Optimal 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 $\mathbb{R}^n$. The gradient of solutions may blow up as $\varepsilon$, the distance between inclusions, approaches to $0$. It was known that the optimal blow up rate in dimension $n = 2$ is of order $\varepsilon^{-1/2}$. It has recently been proved that in dimensions $n \ge 3$, an upper bound of the gradient is of order $\varepsilon^{-1/2 + \beta}$ 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 $n \ge 3$.