Electric field concentration in the presence of an inclusion with eccentric core-shell geometry

TL;DR

This paper investigates the intense electric field concentration around eccentric core-shell inclusions in two-dimensional conductivity problems, deriving asymptotic formulas and integral representations for nearly touching configurations.

Contribution

It introduces new asymptotic and integral formulas for electric field concentration in eccentric core-shell geometries with nearly touching boundaries.

Findings

Derived asymptotic formulas for the solution.

Established integral formulas involving image line charges.

Analyzed gradient blow-up in nearly touching inclusions.

Abstract

In this paper we analyze the gradient blow-up of the solution to the conductivity problem in two dimensions in the presence of an inclusion with eccentric core-shell geometry. Assuming that the core and shell have circular boundaries that are nearly touching, we derive an asymptotic formula for the solution in terms of the single and double layer potentials with image line charges. We also deduce an integral formula with image line charges for the problem relating to two nearly touching separated conductors.