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 by ε, 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 order εfor n = 2, and is of order εfor some β > 0 when dimension n ≥ 3. In this paper, we generalize the above results for insulators with flatter boundaries near touching points.