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Journal of Convex Analysis 17 (2010), No. 3&4, 925--943
Copyright Heldermann Verlag 2010

The Monge-Kantorovich Problem for Distributions and Applications

Guy Bouchitté
Lab. d' Analyse Non Linéaire Appliquée, U.F.R. des Sciences et Techniques, Université du Sud Toulon-Var, Avenue de l'Université, 83957 La Garde, France

Giuseppe Buttazzo
Dip. di Matematica, Università di Pisa, Largo Pontecorvo 5, 56127 Pisa, Italy

Luigi De Pascale
Dip. di Matematica Applicata, Università di Pisa, Via Buonarroti 1/C, 56127 Pisa, Italy


\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.

Keywords: Monge-Kantorovich problem, optimal transportation, transhipment problem, flat norm, minimal connections, Jacobians.

MSC: 49J45, 49J20, 82C70, 90B06

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