Uniqueness for Inverse Elastic Medium Problems

We study the inverse boundary value problem and the inverse scattering problem for a homogeneous and isotropic linear elastic medium in R-3 with an inhomogeneous and anisotropic mass density given by an unknown symmetric matrix. We define the displacement to traction map associated to the boundary value problem and we prove the uniqueness of the mass density from the knowledge of this map. To do this we construct solutions to the corresponding conjugate Faddeev type equation and derive the existence of complex waves solutions to the Lame equation, the analogue to complex geometrical optics solutions in the inverse conductivity problem. We also prove the uniqueness of the mass density in the inverse scattering problem by reducing this problem to an inverse boundary value problem.