  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.
