  We consider a strongly repulsive fermionic gas in a two-dimensional optical
lattice confined by a harmonic trapping potential. To address the strongly
repulsive regime, we consider the $t-J$ Hamiltonian. The presence of the
harmonic trapping potential enables the stabilization of coexisting and
competing phases. In particular, at low temperatures, this allows the
realization of a d-wave superfluid region surrounded by purely (gapless) normal
edges. Solving the Bogoliubov-de Gennes equations and comparing with the local
density approximation, we show that the proximity to the Mott insulator is
revealed by a downturn of the Fermi liquid order parameter at the center of the
trap where the d-wave gap has a maximum. The density profile evolves linearly
with distance.
