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

Abstract

We consider the insulated conductivity problem with two unit balls as insulating inclusions, a distance of order $\varepsilon$ 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 $\varepsilon ^{-(1-\gamma )/2}$ 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 $\gamma$, which in particular approach 1 as n tends to infinity.