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 $\mathbf{X}(\Omega)$ of first order distribution. A particular subclass $\mathbf{X}_0^\sharp(\Omega)$ 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 $\mathbf{X}(\Omega)$ and $\mathbf{X}_0^\sharp(\Omega)$ can be then deduced.