The Monge-Kantorovich problem for distributions and applications

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 $X(Ω)$ of first order distribution. A particular subclass $X_0^\sharp(Ω)$ of such distributions will be considered which includes the infinite sums of dipoles $\sum_k(δ_{p_k}-δ_{n_k})$ 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 $X(Ω)$ and $X_0^\sharp(Ω)$ can be then deduced.

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