  We consider the conductivity problem for a homogeneous body with an inclusion
of a different, but known, conductivity. Our interest concerns the associated
shape derivative, i.e., the derivative of the corresponding electrostatic
potential with respect to the shape of the inclusion. For a smooth inclusion it
is known that the shape derivative is the solution of a specific inhomogeneous
transmission problem. We show that this characterization of the shape
derivative is also valid when the inclusion is a polygonal domain, but due to
singularities at the vertices of the polygon, the shape derivative fails to
belong to $H^1$ in this case.
