Elliptic and parabolic equations in fractured media

The elliptic and the parabolic equations with Dirichlet boundary conditions in fractured media are considered. The fractured media consist of a periodic connected high permeability sub-region and a periodic disconnected matrix block subset with low permeability. Let ϵ∈(0,1]ϵ∈(0,1] denote the size ratio of the matrix blocks to the whole domain and let ω2∈(0,1]ω2∈(0,1] denote the permeability ratio of the disconnected subset to the connected sub-region. It is proved that the W1,pW1,p norm of the elliptic and the parabolic solutions in the high permeability sub-region are bounded uniformly in ω,ϵω,ϵ. However, the W1,pW1,p norm of the solutions in the low permeability subset may not be bounded uniformly in ω,ϵω,ϵ. For the elliptic and the parabolic equations in periodic perforated domains, it is also shown that the W1,pW1,p norm of their solutions are bounded uniformly in ϵ.