Gradient estimates for the perfect conductivity problem in anisotropic media

TL;DR

This paper derives optimal gradient bounds for the electric field in an anisotropic medium with closely spaced perfect conductors, revealing the singular behavior as their separation diminishes.

Contribution

It provides the first sharp gradient estimates for the perfect conductivity problem in anisotropic media, capturing the singularity as inclusions approach each other.

Findings

Optimal upper and lower gradient bounds established

Characterization of electric field singularity in anisotropic media

Results hold in any spatial dimension

Abstract

We study the perfect conductivity problem when two perfectly conducting inclusions are closely located to each other in an anisotropic background medium. We establish optimal upper and lower gradient bounds for the solution in any dimension which characterize the singular behavior of the electric field as the distance between the inclusions goes to zero.