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(Ω) of such distributions will be considered which includes the infinite sums of dipoles ∑ k(δpk − δnk) studied in [28, 29]. 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(Ω) can be then deduced.