In this paper we extend the duality theory of the multi-marginal optimal transport problem for cost functions depending on a decreasing function of the distance (not necessarily bounded). This class of cost functions appears in the context of SCE Density Functional Theory introduced in Strong-interaction limit of density-functional theory by Seidl [Phys. Rev. A 60 (1999) 4387].
Accepté le :
DOI : 10.1051/cocv/2018062
Mots-clés : Multi-marginal optimal transport, repulsive costs, Kantorovich duality
@article{COCV_2019__25__A62_0,
author = {Gerolin, Augusto and Kausamo, Anna and Rajala, Tapio},
title = {Duality theory for multi-marginal optimal transport with repulsive costs in metric spaces},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {25},
year = {2019},
doi = {10.1051/cocv/2018062},
zbl = {1439.49059},
mrnumber = {4023129},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2018062/}
}
TY - JOUR AU - Gerolin, Augusto AU - Kausamo, Anna AU - Rajala, Tapio TI - Duality theory for multi-marginal optimal transport with repulsive costs in metric spaces JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2019 VL - 25 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2018062/ DO - 10.1051/cocv/2018062 LA - en ID - COCV_2019__25__A62_0 ER -
%0 Journal Article %A Gerolin, Augusto %A Kausamo, Anna %A Rajala, Tapio %T Duality theory for multi-marginal optimal transport with repulsive costs in metric spaces %J ESAIM: Control, Optimisation and Calculus of Variations %D 2019 %V 25 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2018062/ %R 10.1051/cocv/2018062 %G en %F COCV_2019__25__A62_0
Gerolin, Augusto; Kausamo, Anna; Rajala, Tapio. Duality theory for multi-marginal optimal transport with repulsive costs in metric spaces. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 62. doi : 10.1051/cocv/2018062. http://www.numdam.org/articles/10.1051/cocv/2018062/
[1] and , Optimal transport with Coulomb cost and the semiclassical limit of density functional theory. J. Éc. Polytech. Math. 4 (2017) 909–934. | DOI | MR | Zbl
[2] , Gamma-Convergence for Beginners, Vol. 22. Clarendon Press, Oxford (2002). | DOI | MR | Zbl
[3] , and , Continuity and estimates for multimarginal optimal transportation problems with singular costs. Appl. Math. Optim. 78 (2018) 185–200. | DOI | MR | Zbl
[4] , and , Optimal-transport formulation of electronic density-functional theory. Phys. Rev. A 85 (2012) 062502. | DOI
[5] , and , Numerical methods for a kohn–sham density functional model based on optimal transport. J. Chem. Theory Comput. 10 (2014) 4360–4368. | DOI
[6] and , Counterexamples to multimarginal optimal transport maps with coulomb cost and radial measures. Math. Models Methods Appl. Sci. 26 (2016) 1025–1049. | DOI | MR | Zbl
[7] , , and , Time-dependent density-functional theory for strongly interacting electrons. Phys. Rev. A 95 (2017) 042505. | DOI
[8] , and , Density functional theory and optimal transportation with coulomb cost. Comm. Pure Appl. Math. 66 (2013) 548–599. | DOI | MR | Zbl
[9] , and , Smoothing of transport plans with fixed marginals and rigorous semiclassical limit of the Hohenberg-Kohn functional. Arch. Ration. Mech. Anal. 228 (2018) 891–922. | DOI | MR | Zbl
[10] , Optimal transport with coulomb cost. approximation and duality. ESAIM: M2AN 49 (2015) 1643–1657. | DOI | Numdam | MR | Zbl
[11] , and , Multimarginal optimal transport maps for 1-dimensional repulsive costs. Can. J. Math. 67 (2015) 350–368. | DOI | MR | Zbl
[12] , and , Topological Optimization and Optimal Transport In the Applied Sciences. De Gruyter (2017).
[13] , , , and , N-density representability and the optimal transport limit of the hohenberg-kohn functional. J. Chem. Phys. 139 (2013) 164109. | DOI
[14] and , Existence of optimal maps in the reflector-type problems. ESAIM: COCV 13 (2007) 93–106. | Numdam | MR | Zbl
[15] , and , Non-existence of optimal transport maps for the multi-marginal repulsive harmonic cost. Preprint (2018). | arXiv | MR
[16] , and , Density-functional theory for strongly interacting electrons. Phys. Rev. Lett. 103 (2009) 166402. | DOI
[17] and , Differentiation of set functions using Vitali coverings. Trans. Amer. Math. Soc. 96 (1960) 185–209. | DOI | MR | Zbl
[18] and , Inhomogeneous electron gas. Phys. Rev. 136 (1964) B864. | DOI | MR
[19] , and , Existence of doubling measures via generalised nested cubes. Proc. Am. Math. Soc. 140 (2012) 3275–3281. | DOI | MR | Zbl
[20] , Duality theorems for marginal problems. Z. Wahrscheinlichkeitstheor. Verw. Geb. 67 (1984) 399–432. | DOI | MR | Zbl
[21] , , , , and , The adiabatic strictly-correlated-electrons functional: kernel and exact properties. Phys. Chem. Chem. Phys. 18 (2016) 21092–21101. | DOI
[22] , 2016 Universal variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solution of the v-representability problem. Proc. Nati. Acad. Sci. 12 (76) 6062–6065. | MR
[23] , Semi-classical limit of the Levy-Lieb functional in density functional theory. C. R. Math. Acad. Sci. Paris 356 (2018) 449–455. | DOI | MR | Zbl
[24] , Density functionals for coulomb systems, in Inequalities. Springer, Berlin, Heidelberg (2002) 269–303. | DOI
[25] and , Strong in Kohn-Sham density functional theory. Phys. Rev. Lett. 109 (2012) 246402. | DOI
[26] , Strong-interaction limit of density-functional theory. Phys. Rev. A, 60 (1999) 4387. | DOI
[27] , , , , and , The strictly-correlated electron functional for spherically symmetric systems revisited. Preprint (2017). | arXiv
[28] , and , Strictly correlated electrons in density-functional theory: a general formulation with applications to spherical densities. Phys. Rev. A 75 (2007) 042511. | DOI
[29] . Optimal Transport: old and New, Vol. 338. Springer Science & Business Media, Berlin, Heidelberg (2008). | Zbl
[30] , On the design of a reflector antenna ii. Calc. Var. Partial Differ. Equ. 20 (2004) 329.–341. | DOI | MR | Zbl
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