Geodesic
The Monge-Kantorovich Problem for Distributions and Applications
Journal of convex analysis, Tome 17 (2010) no. 3, pp. 925-943.

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\def\xsp{{\bf X}(\Omega)} \def\x0s{{\bf X}_0^\sharp (\Omega)} We study the Kantorovich-Rubinstein transhipment problem when the difference between the source and the target is not anymore a balanced measure but belongs to a suitable subspace $\xsp$ of first order distribution. A particular subclass $\x0s$ of such distributions will be considered which includes the infinite sums of dipoles $\sum_k(\delta_{p_k}-\delta_{n_k})$ studied recently by A. C. Ponce ["On the distributions of the form $\sum_i (\delta_{p_i}-\delta_{n_i})$", C. R. Math. Acad. Sci. Paris 336 (2003) 571--576; and "On the distributions of the form $\sum_i (\delta_{p_i}-\delta_{n_i})$", J. Funct. Anal. 210 (2004) 391--435]. In spite of this weakened regularity, it is shown that an optimal transport density still exists among nonnegative finite measures. Some geometric properties of the Banach spaces $\xsp$ and $\x0s$ can be then deduced.
Classification : 49J45, 49J20, 82C70, 90B06
Mots-clés : Monge-Kantorovich problem, optimal transportation, transhipment problem, flat norm, minimal connections, Jacobians 
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     author = {G. Bouchitt\'e and G. Buttazzo and L. De Pascale},
     title = {The {Monge-Kantorovich} {Problem} for {Distributions} and {Applications}},
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G. Bouchitté; G. Buttazzo; L. De Pascale. The Monge-Kantorovich Problem for Distributions and Applications. Journal of convex analysis, Tome 17 (2010) no. 3, pp. 925-943. http://geodesic.mathdoc.fr/item/JCA_2010_17_3_JCA_2010_17_3_a12/