  In this paper, we deal with the inverse problem of the shape reconstruction
of cavities and inclusions embedded in a linear elastic isotropic medium from
boundary displacement's measurements. For, we consider a constrained
minimization problem involving a boundary quadratic misfit functional with a
regularization term that penalizes the perimeter of the cavity or inclusion to
be identified. Then using a phase-field approach we derive a robust algorithm
for the reconstruction of elastic inclusions and of cavities modeled as
inclusions with a very small elasticity tensor.
