Stable Determination of an Inclusion in an Elastic Body by Boundary Measurements

We consider the inverse problem of identifying an unknown inclusion contained in an elastic body by the Dirichlet-to-Neumann map. The body is made by linearly elastic, homogeneous, and isotropic material. The Lame moduli of the inclusion are constant and different from those of the surrounding material. Under mild a priori regularity assumptions on the unknown defect, we establish a logarithmic stability estimate. Main tools are propagation of smallness arguments based on three-spheres inequality for solutions to the Lame system and a refined asymptotic analysis of the fundamental solution of the Lame system in presence of an inclusion which shows surprising features.