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doi: 10.3934/cpaa.2022015
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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 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 & Applied Analysis, 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.  Google Scholar

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

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[29]

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

[30]

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

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

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[29]

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

[30]

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

[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.  Google Scholar

[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.  Google Scholar

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
$ \alpha_{0,\pi} $ and $ \hat\alpha_{0,\pi} $ in the region $ 0\leq \beta \leq \pi $">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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