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

Abstract

We study the insulated conductivity problem with inclusions embedded in a bounded domain in ${\mathbb{R}}^n$. The gradient of solutions may blow up as $\epsilon$, the distance between inclusions, approaches to $0$. It was known that the optimal blow-up rate in dimension $n=2$ is of order ${\epsilon }^{-1/2}$. It has recently been proved that in dimensions $n\ge 3$, an upper bound of the gradient is of order ${\epsilon }^{-1/2+\beta }$ for some $\beta >0$. On the other hand, optimal values of $\beta$ have not been identified. In this paper, we prove that when the inclusions are balls, the optimal value of $\beta$ is $[-(n-1)+\sqrt{(n-1{)}^2+4(n-2)}]/4\in (0,1/2)$ in dimensions $n\ge 3$.