Exact solutions for the insulated and perfect conductivity problems with concentric balls

TL;DR

This paper derives exact solutions for insulated and perfect conductivity problems with concentric balls, revealing no electric field concentration in the former and sharp singularities in the latter, highlighting structural differences in composite materials.

Contribution

It provides the first exact solutions for these problems with concentric balls, showing the distinct behavior of electric fields in insulated versus perfect conductors.

Findings

No electric field concentration in insulated case

Sharp singularity in electric field for perfect conductivity

Concentric balls are optimal for insulated composites

Abstract

The insulated and perfect conductivity problems arising from high-contrast composite materials are considered in all dimensions. The solution and its gradient, respectively, represent the electric potential and field. The novelty of this paper lies in finding exact solutions for the insulated and perfect conductivity problems with concentric balls. Our results show that there appears no electric field concentration for the insulated conductivity problem, while the electric field for the perfect conductivity problem exhibits sharp singularity with respect to the small distance between interfacial boundaries of the interior and exterior balls. This discrepancy reveals that concentric balls is the optimal structure of insulated composites, but not for superconducting composites.