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We study the insulated conductivity problem with inclusions embedded in abounded domain in $\mathbb{R}^n$. The gradient of solutions may blow up as$\varepsilon$, the distance between inclusions, approaches to $0$. It was knownthat 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 $\varepsilon^{-1/2 + \beta}$ forsome $\beta > 0$. On the other hand, optimal values of $\beta$ have not beenidentified. In this paper, we prove that when the inclusions are balls, theoptimal value of $\beta$ is $[-(n-1)+\sqrt{(n-1)^2+4(n-2)}~]/4 \in (0,1/2)$ indimensions $n \ge 3$.

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