no. 1
An optimal matching problem
ESAIM: Control, Optimisation and Calculus of Variations, Tome 11 (2005) no. 1, pp. 57-71

Given two measured spaces (X,μ) and (Y,ν), and a third space Z, given two functions u(x,z) and v(x,z), we study the problem of finding two maps s:X→Z and t:Y→Z such that the images s(μ) and t(ν) coincide, and the integral ∫ X u(x,s(x))dμ-∫ Y v(y,t(y))dν is maximal. We give condition on u and v for which there is a unique solution.

DOI : 10.1051/cocv:2004034
Classification : 05C38, 15A15, 05A15, 15A18
Keywords: optimal transportation, measure-preserving maps 
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     title = {An optimal matching problem},
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     pages = {57--71},
     year = {2005},
     publisher = {EDP Sciences},
     volume = {11},
     number = {1},
     doi = {10.1051/cocv:2004034},
     mrnumber = {2110613},
     zbl = {1106.49054},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/cocv:2004034/}
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Ekeland, Ivar. An optimal matching problem. ESAIM: Control, Optimisation and Calculus of Variations, Tome 11 (2005) no. 1, pp. 57-71. doi: 10.1051/cocv:2004034

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