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We consider a boundary value problem for the conductivity equation in abounded domain containing an inclusion which is nearly touching to the domain'sboundary. We assume that the domain and the inclusion are disks withconductivity jump on the boundary of the inclusion. By using the layerpotential technique and adopting the bipolar coordinates, we derive theasymptotic formulas which explicitly describe the gradient blow-up of thesolution as the distance between the inclusion and the domain's boundary tendsto zero. It turns out that the gradient blow-up term can be identified with theelectric field generated by certain kind of virtual line charges supported online segments outside of the domain; thereby, the gradient blow-up iscompletely characterized in terms of both of Neumann-type and Dirichlet-typeboundary conditions, conductivities and geometric parameters.

Characterization of stress concentration in two-dimensional boundary value problems: Neumann-type and Dirichlet-type