Abstract

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 X(Ω)\mathbf{X}(\Omega) of first order distribution. A particular subclass X0♯(Ω)\mathbf{X}_0^\sharp(\Omega) of such distributions will be considered which includes the infinite sums of dipoles ∑k(δpk−δnk)\sum_k(\delta_{p_k}-\delta_{n_k}) studied recently by A. C. Ponce ["On the distributions of the form ∑i(δpi−δni)\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 ∑i(δpi−δni)\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 X(Ω)\mathbf{X}(\Omega) and X0♯(Ω)\mathbf{X}_0^\sharp(\Omega) can be then deduced.

Author information

Contact details are reproduced from the original publication and may be historical.

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

Keywords

  • Monge-Kantorovich problem
  • optimal transportation
  • transhipment problem
  • flat norm
  • minimal connections
  • Jacobians

Mathematics Subject Classification

49J45, 49J20, 82C70, 90B06

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Suggested citation

G. Bouchitté, G. Buttazzo, L. De Pascale. “The Monge-Kantorovich Problem for Distributions and Applications.” Journal of Convex Analysis 17 (2010), No. 3&4, 925–943. https://doi.org/10.68381/jca17058

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Published by Heldermann Verlag, 2010. Rights now held by Banach Press LLC.