Thomas-Fermi theory with magnetic fields and the Fermi-Amaldi correction

Of concern is a quantum mechanical system having $N_{1}$ (resp. $N_{2}$) spin up (resp. spin down) electrons, in the presence of a potential $V$ and a magnetic field $B.$ When the Fermi-Amaldi correction is incorporated into the Thomas-Fermi energy functional, convexity is lost and the computation of the ground state spin up and down electron densities becomes nontrivial. We discuss the existence of these densities and various approximation procedures for them, via variational calculus, differential equations, and numerical procedures.