Carleman estimates and Calderón's inverse problem on manifolds
Mikko Salo

Abstract: We consider the imaging of anisotropic materials by electrical measurements. This inverse problem arises in Electrical Impedance Tomography (EIT), which has been proposed as a diagnostic method in medical imaging and nondestructive testing. The mathematical model is the anisotropic Calderón problem, which consists in determining a matrix of coefficients in an elliptic equation from boundary measurements of solutions.

In geometric terms, the problem is to determine a Riemannian metric from Cauchy data of harmonic functions on a manifold. Our approach is based on Carleman estimates. We characterize those Riemannian manifolds which admit a special limiting Carleman weight. By using these weights we construct complex geometrical optics solutions to elliptic equations, and prove uniqueness results in inverse problems for a class of Riemannian manifolds.

This is a joint work with D. Dos Santos Ferreira (Paris 13), C. Kenig(Chicago), and G. Uhlmann (Washington).