Non-uniform elliptic equations in convex Lipschitz domains are concerned. The non-smooth domains consist of a periodic connected high permeability sub-region and a periodic disconnected matrix block subset with low permeability. Let ∈ ∈ (0,1] denote the size ratio of the matrix blocks to the whole domain and let ω2 ∈ (0,1] denote the permeability ratio of the disconnected matrix block subset to the connected sub-region. The W1,p norm for p ∈ (1, ∞) of the elliptic solutions in the high permeability sub-region is shown to be bounded uniformly in ω, ∈. However, the W1,p norm of the solutions in the low permeability subset may not be bounded uniformly in ω, ∈. Roughly speaking, if the sources in the low permeability subset are small enough, the solutions in that subset are bounded uniformly in ω, ∈. Otherwise the solutions cannot be bounded uniformly in ω, ∈. Relations between the sources and the variation of the solutions in the low permeability subset are also presented in this work.
|Number of pages||19|
|Journal||Nonlinear Analysis, Theory, Methods and Applications|
|State||Published - 1 Jan 2015|