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 epsilon E (0,1) be the periodic size of domain, epsilon mu E (0,1) the size ratio of a matrix block to the whole domain, and omega 2 E (1, oo) the permeability ratio of the disconnected matrix block subset to the connected sub-region of the domains. This work presented Lipschitz estimate uniformly in epsilon, mu, omega for the Green's functions and the solutions of the elliptic equations.