# American Institute of Mathematical Sciences

April  2014, 34(4): 1465-1480. doi: 10.3934/dcds.2014.34.1465

## Remarks on multi-marginal symmetric Monge-Kantorovich problems

 1 Department of Mathematics, The University of British Columbia, Vancouver BC Canada V6T 1Z2 2 Institut de Mathématiques, UMR 7586 - CNRS, Université Paris Diderot - Paris 7, Paris, France

Received  November 2012 Revised  February 2013 Published  October 2013

Symmetric Monge-Kantorovich transport problems involving a cost function given by a family of vector fields were used by Ghoussoub-Moameni to establish polar decompositions of such vector fields into $m$-cyclically monotone maps composed with measure preserving $m$-involutions ($m\geq 2$). In this note, we relate these symmetric transport problems to the Brenier solutions of the Monge and Monge-Kantorovich problem, as well as to the Gangbo-Święch solutions of their multi-marginal counterparts, both of which involving quadratic cost functions.
Citation: Nassif Ghoussoub, Bernard Maurey. Remarks on multi-marginal symmetric Monge-Kantorovich problems. Discrete and Continuous Dynamical Systems, 2014, 34 (4) : 1465-1480. doi: 10.3934/dcds.2014.34.1465
##### References:
 [1] M. Agueh and G. Carlier, Barycenters in the Wasserstein space, SIAM J. Math. Anal., 43 (2011), 904-924. doi: 10.1137/100805741. [2] R. S. Burachik and B. F. Svaiter,, Maximal monotonicity, conjugation and the duality product, Proc. Amer. Math. Soc., 131 (2003), 2379-2383. doi: 10.1090/S0002-9939-03-07053-9. [3] Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math., 44 (1991), 375-417. doi: 10.1002/cpa.3160440402. [4] G. Carlier and B. Nazaret, Optimal transportation for the determinant, ESAIM Control Optim. Calc. Var., 14 (2008), 678-698. doi: 10.1051/cocv:2008006. [5] S. P. Fitzpatrick, Representing monotone operators by convex functions, in "Workshop/Miniconference on Functional Analysis and Optimization" (Canberra, 1988), Proc. Centre Math. Anal. Austral. Nat. Univ., 20, Austral. Nat. Univ., Canberra, (1988), 59-65. [6] A. Galichon and N. Ghoussoub, Variational representations for N-cyclically monotone vector fields, arXiv:1207.2408, (2012). [7] W. Gangbo and A. Święch, Optimal maps for the multidimensional Monge-Kantorovich problem, Comm. Pure Appl. Math., 51 (1998), 23-45. doi: 10.1002/(SICI)1097-0312(199801)51:1<23::AID-CPA2>3.0.CO;2-H. [8] N. Ghoussoub, Anti-self-dual Lagrangians: Variational resolutions of non self-adjoint equations and dissipative evolutions, AIHP-Analyse Non Linéaire, 24 (2007), 171-205. doi: 10.1016/j.anihpc.2006.02.002. [9] N. Ghoussoub, A variational theory for monotone vector fields, Journal of Fixed Point Theory and Applications, 4 (2008), 107-135. doi: 10.1007/s11784-008-0083-4. [10] N. Ghoussoub, Anti-symmetric Hamiltonians: Variational resolution of Navier-Stokes equations and other nonlinear evolutions, Comm. Pure & Applied Math., 60 (2007), 619-653. doi: 10.1002/cpa.20176. [11] N. Ghoussoub, "Selfdual Partial Differential Systems and their Variational Principles," Springer Monograph in Mathematics, Springer-Verlag, 2008. [12] N. Ghoussoub and A. Moameni, A self-dual polar factorization for vector fields, Comm. Pure. Applied. Math., 66 (2013), 905-933. doi: 10.1002/cpa.21430. [13] N. Ghoussoub and A. Moameni, Symmetric Monge-Kantorovich problems and polar decompositions of vector fields, preprint, arXiv:1302.2886, (2013). [14] P. Millien, "A Functional Analytic Approach to the Selfdual Polar Decomposition," Masters Thesis, UBC, 2011. [15] E. Krauss, A representation of arbitrary maximal monotone operators via subgradients of skew-symmetric saddle functions, Nonlinear Anal., 9 (1985), 1381-1399. doi: 10.1016/0362-546X(85)90097-5. [16] B. Pass, Optimal transportation with infinitely many marginals, J. Funct. Anal., 264 (2013), 947-963. doi: 10.1016/j.jfa.2012.12.002. [17] B. Pass, Ph.D thesis, University of Toronto, 2011. [18] B. Pass, Uniqueness and Monge solutions in the multimarginal optimal transportation problem, SIAM Journal on Mathematical Analysis, 43 (2011), 2758-2775 doi: 10.1137/100804917. [19] B. Pass, On the local structure of optimal measures in the multi-marginal optimal transportation problem, Calculus of Variations and Partial Differential Equations, 43 (2012), 529-536. doi: 10.1007/s00526-011-0421-z. [20] R. R. Phelps, "Convex Functions, Monotone Operators and Differentiability," Second edition, Lecture Notes in Math., 1364, Springer-Verlag, Berlin, 1993. [21] T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, NJ, 1970. [22] B. F. Svaiter, Fixed points in the family of convex representations of a maximal monotone operator, Proc. Amer. Math. Soc., 131 (2003), 3851-3859. doi: 10.1090/S0002-9939-03-07083-7. [23] C. Villani, "Topics in Optimal Transportation," Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI, 2003. doi: 10.1007/b12016. [24] C. Villani, "Optimal Transport. Old and New," Grundlehren der Mathematischen Wissenschaften, 338, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.

