Lp gradient estimate for elliptic equations with high-contrast conductivities in RnRn

Uniform estimate for the solutions of elliptic equations with high-contrast conductivities in RnRn is concerned. The space domain consists of a periodic connected sub-region and a periodic disconnected matrix block subset. The elliptic equations have fast diffusion in the connected sub-region and slow diffusion in the disconnected subset. Suppose ϵ∈(0,1]ϵ∈(0,1] is the diameter of each matrix block and ω2∈(0,1]ω2∈(0,1] is the conductivity ratio of the disconnected matrix block subset to the connected sub-region. It is proved that the W1,pW1,p norm of the elliptic solutions in the connected sub-region is bounded uniformly in ϵ, ω  ; when ϵ≤ωϵ≤ω, the LpLp norm of the elliptic solutions in the whole space is bounded uniformly in ϵ, ω  ; the W1,pW1,p norm of the elliptic solutions in perforated domains is bounded uniformly in ϵ  . However, the LpLp norm of the second order derivatives of the solutions in the connected sub-region may not be bounded uniformly in ϵ, ω.