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We consider the conductivity problem in the presence of adjacent circularinclusions having arbitrary constant conductivity. When two inclusions getcloser and their conductivities degenerate to zero or infinity, the gradient ofthe solution can be arbitrary large. We characterize the gradient blow-up byderiving an explicit formula for the singular term of the solution in terms ofthe Lerch transcendent function. This derivation is valid for inclusions havingarbitrary constant conductivity. We illustrate our results with numericalcalculations.

Characterization of the gradient blow-up of the solution to the conductivity equation in the presence of adjacent circular inclusions