Optimal transport with Coulomb cost. Approximation and duality

De Pascale, Luigi

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 6, pp. 1643-1657.

We revisit the duality theorem for multimarginal optimal transportation problems. In particular, we focus on the Coulomb cost. We use a discrete approximation to prove equality of the extremal values and some careful estimates of the approximating sequence to prove existence of maximizers for the dual problem (Kantorovich’s potentials). Finally we observe that the same strategy can be applied to a more general class of costs and that a classical results on the topic cannot be applied here.

Classification : 49J45, 49N15, 49K30
Mots clés : Multimarginal optimal transportation, Monge−Kantorovich problem, duality theory, Coulomb cost

M. Beiglböck, Ch. Léonard and W. Schachermayer, A general duality theorem for the monge–kantorovich transport problem. Stud. Math. 209 (2012) 2. | DOI | MR | Zbl

A. Braides, Gamma-convergence for Beginners. Vol. 22. Oxford University Press, Oxford (2002). | MR | Zbl

G. Buttazzo, L. De Pascale and P. Gori-Giorgi, Optimal-transport formulation of electronic density-functional theory. Phys. Rev. A 85 (2012) 062502. | DOI

G. Carlier, On a class of multidimensional optimal transportation problems. J. Convex Anal. 10 (2003) 517–530. | MR | Zbl

G. Carlier and B. Nazaret, Optimal transportation for the determinant. ESAIM: COCV 14 (2008) 678–698. | Numdam | MR | Zbl

M. Colombo, L. De Pascale and S. Di Marino, Multimarginal optimal transport maps for 1-dimensional repulsive costs. Canad. J. Math. (2013). | MR

M. Colombo and S. Di Marino, Equality between monge and kantorovich multimarginal problems with coulomb cost. Ann. Mat. Pura Appl. (2013) 1–14. | MR

C. Cotar, G. Friesecke and C. Klüppelberg, Density functional theory and optimal transportation with coulomb cost. Commun. Pure Appl. Math. 66 (2013) 548–599. | DOI | MR | Zbl

G. Dal Maso, An introduction to $\Gamma$-convergence. Springer (1993). | MR | Zbl

G. Friesecke, Ch.B. Mendl, B. Pass, C. Cotar and C. Klüppelberg, N-density representability and the optimal transport limit of the hohenberg−kohn functional. J. Chem. Phys. 139 (2013) 164–109. | DOI

W. Gangbo and A. Swiech, Optimal maps for the multidimensional monge−kantorovich problem. Comm. Pure Appl. Math. 51 (1998) 23–45. | DOI | MR | Zbl

N. Ghoussoub and A. Moameni, A self-dual polar factorization for vector fields. Comm. Pure Appl. Math. 66 (2013) 905–933. | DOI | MR | Zbl

P. Gori-Giorgi and M. Seidl, Density functional theory for strongly-interacting electrons: perspectives for physics and chemistry. Phys. Chem. Chem. Phys. 12 (2010) 14405–14419. | DOI

P. Gori-Giorgi, M. Seidl and G. Vignale, Density-functional theory for strongly interacting electrons. Phys. Rev. Lett. 103 (2009) 166402. | DOI

H. Heinich, Problème de monge pour n probabilités. C. R. Math. 334 (2002) 793–795. | DOI | MR | Zbl

P. Hohenberg and W. Kohn, Inhomogeneous electron gas. Phys. Rev. 136 (1964) B864. | DOI | MR

H.G. Kellerer, Duality theorems for marginal problems. Probab. Theory Relat. Fields 67 (1984) 399–432. | MR | Zbl

Walter Kohn and Lu Jeu Sham, Self-consistent equations including exchange and correlation effects. Phys. Rev. 140 (1965) A1133. | DOI | MR

E.H. Lieb. Density functionals for coulomb systems. Int J. Quantum Chem. 24 (1983) 243–277. | DOI

Ch.B. Mendl and L. Lin, Kantorovich dual solution for strictly correlated electrons in atoms and molecules. Phys. Rev. B 87 (2013) 125106. | DOI

B. Pass, Uniqueness and monge solutions in the multimarginal optimal transportation problem. SIAM J. Math. Anal. 43 (2011) 2758–2775. | DOI | MR | Zbl

B. Pass, On the local structure of optimal measures in the multi-marginal optimal transportation problem. Calc. Var. Partial Differ. Equ. 43 (2012) 529–536. | DOI | MR | Zbl

S.T. Rachev and L. Rüschendorf, Mass transportation problems. Probab. Appl. Springer-Verlag (1998), Vol. I. | MR

M. Seidl, Strong-interaction limit of density-functional theory. Phys. Rev. A 60 (1999) 4387. | DOI

M. Seidl, P. Gori-Giorgi and A. Savin, Strictly correlated electrons in density-functional theory: A general formulation with applications to spherical densities. Phys. Rev. A 75 (2007) 042511. | DOI

M. Seidl, J.P. Perdew and M. Levy, Strictly correlated electrons in density-functional theory. Phys. Rev. A 59 (1999) 51. | DOI