Boundary determination for hybrid imaging from a single measurement

Abstract

We recover the conductivity σ at the boundary of a domain from a combination of Dirichlet and Neumann boundary data and generalized power/current density data at the boundary, from a single quite arbitrary set of data, in AET or CDII. The argument is elementary, algebraic and local. More generally, we consider the variable exponent p ⁢ ( ⋅ ) {p(\,\cdot\,)} -Laplacian as a forward model with the interior density data σ ⁢ | ∇ ⁡ u | q {\sigma|\nabla u|^{q}} , and find out that single measurement specifies the boundary conductivity when p - q ≥ 1 {p-q\geq 1} , and otherwise the measurement specifies two alternatives. We present heuristics for selecting between these alternatives. Both p and q may depend on the spatial variable x, but they are assumed to be a priori known. We illustrate the practical situations with numerical examples with the code available.