  We present some progress in the direction of determining the semiclassical
limit of the Hoenberg-Kohn universal functional in Density Functional Theory
for Coulomb systems. In particular we give a proof of the fact that for Bosonic
systems with an arbitrary number of particles the limit is the multimarginal
optimal transport problem with Coulomb cost and that the same holds for
Fermionic systems with 2 or 3 particles. Comparisons with previous results are
reported . The approach is based on some techniques from the optimal
transportation theory.

Analysis of PDEs