  We investigate the statics and dynamics of spatial phase segregation process
of a mixture of fermion atoms in a harmonic trap using the density functional
theory. The kinetic energy of the fermion gas is written in terms of the
density and its gradients. Several cases have been studied by neglecting the
gradient terms (the Thomas-Fermi limit) which are then compared with the
Monte-Carlo results using the full gradient corrected kinetic energy. A linear
instability analysis has been performed using the random-phase approximation.
Near the onset of instability, the fastest unstable mode for spinodal
decomposition is found to occur at $q=0$. However, in the strong coupling
limit, many more modes with $q\approx K_F$ decay with comparable time scales.
