Optimal transport with Coulomb cost and the semiclassical limit of density functional theory

We present some progress in the direction of determining the semiclassical limit of the Levy-Lieb or Hohenberg-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 two or three particles. Comparisons with previous results are reported. The approach is based on some techniques from the optimal transportation theory.