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April  2022, 21(4): 1189-1208. doi: 10.3934/cpaa.2022015

On deterministic solutions for multi-marginal optimal transport with Coulomb cost

1. 

Department of Mathematics and Statistics, University of Jyväskylä, PO Box 35, FI-40014 University of Jyväskylä, Finland

2. 

Dipartimento di Matematica e Informatica, Università di Firenze, Viale Morgagni 67/a, 50134 Firenze, Italy

*Corresponding author

Received  January 2021 Revised  October 2021 Published  April 2022 Early access  December 2021

Fund Project: The second and third authors are partially supported by the project:Alcuni problemi di trasporto ottimo ed applicazioni of GNAMPA-INDAM, the second author is partially supported by Fondi di Ateneo of the University of Firenze, the third author was partially supported by the project Contemporary topics on multi-marginal optimal mass transportation, funded by the Finnish Postdoctoral Pool (Suomen Kulttuurisäätiö)

In this paper we study the three-marginal optimal mass transportation problem for the Coulomb cost on the plane $ \mathbb R^2 $. The key question is the optimality of the so-called Seidl map, first disproved by Colombo and Stra. We generalize the partial positive result obtained by Colombo and Stra and give a necessary and sufficient condition for the radial Coulomb cost to coincide with a much simpler cost that corresponds to the situation where all three particles are aligned. Moreover, we produce an infinite class of regular counterexamples to the optimality of this family of maps.

Citation: Ugo Bindini, Luigi De Pascale, Anna Kausamo. On deterministic solutions for multi-marginal optimal transport with Coulomb cost. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1189-1208. doi: 10.3934/cpaa.2022015
References:
[1]

M. BeiglböckC. Léonard and W. Schachermayer, A general duality theorem for the Monge-Kantorovich transport problem, Stud. Math., 209 (2012), 151-167.  doi: 10.4064/sm209-2-4.

[2] A. Braides, Gamma-Convergence for Beginners, Clarendon Press, 2002.  doi: 10.1093/acprof:oso/9780198507840.001.0001.
[3]

G. Buttazzo, L. De Pascale and Paola Gori-Giorgi, Optimal-transport formulation of electronic density-functional theory, Phys. Rev. A, 85 (2012), 11 pp. doi: 10.1103/PhysRevA.85.062502.

[4]

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

[5]

G. CarlierC. Jimenez and F. Santambrogio, Optimal transportation with traffic congestion and wardrop equilibria, SIAM J. Contr. Optim., 47 (2008), 1330-1350.  doi: 10.1137/060672832.

[6]

M. ColomboL. De Pascale and S. Di Marino, Multimarginal optimal transport maps for 1-dimensional repulsive costs, Canad. J. Math., 67 (2013), 350-368.  doi: 10.4153/CJM-2014-011-x.

[7]

M. Colombo and S. Di Marino, Equality between Monge and Kantorovich multimarginal problems with coulomb cost, Ann. Mate. Pura Appl., 194 (2015), 307-320.  doi: 10.1007/s10231-013-0376-0.

[8]

M. Colombo and F. Stra, Counterexamples in multimarginal optimal transport with Coulomb cost and spherically symmetric data, Math. Models Methods Appl. Sci., 26 (2016), 1025-1049.  doi: 10.1142/S021820251650024X.

[9]

C. CotarG. Friesecke and C. Klüppelberg, Density functional theory and optimal transportation with Coulomb cost, Commun. Pure Appl. Math., 66 (2013), 548-599.  doi: 10.1002/cpa.21437.

[10]

G. Dal Maso, An Introduction to $\Gamma$-Convergence, Springer Science & Business Media, 2012. doi: 10.1007/978-1-4612-0327-8.

[11]

L. De Pascale, Optimal transport with Coulomb cost. Approximation and duality, ESAIM: Math. Model. Numer. Anal., 49 (2015), 1643-1657.  doi: 10.1051/m2an/2015035.

[12]

L. De Pascale, On $c$-cyclical monotonicity for optimal transport problem with Coulomb cost, Euro. J. Appl. Math., 30 (2019), 1210-1219.  doi: 10.1017/s0956792519000111.

[13]

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, J. Chem. Phys., 139 (2013), 13 pp. doi: 10.1063/1.4821351.

[14]

W. Gangbo and A. Świech, Optimal maps for the multidimensional Monge-Kantorovich problem, Commun. Pure Appl. Math., 51 (1998), 23-45.  doi: 10.1002/(SICI)1097-0312(199801)51:1<23::AID-CPA2>3.0.CO;2-H.

[15]

N. Ghoussoub and B. Maurey, Remarks on multi-marginal symmetric Monge-Kantorovich problems, Discret. Contin. Dynam. Syst. A, 34 (2014), 1465-1480.  doi: 10.3934/dcds.2014.34.1465.

