
Journal of Convex Analysis 17 (2010), No. 3&4, 925943 Copyright Heldermann Verlag 2010 The MongeKantorovich Problem for Distributions and Applications Guy Bouchitté Lab. d' Analyse Non Linéaire Appliquée, U.F.R. des Sciences et Techniques, Université du Sud ToulonVar, Avenue de l'Université, 83957 La Garde, France bouchitte@univtln.fr Giuseppe Buttazzo Dip. di Matematica, Università di Pisa, Largo Pontecorvo 5, 56127 Pisa, Italy buttazzo@dm.unipi.it Luigi De Pascale Dip. di Matematica Applicata, Università di Pisa, Via Buonarroti 1/C, 56127 Pisa, Italy depascal@dm.unipi.it [Abstractpdf] \def\xsp{{\bf X}(\Omega)} \def\x0s{{\bf X}_0^\sharp (\Omega)} We study the KantorovichRubinstein 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) 571576; and "On the distributions of the form $\sum_i (\delta_{p_i}\delta_{n_i})$", J. Funct. Anal. 210 (2004) 391435]. 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: MongeKantorovich problem, optimal transportation, transhipment problem, flat norm, minimal connections, Jacobians. MSC: 49J45, 49J20, 82C70, 90B06 [ Fulltextpdf (200 KB)] for subscribers only. 