Hostname: page-component-6bf8c574d5-8gtf8 Total loading time: 0 Render date: 2025-02-26T17:07:15.306Z Has data issue: false hasContentIssue false
  • English
  • Français

Multimarginal Optimal Transport Maps for One–dimensional Repulsive Costs

Published online by Cambridge University Press:  20 November 2018

Maria Colombo ,
Luigi De Pascale  and
Simone Di Marino
Maria Colombo
Affiliation:
Scuola Normale Superiore, 56126 Pisa, Italy. e-mail: [email protected]@sns.it
Luigi De Pascale
Affiliation:
Dipartimento di Matematica, Universitá di Pisa, Pisa, Italy. e-mail: [email protected]
Simone Di Marino
Affiliation:
Scuola Normale Superiore, 56126 Pisa, Italy. e-mail: [email protected]@sns.it
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study a multimarginal optimal transportation problem in one dimension. For a symmetric, repulsive cost function, we show that, given a minimizing transport plan, its symmetrization is induced by a cyclical map, and that the symmetric optimal plan is unique. The class of costs that we consider includes, in particular, the Coulomb cost, whose optimal transport problem is strictly related to the strong interaction limit of Density Functional Theory. In this last setting, our result justifies some qualitative properties of the potentials observed in numerical experiments.

Keywords

49Q2049K30Monge–Kantorovich problemoptimal transport problemcyclical monotonicity
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 67 , Issue 2 , April 2015 , pp. 350 - 368
Copyright
Copyright © Canadian Mathematical Society 2015

References

[1] Ambrosio, L., Lecture notes on optimal transport problems. In: Mathematical aspects of evolving interfaces (Funchal, 2000), Lecture Notes in Math., 1812, Springer, Berlin, 2003, pp. 1–52.Google Scholar
[2] Ambrosio, L., Kirchheim, B., and Pratelli, A., Existence of optimal transport maps for crystalline norms. Duke Math. J. 125(2004), no. 2, 207–241. http://dx.doi.org/10.1215/S0012-7094-04-12521-7 Google Scholar
[3] Buttazzo, G., De Pascale, L., and Gori–Giorgi, P., Optimal transport formulation of electronic density–functional theory. Phys. Rev. A 85 (2012), 062502.Google Scholar
[4] Champion, T. and De Pascale, L., On the twist condition and c–monotone transport plans. Discrete Contin. Dyn. Sys. 34(2014), no. 4, 1339–1353. http://dx.doi.org/10.3934/dcds.2014.34.1339 Google Scholar
[5] Colombo, M., Di Marino, S., Equality between Monge and Kantorovich multimarginal problems with Coulomb cost..Ann. Mat. Pura Appl., to appear. http://dx.doi.org/10.1007/s10231-013-0376-0 Google Scholar
[6] Cotar, C., Friesecke, G., Klüppelberg, C., Density functional theory and optimal transportation with Coulomb cost. Comm. Pure Appl. Math. 66(2013), no. 4, 548–599.http://dx.doi.org/10.1002/cpa.21437 Google Scholar
[7] Cotar, C., Friesecke, G., Pass, B., Infinite–body optimal transport with Coulomb Cost. http://arxiv:1307.6540 Google Scholar
[8] Di Marino, S., Trasporto ottimo e problemi di evoluzione per sistemi di particelle. http://cvgmt.sns.it/paper/1862/ Google Scholar
[9] Fathi, A. and Figalli, A., Optimal transportation on non-compact manifolds. Israel J. Math. 175(2010), 1–59. http://dx.doi.org/10.1007/s11856-010-0001-5 Google Scholar
[10] Gangbo, W. and McCann, R. J., The geometry of optimal transportation. Acta Math. 177(1996), no. 2, 113–161. http://dx.doi.org/10.1007/BF02392620 Google Scholar
[11] Ghoussoub, N. and Maurey, B., Remarks on multidimensional symmetric Monge–Kantorovic problems. Discrete Contin. Dyn. Syst. 34(2014), no. 4, 1465–1480.Google Scholar
[12] Ghoussoub, N. and Moameni, A., Symmetric Monge–Kantorovich problems and polar decompositions of vector fields.. http://arxiv:1302.2886 Google Scholar
[13] Malet, F., Mirtschink, A., Cremon, J. C., Reimann, S. M., and Gori-Giorgi, P., Kohn-Sham density functional theory for quantum wires in arbitrary correlation regimes. Phys. Rev. B 87(2013), 115146.Google Scholar
[14] Mendl, C. B. and Lin, L., Towards the Kantorovich dual solution for strictly correlated electrons in atoms and molecules.. arxiv:1210.7117 Google Scholar
[15] Pass, B., On the local structure of optimal measures in the multi–marginal optimal transportation problem. Calc. Var. Partial Differential Equations 43(2012), no. 3–4, 529–536.http://dx.doi.org/10.1007/s00526-011-0421-z Google Scholar
[16] Pass, B., Uniqueness and Monge solutions in the multimarginal optimal transportation problem. SIAM J. Math. Anal. 43(2011), no. 6, 2758–2775. http://dx.doi.org/10.1137/100804917 Google Scholar
[17] Pass, B., Remarks on the semi–classical Hohenberg–Kohn functional. Nonlinearity 26(2013), no. 9. 2731–2744. http://dx.doi.org/10.1088/0951-7715/26/9/2731 Google Scholar
[18] Pratelli, A., On the equality between Monge's infimum and Kantorovich's minimum in optimal mass transportation. Ann. Inst. H. Poincaré Probab. Statist. 43(2007), no. 1, 1–13.http://dx.doi.org/10.1016/j.anihpb.2005.12.001 Google Scholar
[19] Seidl, M., Strong–interaction limit of density–functional theory. Phys. Rev. A 60(1999), 4387.Google Scholar
[20] Smith, C. and Knott, M., On Hoeffding–Fréchet bounds and cyclic monotone relations. J. Multivariate Anal. 40(1992), no. 2, 328–334. http://dx.doi.org/10.1016/0047-259X(92)90029-F Google Scholar
[21] Villani, C., Optimal transport. Old and new. Grundlehren der Mathematischen Wissenschaften, 338, Springer–Verlag, Berlin, 2009.Google Scholar