[16]

N. Ghoussoub and A. Moameni, A self-dual polar factorization for vector fields, Commun. Pure Appl. Math., 66 (2013), 905-933.  doi: 10.1002/cpa.21430.

[17]

N. Ghoussoub and A. Moameni, Symmetric Monge-Kantorovich problems and polar decompositions of vector fields, Geometric Funct. Anal., 24 (2014), 1129-1166.  doi: 10.1007/s00039-014-0287-2.

[18]

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. 

[19]

P. Gori-Giorgi, M. Seidl and G. Vignale, Density-functional theory for strongly interacting electrons, Phys. Rev. Lett., 103 (2009), 4 pp. doi: 10.1103/PhysRevLett.103.166402.

[20]

H. Heinich, Problème de Monge pour n probabilités, CR Math., 334 (2002), 793-795.  doi: 10.1016/S1631-073X(02)02341-5.

[21]

P. Hohenberg and W. Kohn, Inhomogeneous electron gas, Phys. rev., 136 (1964), 809-811. 

[22]

H. G. Kellerer, Duality theorems for marginal problems, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 67 (1984), 399–432. doi: 10.1007/BF00532047.

[23]

W. Kohn and L. J. Sham, Self-consistent equations including exchange and correlation effects, Phys. Rev., 140 (1965), 133-1138. 

[24]

E. H. Lieb, Density functionals for Coulomb systems, in Inequalities, Springer, 2002.

[25]

C. B. Mendl and L. Lin, Kantorovich dual solution for strictly correlated electrons in atoms and molecules, Phys. Rev. B, 87 (2013), 6 pp.

[26]

B. Pass., Uniqueness and Monge solutions in the multimarginal optimal transportation problem, SIAM J. Math. Anal., 43 (2011), 2758-2775.  doi: 10.1137/100804917.

[27]

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: 10.1007/s00526-011-0421-z.

[28]

B. Pass, Remarks on the semi-classical Hohenberg-Kohn functional, Nonlinearity, 26 (2013), 15 pp. doi: 10.1088/0951-7715/26/9/2731.

[29]

S. T. Rachev and L. Rüschendorf, Mass Transportation Problems: Volume I: Theory, Springer Science & Business Media, 1998.

[30]

M. Seidl, Strong-interaction limit of density-functional theory, Phys. Rev. A, 60 (1999), 9 pp. doi: 10.1103/PhysRevA.60.4387.

[31]

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), 12 pp. doi: 10.1103/PhysRevA.75.042511.

[32]

M. Seidl, J. P. Perdew and M. Levy, Strictly correlated electrons in density-functional theory, Phys. Rev. A, 59 (1999), 4 pp. doi: 10.1103/PhysRevA.59.51.

show all references

References:
[1]

M. BeiglböckC. Léonard and W. Schachermayer, A general duality theorem for the Monge-Kantorovich transport problem, Stud. Math., 209 (2012), 151-167.  doi: 10.4064/sm209-2-4.

[2] A. Braides, Gamma-Convergence for Beginners, Clarendon Press, 2002.  doi: 10.1093/acprof:oso/9780198507840.001.0001.
[3]

G. Buttazzo, L. De Pascale and Paola Gori-Giorgi, Optimal-transport formulation of electronic density-functional theory, Phys. Rev. A, 85 (2012), 11 pp. doi: 10.1103/PhysRevA.85.062502.

[4]

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

[5]

G. CarlierC. Jimenez and F. Santambrogio, Optimal transportation with traffic congestion and wardrop equilibria, SIAM J. Contr. Optim., 47 (2008), 1330-1350.  doi: 10.1137/060672832.

[6]

M. ColomboL. De Pascale and S. Di Marino, Multimarginal optimal transport maps for 1-dimensional repulsive costs, Canad. J. Math., 67 (2013), 350-368.  doi: 10.4153/CJM-2014-011-x.

[7]

M. Colombo and S. Di Marino, Equality between Monge and Kantorovich multimarginal problems with coulomb cost, Ann. Mate. Pura Appl., 194 (2015), 307-320.  doi: 10.1007/s10231-013-0376-0.

[8]

M. Colombo and F. Stra, Counterexamples in multimarginal optimal transport with Coulomb cost and spherically symmetric data, Math. Models Methods Appl. Sci., 26 (2016), 1025-1049.  doi: 10.1142/S021820251650024X.

[9]

C. CotarG. Friesecke and C. Klüppelberg, Density functional theory and optimal transportation with Coulomb cost, Commun. Pure Appl. Math., 66 (2013), 548-599.  doi: 10.1002/cpa.21437.

[10]

G. Dal Maso, An Introduction to $\Gamma$-Convergence, Springer Science & Business Media, 2012. doi: 10.1007/978-1-4612-0327-8.

