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.
Mots clés : optimal transportation map, convex constraint, Monge transportation problem
@article{COCV_2013__19_4_1064_0,
author = {Chen, Ping and Jiang, Feida and Yang, Xiaoping},
title = {Two dimensional optimal transportation problem for a distance cost with a convex constraint},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1064--1075},
publisher = {EDP-Sciences},
volume = {19},
number = {4},
year = {2013},
doi = {10.1051/cocv/2013045},
mrnumber = {3182681},
zbl = {1282.49036},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2013045/}
}
TY - JOUR AU - Chen, Ping AU - Jiang, Feida AU - Yang, Xiaoping TI - Two dimensional optimal transportation problem for a distance cost with a convex constraint JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2013 SP - 1064 EP - 1075 VL - 19 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2013045/ DO - 10.1051/cocv/2013045 LA - en ID - COCV_2013__19_4_1064_0 ER -
%0 Journal Article %A Chen, Ping %A Jiang, Feida %A Yang, Xiaoping %T Two dimensional optimal transportation problem for a distance cost with a convex constraint %J ESAIM: Control, Optimisation and Calculus of Variations %D 2013 %P 1064-1075 %V 19 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2013045/ %R 10.1051/cocv/2013045 %G en %F COCV_2013__19_4_1064_0
Chen, Ping; Jiang, Feida; Yang, Xiaoping. Two dimensional optimal transportation problem for a distance cost with a convex constraint. ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 4, pp. 1064-1075. doi : 10.1051/cocv/2013045. http://www.numdam.org/articles/10.1051/cocv/2013045/
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