  We study population imbalanced Fermi mixtures under quasi-two-dimensional
confinement at zero temperature. Using mean-field theory and the local-density
approximation, we study the ground state configuration throughout the BEC-BCS
crossover. We find the trapped system to be either fully normal or to consist
of a superfluid core surrounded by a normal shell, which is itself either fully
or partially polarized. Upon changing the trap imbalance, the trap
configuration may undergo continuous transitions between the different ground
states. Finally, we argue that thermal equilibration throughout the trap will
be considerably slowed down at low temperatures when a superfluid phase is
present.
