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