We consider a boundary value problem of the anti-plane elasticity in a domain
containing an inclusion which is nearly touching to the domain's boundary. We
assume that the domain and the inclusion are disks. By using the boundary
integral formulation for the interface problem 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. We also consider the boundary value problem
for the Lam\'{e} system in a circular domain containing a circular hole. We
show that the stress tensor blows up under the uniform boundary traction, as
the distance tends to zero. Additionally, we provide a Fourier series solution
in bipolar coordinates for the Lam\'{e} system in the whole plane with an
inclusion of core-shell geometry.