  We construct the complete $U$-$\mu$ phase diagram for harmonically confined
ultracold fermionic atoms with repulsive and attractive interactions. ($\mu$ is
the chemical potential and $U$ the interaction strength.) Our approach is based
on density-functional theory, and employs analytical expressions for the
kinetic and correlation energy functionals. For repulsive interactions our
calculations confirm previous numerical studies, and complement these by closed
expressions for all phase boundaries and characteristic lines of the phase
diagram, and by providing an explanation and solution for difficulties
encountered in earlier density-functional work on the same system. For
attractive interactions we propose a new and accurate interpolation for the
correlation energy, and use it to extend the phase diagram to $U<0$.
