  We predict the phase separations of two-dimensional Fermi gases with
repulsive contact-type interactions between two spin components. Using
density-potential functional theory with systematic semiclassical
approximations, we address the long-standing problem of itinerant
ferromagnetism in realistic settings. We reveal a universal transition from the
paramagnetic state at small repulsive interactions towards ferromagnetic
density profiles at large interaction strengths, with intricate particle-number
dependent phases in between. Building on quantum Monte Carlo results for
uniform systems, we benchmark our simulations against Hartree-Fock calculations
for a small number of trapped fermions. We thereby demonstrate that our
employed corrections to the mean-field interaction energy and especially to the
Thomas-Fermi kinetic energy functional are necessary for reliably predicting
properties of trapped mesoscopic Fermi gases. The density patterns of the
ground state survive at low finite temperatures and confirm the Stoner-type
polarization behavior across a universal interaction parameter, albeit with
substantial quantitative differences that originate in the trapping potential
and the quantum-corrected kinetic energy. We also uncover a zoo of metastable
configurations that are energetically comparable to the ground-state density
profiles and are thus likely to be observed in experiments. We argue that our
density-functional approach can be easily applied to interacting
multi-component Fermi gases in general.
