Uniform bound and convergence for elliptic homogenization problems

Uniform bound and convergence for the solutions of elliptic homogenization problems are concerned. The problem domain has a periodic microstructure; it consists of a connected subregion with high permeability and a disconnected matrix block subset with low permeability. Let ϵ∈ (0 , 1) denote the size ratio of the period to the whole domain, and let ω²∈ (0 , 1) denote the permeability ratio of the disconnected matrix block subset to the connected subregion. For elliptic equations with diffusion depending on the permeability, the elliptic solutions are smooth in the connected subregion but change rapidly in the disconnected matrix block subset. More precisely, the solutions in the connected subregion can be bounded uniformly in ω, ϵ in Hölder norm, but not in the matrix block subset. It is known that the elliptic solutions converge to a solution of some homogenized elliptic equation as ω, ϵ converge to 0. In this work, the L^p convergence rate for p∈ (2 , ∞] is derived. Depending on strongly coupled or weakly coupled case, the convergence rate is related to the factors ω,ϵ,ωϵ for the former and related to the factors ω, ϵ for the latter.