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
bouchitte@univ-tln.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
[Abstract-pdf]
\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
[ Fulltext-pdf (200 KB)] for subscribers only.