Elliptic equations in highly heterogeneous porous media

Uniform estimate and convergence for highly heterogeneous elliptic equations are concerned. The domain considered consists of a connected fractured subregion (with high permeability) and a disconnected matrix block subregion (with low permeability). Let ε denote the size ratio of one matrix block to the whole domain and let the permeability ratio of the matrix block region to the fractured region be of the order ε2. In the fractured region, uniform Hölder and uniform Lipschitz estimates in ε of the elliptic solutions are derived; the convergence of the solutions in L∞ norm is obtained as well.