This work deals with a non linear inverse problem of reconstructing
an unknown boundary γ, the boundary conditions prescribed on γ being of Signorini type,
by using boundary measurements. The problem is turned into an optimal shape design one, by constructing
a Kohn & Vogelius-like cost function, the only minimum of which is proved to be the unknown boundary.
Furthermore, we prove that the derivative of this cost function with respect to a direction θ
depends only on the state u0, and not on its Lagrangian derivative u1(θ).