L^p gradient estimate for elliptic equations with high-contrast conductivities in Rⁿ

Uniform estimate for the solutions of elliptic equations with high-contrast conductivities in Rⁿ 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] is the diameter of each matrix block and ω²∈(0, 1] is the conductivity ratio of the disconnected matrix block subset to the connected sub-region. It is proved that the W^1,p norm of the elliptic solutions in the connected sub-region is bounded uniformly in ε, ω when ε≤ω, the L^p norm of the elliptic solutions in the whole space is bounded uniformly in ε, ω the W^1,p norm of the elliptic solutions in perforated domains is bounded uniformly in ε. However, the L^p norm of the second order derivatives of the solutions in the connected sub-region may not be bounded uniformly in ε, ω.