  We analyze the ground state phases of two-component ($\sigma \equiv \lbrace
\uparrow, \downarrow \rbrace$) population- and mass-balanced ($N_\uparrow =
N_\downarrow$ and $m_\uparrow = m_\downarrow$) but trap-imbalanced
($\omega_\uparrow \ne \omega_\downarrow$) fermion mixtures as a function of
interaction strength from the weak attraction Bardeen-Cooper-Schrieffer (BCS)
to the strong attraction Bose-Einstein condensation (BEC) limit. In the BCS
limit, we find that the unpolarized superfluid (UPS) fermions exist away from
the central core of the trapping potentials, and are surrounded by partially
polarized normal (P$\sigma$PN) fermions. As the interactions increase towards
unitarity, we find that the central P$\sigma$PN core first transitions to a
UPS, and then expands towards the edges until the entire mixture becomes a UPS
in the BEC limit.
