• Coulomb cost
• Density Functional Theory
• duality theory
• Monge-Kantorovich problem
• Multimarginal optimal transportation

We prove that optimal Coulomb transportation cost is the semiclassical limit of the N-particles Hohenberg-Kohn functionals, both in the symmetric (bosonic) and the antisymmetric (fermionic) case.
The first result was already known in the case N=2, but we are able to generalize it to every N thanks to a new result of Buttazzo, Champion and De Pascale.
The antysimmetric case was proved for N=2, but the proof was impossibile to generalize to a higher number of particles. Here we provide a new class of antisymmetric functions for N=2,3, and some useful constructions which have hope to be generalized to higher N.
Also, a new tecnique to rescue the marginals of a transport plan smoothed by convolution is provided. This simplifies the construction of a recovering sequence to prove the limsup inequality, and may be applied to this general case: if P is a probability measure on R^{Nd}, even with not equal marginals, we provide a family of regular probabilities {P_\eps}_{\eps>0}, with the same marginals of P, such that P_\eps converges weakly to P, with an explicit control over the kinetic energy (i.e., the H^1 norm) of P_\eps.