefficients was concerned. Problem domains were periodic and consisted of a connected region with normal permeability and a disconnected matrix block subset with high permeability. Coefficients of the elliptic equations depending on the permeability of the domains were highly oscillatory. Let $ϵ\in (0,1)$ be the periodic size of domain, $ϵ\mu \in (0,1)$ the size ratio of a matrix block to the whole domain, and ${\omega }^2\in (1,\mathrm{\infty })$ the permeability ratio of the disconnected matrix block subset to the connected sub-region of the domains. This work presented Lipschitz estimate uniformly in $ϵ,\mu ,\omega$ for the Green's functions and the solutions of the elliptic equations.