Fault detection in an anisotropic elastic medium

We study a two-dimensional model of elastic dislocations motivated by applications in geophysics. In this model, the displacement satis es the system of linear elasticity with mixed displacement-traction homogeneous boundary conditions in the complement of an open curve in a bounded planar domain, and has a speci ed jump, the slip, across the curve, while the traction is continuous there. The sti ness tensor is allowed to be anisotropic and inhomogeneous. We prove well-posedness of the direct problem in a variational setting, assuming the coe cients are Lipschitz continuous. Using unique continuation arguments, we then establish uniqueness in the inverse problem of determining the dislocation curve and the slip from a single measurement of the displacement on an open patch of the traction-free part of the boundary. Uniqueness holds when the elasticity operators admits a suitable decomposition and the curve satis es additional geometric assumptions. This work extends the results in Arch. Ration. Mech. Anal., 236(1):71 111, 2020, and in Preprint arXiv:2004.00321 to the anisotropic case.