Calderón's inverse problem asks whether a conductivity inside a domain can be recovered from its Dirichlet-to-Neumann
operator, which records the boundary flux produced by every imposed boundary
potential. For a positive scalar conductivity 
on 
, the potential satisfies
 |
with prescribed boundary values. Sylvester and Uhlmann (1987) proved uniqueness for smooth scalar conductivities in dimensions at least three, while Astala and Päivärinta (2006) proved planar uniqueness for uniformly positive bounded measurable scalar conductivities.
For anisotropic conductivities, the analogous geometric formulation seeks a Riemannian metric. A diffeomorphism fixing the measured boundary leaves the data unchanged, so uniqueness can hold only up to this transformation.
