  We deal with the geometrical inverse problem of the shape reconstruction of
cavities in a bounded linear isotropic medium by means of boundary data. The
problem is addressed from the point of view of optimal control: the goal is to
minimize in the class of Lipschitz domains a Kohn-Vogelius type functional with
a perimeter regularization term which penalizes the perimeter of the cavity to
be reconstructed. To solve numerically the optimization problem, we use a
phase-field approach, approximating the perimeter functional with a
Modica-Mortola relaxation and modeling the cavity as an inclusion with a very
small elastic tensor. We provide a detailed analysis showing the robustness of
the algorithm through some numerical experiments.
