Gradient Estimates of Solutions to the Conductivity Problem with Flatter Insulators

We study the insulated conductivity problem with inclusions embedded in a bounded domain in R^(n).When the distance of inclusions,denoted byε,goes to 0,the gradient of solutions may blow up.When two inclusions are strictly convex,it was known that an upper bound of the blow-up rate is of orderε^(-1/2)for n=2,and is of orderε^(-1/2+β)for someβ>0 when dimension n≥3.In this paper,we generalize the above results for insulators with flatter boundaries near touching points.