  Following recent experiments, we consider current flow in two dimensional
electronic systems in the quantum Hall regime where a gradient in the electron
density induces a spatial variation in the Hall resistivity. Describing the
system in terms of a spatially varying local resistivity tensor, we show that
in such a system the current density is generically non-uniform, with the
current being pushed towards one side of the sample. We show that, for
sufficiently large density gradient, the voltage along that side is determined
by the change of the Hall resistivity, and is independent of the microscopic
longitudinal resistivity, while the voltage on the opposite side is small and
determined by both longitudinal and Hall resistivities. We solve some
particular models in detail, and propose ways by which the microscopic
longitudinal resistivity may be extracted from measurements of the longitudinal
voltage on both sides of the sample.
