  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.
