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(Omega) of first order distribution. A particular subclass X0^sharp(Omega) of such distributions will be considered which includes the infinite sums of dipoles sumk(δpk-δnk) studied in citeP1, 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 X0^sharp(Omega) can be then deduced.