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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 |
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] | |
[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. |
show all references
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] | |
[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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