The insulated conductivity problem, effective gradient estimates and the maximum principle

We consider the insulated conductivity problem with two unit balls as insulating inclusions, a distance of order $\epsilon$ apart. The solution $u$ represents the electric potential. In dimensions $n\ge 3$ it is an open problem to find the optimal bound on the gradient of $u$, the electric field, in the narrow region between the insulating bodies. Li-Yang recently proved a bound of order ${\epsilon }^{-(1-\gamma )\mathrm{/}2}$ for some $\gamma >0$. In this paper we use a direct maximum principle argument to sharpen the Li-Yang estimate for $n\ge 4$. Our method gives effective lower bounds on the best constant $\gamma$, which in particular approach $1$ as $n$ tends to infinity.