  The aim of hybrid inverse problems such as Acousto-Electric Tomography or
Current Density Imaging is the reconstruction of the electrical conductivity in
a domain that can only be accessed from its exterior. In the inversion
procedure, the solutions to the conductivity equation play a central role. In
particular, it is important that the Jacobian of the solutions is
non-vanishing. In the present paper we address a two-dimensional limited view
setting, where only a part of the boundary of the domain can be controlled by a
non-zero Dirichlet condition, while on the remaining boundary there is a zero
Dirichlet condition. For this setting, we propose sufficient conditions on the
boundary functions so that the Jacobian of the corresponding solutions is
non-vanishing. In that regard we allow for discontinuous boundary functions,
which requires the use of solutions in weighted Sobolev spaces. We implement
the procedure of reconstructing a conductivity from power density data
numerically and investigate how this limited view setting affects the Jacobian
and the quality of the reconstructions.
