The following inverse problem stated by Calderon has many potential practical applications and had received a lot of attention in the past decades, see [Uh]. This is the problem of recovering the conductivity of a body from its Dirichlet-to-Neumann operator.
Given a domain with positive measurable function on it, the Dirichlet-to-Neumann operator connects the Dirichlet and Neumann boundary values of -harmonic functions in , defined on the domain. It is a pseudo-differential operator of order 1.
One can only recover one number (an integral of reciprocal of conductivity) in 1D. The problem is overdetermined in dimensions higher than 2. It was settled in these cases @ [KV] & [SU].
The dimensions of the measurement parameter and the unknown one fit precisely in the case of 2D. It was recently proved that the Dirichlet-to-Neumann map uniquely determines the conductivity in a 2D simply connected bounded domain if the conductivity is in a weighted space including differentiable functions in [N], and if it's measurable and bounded from 0 and infinity in [AP].
In the 2D case one can make sense of the operator for measurable conductivities (vs. differentiable), (see [AP]), using the Hilbert transform, defined later in the book, b/w the boundary values of harmonic functions and their conjugates.
In this book the problem is also stated for the planar electrical networks, that allows a different approach for solution of the inverse problem through discretization technique, that was implemented by Druskin et al. @ Schlumberger-Doll Research, see [BDK] and [BDMV].