The Monge-Kantorovich Problem for Distributions and Applications

\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.