The Monge-Kantorovich problem for distributions and applications

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 $X(\Omega )$ of first order distribution. A particular subclass $X_0^{♯}(\Omega )$ of such distributions will be considered which includes the infinite sums of dipoles ${\sum }_k({\delta }_{p_k}-{\delta }_{n_k})$ studied in \cite{P1, P2}. 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 $X(\Omega )$ and $X_0^{♯}(\Omega )$ can be then deduced.

arXiv comment15 pages, 0 figures, The final publication is available at http://www.heldermann.de/JCA/jcacover.htm