Hostname: page-component-77c89778f8-cnmwb Total loading time: 0 Render date: 2024-07-16T14:00:24.398Z Has data issue: false hasContentIssue false
  • English
  • Français

Two dimensional optimal transportation problem for a distance cost with a convex constraint∗

Published online by Cambridge University Press:  04 July 2013

Ping Chen ,
Feida Jiang  and
Xiaoping Yang
Ping Chen
Affiliation:
School of Science, Nanjing University of Science and Technology, Nanjing 210094, P.R. China. chenping200517@126.com; jfd2001@163.com; yangxp@mail.njust.edu.cn School of Mathematics and Computer Science, Anhui Normal University, Wuhu 241000, P.R. China
Feida Jiang
Affiliation:
School of Science, Nanjing University of Science and Technology, Nanjing 210094, P.R. China. chenping200517@126.com; jfd2001@163.com; yangxp@mail.njust.edu.cn
Xiaoping Yang
Affiliation:
School of Science, Nanjing University of Science and Technology, Nanjing 210094, P.R. China. chenping200517@126.com; jfd2001@163.com; yangxp@mail.njust.edu.cn
Get access

Abstract

We first prove existence and uniqueness of optimal transportation maps for the Monge’s problem associated to a cost function with a strictly convex constraint in the Euclidean plane ℝ2. The cost function coincides with the Euclidean distance if the displacement y − x belongs to a given strictly convex set, and it is infinite otherwise. Secondly, we give a sufficient condition for existence and uniqueness of optimal transportation maps for the original Monge’s problem in ℝ2. Finally, we get existence of optimal transportation maps for a cost function with a convex constraint, i.e. y − x belongs to a given convex set with at most countable flat parts.

Keywords

Optimal transportation mapconvex constraintMonge transportation problem
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 4 , October 2013 , pp. 1064 - 1075
Copyright
© EDP Sciences, SMAI, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

L. Ambrosio, Lectures notes on optimal transport problems. In Mathematical aspects of evolving interfaces, CIME summer school in Madeira (Pt), vol. 1812, edited by P. Colli and J. Rodrigues, Springer (2003) 1–52.
Ambrosio, L. and Rigot, S., Optimal mass transportation in the Heisenberg group. J. Funct. Anal. 208 (2004) 261–301. Google Scholar
Bianchini, S. and Cavalletti, F., The monge problem in geodesic spaces. IMA Vol. Math. Appl. 153 (2011) 217–233. Google Scholar
Caffarelli, L.A., Feldman, M. and McCann, R.J., Constructing optimal maps for Morge’s transport problem as a limit of strictly convex costs. J. Amer. Math. Soc. 15 (2001) 1–26. Google Scholar
Carlier, G., De Pascale, L. and Santambrogio, F., A strategy for non-strictly convex transport cost and the example of  || x − y || p in R2. Commun. Math. Sci. 8 (2010), 931–941. Google Scholar
Champion, T. and De Pascale, L., The monge problem for strictly convex norms in Rd. J. Eur. Math. Soc. 12 (2010) 1355–1369. Google Scholar
Champion, T. and De Pascale, L., The monge problem in Rd. Duke Math. J. 157 (2011) 551–572. Google Scholar
Champion, T., De Pascale, L. and Juutinen, P., The ∞ − Wasserstein distance: local solutions and existence of optimal transport maps. SIAM J. Math. Anal. 40 (2008) 1–20. Google Scholar
P. Chen, F. Jiang and X.-P. Yang, Optimal transportation in Rn for a distance cost with convex constraints. To appear.
De Pascale, L. and Rigot, S., Monge’s transport problem in the Heisenberg group. Adv. Calc. Var. 4 (2010) 195–227. Google Scholar
Evans, L.C. and Gangbo, W., Differantial equations methods for the Monge–Kantorovich mass transfer problem. Mem. Amer. Math. Soc. 137 (1999) 1–66. Google Scholar
Jimenez, C. and Santambrogio, F., Optimal transportation for a quadratic cost with convex constrains and applications. J. Math. Pures Appl. 98 (2012) 103–113. Google Scholar
Kantorovich, L.V., On the translocation of masses. Dokl. Akad. Nauk. USSR 37 (1942) 199–201.Google Scholar
Kantorovich, L., On a problem of Monge (in Russian). Uspekhi Mat. Nauk. 3 (1948) 225–226.Google Scholar
R.J. McCann, N. Guillen, Five lectures on optimal transportation: geometry, regularity and applications. To appear in Proc. of the Séminaire de Mathématiques Supérieure (SMS) held in Montréal, QC, June 27-July 8, 2011.
G. Monge, Mémoire sur la théorie des déblais et des Remblais. Histoire de l’Académie Royal des Sciences de Paris (1781) 666–704.
Sudakov, V.N., Geometric problems in the theory of infinite-dimensional probability distributions. Proc. Steklov Inst. Math. 141 (1979) 1–178. Google Scholar
Trudinger, N.S. and Wang, X.-J., On the Monge mass transfer problem. Calc. Var. Partial Differ. Equ. 13 (2001) 19–31. Google Scholar
C. Villani, Topics in optimal transportation. Graduate Studies in Mathematics, J. Amer. Math. Soc. Providence, RI 58 (2003).
C. Villani, Optimal transport, old and new. Springer Verlag (2008).