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

TL;DR

This paper improves gradient estimates for the electric potential in the insulated conductivity problem with two close insulating spheres, providing sharper bounds and insights into the behavior as the dimension increases.

Contribution

It introduces a direct maximum principle method to refine existing bounds on the electric field gradient in high dimensions for the insulated conductivity problem.

Findings

Sharpened bounds on the electric field gradient for $n \\ge 4$

Effective lower bounds on the constant approaching 1 as dimension increases

Method applicable to high-dimensional insulated conductivity problems

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 the best constant $\gamma$, which in particular approach $1$ as $n$ tends to infinity.