[11]

L. De Pascale, Optimal transport with Coulomb cost. Approximation and duality, ESAIM: Math. Model. Numer. Anal., 49 (2015), 1643-1657.  doi: 10.1051/m2an/2015035.

[12]

L. De Pascale, On $c$-cyclical monotonicity for optimal transport problem with Coulomb cost, Euro. J. Appl. Math., 30 (2019), 1210-1219.  doi: 10.1017/s0956792519000111.

[13]

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, J. Chem. Phys., 139 (2013), 13 pp. doi: 10.1063/1.4821351.

[14]

W. Gangbo and A. Świech, Optimal maps for the multidimensional Monge-Kantorovich problem, Commun. Pure Appl. Math., 51 (1998), 23-45.  doi: 10.1002/(SICI)1097-0312(199801)51:1<23::AID-CPA2>3.0.CO;2-H.

[15]

N. Ghoussoub and B. Maurey, Remarks on multi-marginal symmetric Monge-Kantorovich problems, Discret. Contin. Dynam. Syst. A, 34 (2014), 1465-1480.  doi: 10.3934/dcds.2014.34.1465.

[16]

N. Ghoussoub and A. Moameni, A self-dual polar factorization for vector fields, Commun. Pure Appl. Math., 66 (2013), 905-933.  doi: 10.1002/cpa.21430.

[17]

N. Ghoussoub and A. Moameni, Symmetric Monge-Kantorovich problems and polar decompositions of vector fields, Geometric Funct. Anal., 24 (2014), 1129-1166.  doi: 10.1007/s00039-014-0287-2.

[18]

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. 

[19]

P. Gori-Giorgi, M. Seidl and G. Vignale, Density-functional theory for strongly interacting electrons, Phys. Rev. Lett., 103 (2009), 4 pp. doi: 10.1103/PhysRevLett.103.166402.

[20]

H. Heinich, Problème de Monge pour n probabilités, CR Math., 334 (2002), 793-795.  doi: 10.1016/S1631-073X(02)02341-5.

[21]

P. Hohenberg and W. Kohn, Inhomogeneous electron gas, Phys. rev., 136 (1964), 809-811. 

[22]

H. G. Kellerer, Duality theorems for marginal problems, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 67 (1984), 399–432. doi: 10.1007/BF00532047.

[23]

W. Kohn and L. J. Sham, Self-consistent equations including exchange and correlation effects, Phys. Rev., 140 (1965), 133-1138. 

[24]

E. H. Lieb, Density functionals for Coulomb systems, in Inequalities, Springer, 2002.

[25]

C. B. Mendl and L. Lin, Kantorovich dual solution for strictly correlated electrons in atoms and molecules, Phys. Rev. B, 87 (2013), 6 pp.

[26]

B. Pass., Uniqueness and Monge solutions in the multimarginal optimal transportation problem, SIAM J. Math. Anal., 43 (2011), 2758-2775.  doi: 10.1137/100804917.

[27]

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: 10.1007/s00526-011-0421-z.

[28]

B. Pass, Remarks on the semi-classical Hohenberg-Kohn functional, Nonlinearity, 26 (2013), 15 pp. doi: 10.1088/0951-7715/26/9/2731.

[29]

S. T. Rachev and L. Rüschendorf, Mass Transportation Problems: Volume I: Theory, Springer Science & Business Media, 1998.

[30]

M. Seidl, Strong-interaction limit of density-functional theory, Phys. Rev. A, 60 (1999), 9 pp. doi: 10.1103/PhysRevA.60.4387.

[31]

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), 12 pp. doi: 10.1103/PhysRevA.75.042511.

[32]

M. Seidl, J. P. Perdew and M. Levy, Strictly correlated electrons in density-functional theory, Phys. Rev. A, 59 (1999), 4 pp. doi: 10.1103/PhysRevA.59.51.

Figure 1.  The relative position of the graps of $ g_{12} $ and $ g_{13} $ on the interval $ [0,\pi] $. However the strict inequality between the two maximal values is not proved. See Lemma 2.1
Figure 2.  A graphical understanding of Lemma 2.2: the function $ h(t) $ stays below two segments
Figure 3.  In blue, the "butterfly" region of admissible solutions to optimality conditions (2.3). In black and orange, a plot of the curves $ \alpha_{0,\pi} $ and $ \hat\alpha_{0,\pi} $ in the region $ 0\leq \beta \leq \pi $
Figure 4.  A graphical understanding of Lemma 2.3: the function $ \alpha_\pi(\beta) $ is confined by $ \pi \leq \alpha_\pi(\beta) \leq \pi + \alpha_\pi'(0)\beta $, and similarly $ \pi + \widehat{\alpha}_\pi'(0)\beta \leq \widehat{\alpha}_\pi(\beta) \leq \pi + \beta $. This implies that the intersection between $ \alpha_\pi $ and $ \widehat{\alpha}_\pi $ is only at $ \beta = 0 $
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