M. Beiglböck, C. Léonard, and W. Schachermayer, A general duality theorem for the Monge???Kantorovich transport problem, Studia Mathematica, vol.209, issue.2, 2012.
DOI : 10.4064/sm209-2-4

G. Buttazzo, L. De-pascale, and P. Gori-giorgi, Optimal-transport formulation of electronic density-functional theory, Physical Review A, vol.85, issue.6, p.62502, 2012.
DOI : 10.1103/PhysRevA.85.062502

G. Carlier, On a class of multidimensional optimal transportation problems, J. Convex Anal, vol.10, issue.2, p.517530, 2003.

G. Carlier and B. Nazaret, Optimal transportation for the determinant, ESAIM: Control, Optimisation and Calculus of Variations, vol.14, issue.4, p.678698, 2008.
DOI : 10.1051/cocv:2008006

URL : https://hal.archives-ouvertes.fr/hal-00118459

M. Colombo, L. De-pascale, and S. D. Marino, Multimarginal Optimal Transport Maps for One-dimensional Repulsive Costs, Journal canadien de math??matiques, vol.67, issue.2, 2013.
DOI : 10.4153/CJM-2014-011-x

M. Colombo and S. D. Marino, Equality between Monge and Kantorovich multimarginal problems with Coulomb cost, Annali di Matematica Pura ed Applicata (1923 -), vol.75, issue.61, p.114, 2013.
DOI : 10.1007/s10231-013-0376-0

C. Cotar, G. Friesecke, and C. Klüppelberg, Density Functional Theory and Optimal Transportation with Coulomb Cost, Communications on Pure and Applied Mathematics, vol.12, issue.3, p.548599, 2013.
DOI : 10.1002/cpa.21437

L. Pascale, Optimal transport with Coulomb cost. Approximation and duality, ESAIM: Mathematical Modelling and Numerical Analysis, vol.49, issue.6, p.16431657, 2015.
DOI : 10.1051/m2an/2015035

D. Marino, A. Gerolin, and L. Nenna, Optimal Transportation Theory with Repulsive Costs
URL : https://hal.archives-ouvertes.fr/hal-01163737

G. Friesecke, C. B. Mendl, B. Pass, C. Cotar, and C. Klüppelberg, N-density representability and the optimal transport limit of the Hohenberg-Kohn functional, The Journal of Chemical Physics, vol.139, issue.16, p.139164109, 2013.
DOI : 10.1063/1.4821351

W. Gangbo and A. Swiech, Optimal maps for the multidimensional Monge-Kantorovich problem, Communications on Pure and Applied Mathematics, vol.51, issue.1, p.2345, 1998.
DOI : 10.1002/(SICI)1097-0312(199801)51:1<23::AID-CPA2>3.0.CO;2-H

N. Ghoussoub and A. Moameni, A self-dual polar factorization for vector elds, Comm. Pure Appl. Math, vol.66, issue.6, p.905933, 2013.

P. Gori-giorgi and M. Seidl, Density functional theory for strongly-interacting electrons: perspectives for physics and chemistry, Physical Chemistry Chemical Physics, vol.119, issue.43, p.1440514419, 2010.
DOI : 10.1039/c0cp01061h

P. Gori-giorgi, M. Seidl, and G. Vignale, Density-functional theory for strongly interacting electrons . Physical review letters, p.166402, 2009.

H. Heinich, Probl??me de Monge pour probabilit??s, Comptes Rendus Mathematique, vol.334, issue.9, pp.793-795, 2002.
DOI : 10.1016/S1631-073X(02)02341-5

P. Hohenberg and W. Kohn, Inhomogeneous Electron Gas, Physical Review, vol.136, issue.3B, p.864, 1964.
DOI : 10.1103/PhysRev.136.B864

H. G. Kellerer, Duality theorems for marginal problems. Probab. Theory Related Fields, p.399432, 1984.

W. Kohn and L. J. Sham, Self-Consistent Equations Including Exchange and Correlation Effects, Physical Review, vol.140, issue.4A, p.1133, 1965.
DOI : 10.1103/PhysRev.140.A1133

E. H. Lieb, Density functionals for coulomb systems, International journal of quantum chemistry, vol.24, issue.3, p.243277, 1983.

C. B. Mendl and L. Lin, Kantorovich dual solution for strictly correlated electrons in atoms and molecules, Physical Review B, vol.87, issue.12, p.125106, 2013.
DOI : 10.1103/PhysRevB.87.125106

B. Pass, Uniqueness and Monge Solutions in the Multimarginal Optimal Transportation Problem, SIAM Journal on Mathematical Analysis, vol.43, issue.6, p.27582775, 2011.
DOI : 10.1137/100804917

B. Pass, On the local structure of optimal measures in the multi-marginal optimal transportation problem, Calculus of Variations and Partial Differential Equations, vol.40, issue.1, pp.3-4529536, 2012.
DOI : 10.1007/s00526-011-0421-z

S. T. Rachev and L. Rüschendorf, Mass transportation problems, Theory. Probability and its Applications, 1998.

M. Seidl, Strong-interaction limit of density-functional theory, Physical Review A, vol.60, issue.6, p.4387, 1999.
DOI : 10.1103/PhysRevA.60.4387

M. Seidl, P. Gori-giorgi, and A. Savin, Strictly correlated electrons in density-functional theory: A general formulation with applications to spherical densities, Physical Review A, vol.75, issue.4, p.42511, 2007.
DOI : 10.1103/PhysRevA.75.042511

M. Seidl, J. P. Perdew, and M. Levy, Strictly correlated electrons in density-functional theory, Physical Review A, vol.59, issue.1, p.51, 1999.
DOI : 10.1103/PhysRevA.59.51

B. Largo, Pontecorvo 5, 56127 Pisa -ITALY buttazzo@dm.unipi.it http

B. Largo, Pontecorvo 5, 56127 Pisa -ITALY depascal@dm.unipi.it http