PARTIAL UNIQUENESS AND OBSTRUCTION TO UNIQUENESS IN INVERSE PROBLEMS FOR ANISOTROPIC ELASTIC MEDIA

ANNA L. MAZZUCATO LIZABETH V. RACHELE

Abstract. We consider the dynamic inverse problem of uniquely determining the material parameters of bounded, three-dimensional elastic objects from displacement and traction measurements made at their surfaces. These surface measurements are modeled by the Dirichlet-to-Neumann map on a finite time interval. We first construct an obstruction to uniqueness for the inverse problem by transforming the medium via diffeomorphisms that fix the boundary to first order. We study the equivalence classes of elastic media under the action of pullback via these diffeomorphisms. This action leaves the Dirichlet-to-Neumann map invariant. We then show that any uniqueness result for a particular type of elastic media extends to (partial) uniqueness on the entire class. For example, we establish that uniqueness for isotropic elastic media implies the unique determination by surface measurements of three parameters of some anisotropic elastic media.