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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
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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

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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

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