  An inverse problem of elasticity of $n$ elastic inclusions embedded into an
elastic half-plane is analyzed. The boundary of the half-plane is free of
traction. The half-plane and the inclusions are subjected to antiplane shear,
and the conditions of ideal contact hold in the interfaces between the
inclusions and the half-plane. The shapes of the inclusions are not prescribed
and have to be determined by enforcing uniform stresses inside the inclusions.
The method of conformal mappings from a slit domain onto the $(n+1)$-connected
physical domain is worked out. It is shown that to recover the map and
therefore the inclusions shapes, one needs to solve a vector Riemann-Hilbert
problem on a genus-$n$ hyperelliptic surface. In a particular case of loading
of a single inclusion in a half-plane, the problem is equivalent to two scalar
Riemann-Hilbert problems on two slits on an elliptic surface. In addition to
three parameters of the model the conformal map possesses a free geometric
parameter. Results of numerical tests which show the impact of these parameters
on the inclusion shape are presented.
