Gradient estimates of solutions to the insulated conductivity problem in dimension greater than two

We study the insulated conductivity problem with inclusions embedded in a bounded domain in R-n. The gradient of solutions may blow up as epsilon, the distance between inclusions, approaches to 0. An upper bound for the blow up rate was proved to be of order epsilon(-1/2). The upper bound was known to be sharp in dimension n = 2. However, whether this upper bound is sharp in dimension n >= 3 has remained open. In this paper, we improve the upper bound in dimension n >= 3 to be of order epsilon(-1/2+beta), for some beta > 0.