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 $X(\mathrm{\Omega })$ of first order distribution. A particular subclass $X_0^{\mathrm{♯}}(\mathrm{\Omega })$ 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 $X(\mathrm{\Omega })$ and $X_0^{\mathrm{♯}}(\mathrm{\Omega })$ can be then deduced.