Boundary determination for hybrid imaging from a single measurement

We recover the conductivity $\sigma$ at the boundary of a domain from a combination of interior and boundary data, with a single quite arbitrary measurement, in AET or CDII. The argument is elementary and local. More generally, we consider the variable exponent $p(\cdot)$-Laplacian as a forward model with the interior data $\sigma |\nabla u|^q$, and find out that single measurement specifies the boundary conductivity when $p-q \ge 1$, 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.