  We study the effects of electrostatic gating on the lateral distribution of
charge carriers in two dimensional devices, in a non-linear dielectric
environment. We compute the charge distribution using the Thomas-Fermi
approximation to model the electrostatics of the system. The electric field
lines generated by the gate are focused at the edges of the device, causing an
increased depletion near the edges, compared to the center of the device. This
effect strongly depends on the dimensions of the device, and the non-linear
dielectric constant of the substrate. We experimentally demonstrate this effect
using scanning superconducting interference device (SQUID) microscopy images of
current distributions in gated LaAlO$_3$/SrTiO$_3$ heterostructures.
