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 ε apart. The solution u represents the electric potential. In dimensions n≥ 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 ε^-(1-γ)/2 for some γ> 0. In this paper we use a direct maximum principle argument to sharpen the Li-Yang estimate for n≥ 4. Our method gives effective lower bounds on γ, which in particular approach 1 as n tends to infinity.