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##### References:
 [1] M. Agueh and G. Carlier, Barycenters in the Wasserstein space, SIAM J. Math. Anal., 43 (2011), 904-924. doi: 10.1137/100805741. [2] R. S. Burachik and B. F. Svaiter,, Maximal monotonicity, conjugation and the duality product, Proc. Amer. Math. Soc., 131 (2003), 2379-2383. doi: 10.1090/S0002-9939-03-07053-9. [3] Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math., 44 (1991), 375-417. doi: 10.1002/cpa.3160440402. [4] G. Carlier and B. Nazaret, Optimal transportation for the determinant, ESAIM Control Optim. Calc. Var., 14 (2008), 678-698. doi: 10.1051/cocv:2008006. [5] S. P. Fitzpatrick, Representing monotone operators by convex functions, in "Workshop/Miniconference on Functional Analysis and Optimization" (Canberra, 1988), Proc. Centre Math. Anal. Austral. Nat. Univ., 20, Austral. Nat. Univ., Canberra, (1988), 59-65. [6] A. Galichon and N. Ghoussoub, Variational representations for N-cyclically monotone vector fields, arXiv:1207.2408, (2012). [7] W. Gangbo and A. Święch, Optimal maps for the multidimensional Monge-Kantorovich problem, Comm. Pure Appl. Math., 51 (1998), 23-45. doi: 10.1002/(SICI)1097-0312(199801)51:1<23::AID-CPA2>3.0.CO;2-H. [8] N. Ghoussoub, Anti-self-dual Lagrangians: Variational resolutions of non self-adjoint equations and dissipative evolutions, AIHP-Analyse Non Linéaire, 24 (2007), 171-205. doi: 10.1016/j.anihpc.2006.02.002. [9] N. Ghoussoub, A variational theory for monotone vector fields, Journal of Fixed Point Theory and Applications, 4 (2008), 107-135. doi: 10.1007/s11784-008-0083-4. [10] N. Ghoussoub, Anti-symmetric Hamiltonians: Variational resolution of Navier-Stokes equations and other nonlinear evolutions, Comm. Pure & Applied Math., 60 (2007), 619-653. doi: 10.1002/cpa.20176. [11] N. Ghoussoub, "Selfdual Partial Differential Systems and their Variational Principles," Springer Monograph in Mathematics, Springer-Verlag, 2008. [12] N. Ghoussoub and A. Moameni, A self-dual polar factorization for vector fields, Comm. Pure. Applied. Math., 66 (2013), 905-933. doi: 10.1002/cpa.21430. [13] N. Ghoussoub and A. Moameni, Symmetric Monge-Kantorovich problems and polar decompositions of vector fields, preprint, arXiv:1302.2886, (2013). [14] P. Millien, "A Functional Analytic Approach to the Selfdual Polar Decomposition," Masters Thesis, UBC, 2011. [15] E. Krauss, A representation of arbitrary maximal monotone operators via subgradients of skew-symmetric saddle functions, Nonlinear Anal., 9 (1985), 1381-1399. doi: 10.1016/0362-546X(85)90097-5. [16] B. Pass, Optimal transportation with infinitely many marginals, J. Funct. Anal., 264 (2013), 947-963. doi: 10.1016/j.jfa.2012.12.002. [17] B. Pass, Ph.D thesis, University of Toronto, 2011. [18] B. Pass, Uniqueness and Monge solutions in the multimarginal optimal transportation problem, SIAM Journal on Mathematical Analysis, 43 (2011), 2758-2775 doi: 10.1137/100804917. [19] B. Pass, On the local structure of optimal measures in the multi-marginal optimal transportation problem, Calculus of Variations and Partial Differential Equations, 43 (2012), 529-536. doi: 10.1007/s00526-011-0421-z. [20] R. R. Phelps, "Convex Functions, Monotone Operators and Differentiability," Second edition, Lecture Notes in Math., 1364, Springer-Verlag, Berlin, 1993. [21] T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, NJ, 1970. [22] B. F. Svaiter, Fixed points in the family of convex representations of a maximal monotone operator, Proc. Amer. Math. Soc., 131 (2003), 3851-3859. doi: 10.1090/S0002-9939-03-07083-7. [23] C. Villani, "Topics in Optimal Transportation," Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI, 2003. doi: 10.1007/b12016. [24] C. Villani, "Optimal Transport. Old and New," Grundlehren der Mathematischen Wissenschaften, 338, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.
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