Sources recovery from boundary data: A model related to electroencephalography

We are concerned with an inverse problem related to sources detection from boundary data in a 2D medium with piecewise constant conductivity. It stands as a 2D version of the inverse problem of electroencephalography, where pointwise sources model epilepsy foci, with the so-called multi-layer spherical model of the head (scalp, skull, brain). When overdetermined electrical measurements (potential and current flux) are available on the scalp, one wants to recover the current sources (conductivity defaults) located in the brain (inner boundary). This recovery issue reduces to a number of inverse problems, where the sources identification process makes use of best rational approximation in the disk, whereas the preliminary cortical mapping step (Cauchy type issue) relies on best constrained harmonic or analytic approximation in an annulus (bounded extremal problems).