no. 3
Transport problems and disintegration maps
ESAIM: Control, Optimisation and Calculus of Variations, Volume 19 (2013) no. 3, pp. 888-905

By disintegration of transport plans it is introduced the notion of transport class. This allows to consider the Monge problem as a particular case of the Kantorovich transport problem, once a transport class is fixed. The transport problem constrained to a fixed transport class is equivalent to an abstract Monge problem over a Wasserstein space of probability measures. Concerning solvability of this kind of constrained problems, it turns out that in some sense the Monge problem corresponds to a lucky case.

DOI: 10.1051/cocv/2012037
Classification: 37J50, 49Q20, 49Q15
Keywords: optimal mass transportation theory, Monge − Kantorovich problem, calculus of variations, shape analysis, geometric measure theory 
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Granieri, Luca; Maddalena, Francesco. Transport problems and disintegration maps. ESAIM: Control, Optimisation and Calculus of Variations, Volume 19 (2013) no. 3, pp. 888-905. doi: 10.1051/cocv/2012037

[1] P.T. Abdellaoui and H. Heinich, Caracterisation d'une solution optimale au probleme de Monge − Kantorovich. Bull. Soc. Math. France 127 (1999) 429-443. | Zbl | MR | Numdam

[2] N. Ahmad, H.K. Kim and R.J. Mccann, Optimal transportation, topology and uniqueness. Bull. Math. Sci. 1 (2011) 13-32.

[3] L. Ambrosio, Lecture Notes on Transport Problems, in Mathematical Aspects of Evolving Interfaces. Lect. Notes Math. vol. 1812. Springer, Berlin (2003) 1-52. | Zbl | MR

[4] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, New York (2000). | Zbl | MR

[5] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lect. Notes Math. ETH Zürich, Birkhäuser (2005). | Zbl | MR

[6] P. Bernard, Young measures, superposition and transport. Indiana Univ. Math. J. 57 (2008) 247-276. | Zbl | MR

[7] G. Carlier and A. Lachapelle, A Planning Problem Combining Calculus of Variations and Optimal Transport. Appl. Math. Optim. 63 (2011) 1-9. | Zbl | MR

[8] J.A. Cuesta-Albertos and A. Tuero-Diaz, A characterization for the Solution of the Monge − Kantorovich Mass Transference Problem. Statist. Probab. Lett. 16 (1993) 147-152. | Zbl | MR

[9] I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: Lp spaces. Springer (2007). | Zbl | MR

[10] W. Gangbo, The Monge Transfer Problem and its Applications. Contemp. Math. 226 (1999) 79-104. | Zbl | MR

[11] J. Gonzalez-Hernandez and J. Gonzalez-Hernandez, Extreme Points of Sets of Randomized Strategies in Constrained Optimization and Control Problems. SIAM J. Optim. 15 (2005) 1085-1104. | Zbl | MR

[12] J. Gonzalez-Hernandez, J. Rigoberto Gabriel and J. Gonzalez-Hernandez, On Solutions to the Mass Transfer Problem. SIAM J. Optim. 17 (2006) 485-499. | Zbl

[13] L. Granieri, Optimal Transport and Minimizing Measures. LAP Lambert Academic Publishing (2010).

[14] L. Granieri and F. Maddalena, A Metric Approach to Elastic reformations, preprint (2012), on http://cvgmt.sns.it. | MR

[15] V. Levin, Abstract Cyclical Monotonicity and Monge Solutions for the General Monge − Kantorovich Problem. Set-Valued Anal. 7 (1999) 7-32. | Zbl | MR

[16] M. Mcasey and L. Mou, Optimal Locations and the Mass Transport Problem. Contemp. Math. 226 (1998) 131-148. | Zbl | MR

[17] A. Pratelli, Existence of optimal transport maps and regularity of the transport density in mass transportation problems, Ph.D. Thesis, Scuola Normale Superiore, Pisa (2003).

[18] S.T. Rachev and L. Ruschendorf, Mass Transportation Problems, Probab. Appl. Springer-Verlag, New York I (1998).

[19] C. Villani, Topics in Mass Transportation. Grad. Stud. Math., vol. 58. AMS, Providence, RI (2004). | MR

[20] C. Villani, Optimal Transport, Old and New. Springer (2009). | Zbl | MR

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