  Linear and non-linear transport properties through an atomic-size point
contact based on oxides two-dimensional electron gas is examined using the
tight-binding method and the $\mathbf{k\cdot p}$ approach. The ballistic
transport is analyzed in contacts realized at the (001) interface between band
insulators $LaAlO_3$ and $SrTiO_3$ by using the Landauer-B\"uttiker method for
many sub-bands derived from three Ti 3d orbitals ($d_{yz}$, $d_{zx}$ and
$d_{xy}$) in the presence of an out-of-plane magnetic field. We focus
especially on the role played by the atomic spin-orbit coupling and the
inversion symmetry breaking term pointing out three transport regimes: the
first, at low energies, involving the first $d_{xy}$-like sub-bands, where the
conductance quantization is robust; a second one, at intermediate energies,
entailing further $d_{xy}$-like sub-bands, where the sub-band splitting induced
by the magnetic field is quenched; the third one, where the mixing between
light $d_{xy}$-like, heavy $d_{yz}$-like and $d_{zx}$-like sub-bands is so
strong that the conductance plateaus turn out to be very narrow. Very good
agreement is found with recent experiments exploring the transport properties
at low energies.
