L-p gradient estimate for elliptic equations with high-contrast conductivities in R-n

Abstract

Uniform estimate for the solutions of elliptic equations with high-contrast conductivities R-n 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 epsilon is an element of (0, 1] is the diameter of each matrix block and omega(2) is an element of a (0, 1] is the conductivity ratio of the disconnected matrix block subset to the connected sub-region. It is proved that the W-1,W-p norm of the elliptic solutions in the connected sub-region is bounded uniformly in epsilon, is an element of, on when epsilon <= is an element of, the L-p norm of the elliptic solutions in the whole space is bounded uniformly in epsilon, omega; the W-1,W-p norm of the elliptic solutions in perforated domains is bounded uniformly in epsilon. However, the L-p norm of the second order derivatives of the solutions in the connected sub-region may not be bounded uniformly in epsilon, omega. (C) 2016 Elsevier Inc. All rights reserved.