Lipschitz estimate for elliptic equations with oscillatory coefficients

Abstract

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}\in(0, 1) $ be the periodic size of domain, $ {\epsilon}\mu\in(0, 1) $ the size ratio of a matrix block to the whole domain, and $ \omega^2\in(1, \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 $ {\epsilon}, \mu, \omega $ for the Green's functions and the solutions of the elliptic equations.

Keywords

• lipschitz estimate
• permeability
• periodic size
• green's function
• oscillatory coefficient