The purpose of this project is to study uniqueness for the dynamic inverse problem for bounded, three-dimensional, anisotropic elastic media, that is, to address the question: do displacement-traction measurements at the surface of an elastic object uniquely determine the density and elastic properties of the interior? In particular, the aim is to study whether the theory predicts any ambiguities in the sensing of the internal properties from surface measurements, and, if so, if it is possible to characterize features of the elastic media that are ``hidden'' from the surface. The PI plans (in joint work with Anna Mazzucato) to present such an ambiguity, an obstruction to uniqueness, by transforming the medium via diffeomorphisms that fix the boundary to first order. That is, the plan is to show that elastic media have the same Dirichlet-to-Neumann map if they lie in the same orbit under the action of pullback via these diffeomorphisms. A consequence of the obstruction to uniqueness is that the parameter identification problem for large classes of anisotropic elastic media may be solved, in part, if it has been solved for simpler classes. In particular, the PI plans to extend uniqueness results for isotropic elastodynamics to classes of (possibly composite) anisotropic elastic media. To apply the partial uniqueness result, it is important to develop tools to identify whether a given elastic medium is in the orbit of a certain class of elastic media (for example, isotropic). The PI plans to begin addressing this problem by giving a pointwise characterization of the orbits of general anisotropic elastic media under this action.
Imaging technologies aid physicians in detecting and diagnosing abnormal tissue. Techniques using ultrasound, for example, have been developed recently for sensing how the interior points of biological tissues respond when movement is initiated at the surface. Information about how the tissue moves can be used to identify regions that are stiffer than their surroundings, and since tumors are often encapsulated in tissue which is stiffer than normal, this tool can have important medical applications. Since stiffness is an elastic feature, mathematicians can apply methods from the field of differential equations to contribute to this study of this inverse problem. The PI plans to work with undergraduate and graduate students from the host institution to present graphical results and instructive text on the internet for this inverse problem. This award is supported by the NSF ADVANCE Program. The overall mission of the ADVANCE Program is to increase the participation of women in the scientific and engineering workforce through the increased representation and advancement of women in academic science and engineering careers.