Reconstruction of Planar Conductivities in Subdomains from Incomplete Data

We consider the problem of recovering a sufficiently smooth isotropic conductivity from interior knowledge of the magnitude of the current density field $|J|$ generated by an imposed voltage potential f on the boundary. In any dimension $n\geq2$, we show that equipotential sets are global area minimizers in the conformal metric determined by $|J|$. In two dimensions, assuming the boundary voltage is almost two-to-one, we prove uniqueness of the minimization problem. This yields two results on reconstruction from incomplete data. In the first case, $|J|$ is known in all of $\Omega$, but the almost two-to-one f is know only on subintervals of the boundary. The second case assumes that $|J|$ is known only in an appropriate subdomain $\tilde{\Omega}$: our method works provided that $\tilde{\Omega}$ contains entire equipotential curves joining boundary points. Based on solving two point boundary value problems for the geodesic system, we give a procedure to determine whether $\tilde{\Omega}$ satisfies this pro...