Stratified Monge-Kantorovich optimal transport problems

In this paper, we investigate Monge-Kantorovich problems for which the absolute continuity of marginals is relaxed. For X, Y^n+1 let (X, BX, ) and (Y, BY, ) be two Borel probability spaces, c: X Y be a cost function, and consider the problem align*MKPMKPEQ \ₗ ₘ c (x, y) \, d\: \ (, ) \. align* Inspired by the seminal paper GANGBOMCCANN2 with applications in shape recognition problem, we first consider MKPEQ for the cost c (x, y) =h (x-y) with h strictly convex defined on the multi-layers target space align* X=X\x\, Y=₊=₁K (Y₊ \yₖ\), align* where X, Y₊ R^n for k \1, , K\, x R, and \y₁,. . . , yK\ R. Here, we assume that |Xⁿ (the Lebesgue measure on Rⁿ), but is singular w. r. t. L^n+1. When K=1, this translates to the standard MKPEQ for which the unique solution is concentrated on a map. We show that for K 2, the solution is still unique but it concentrates on the graph of several maps. Next, we study MKPEQ for a closed subset X R^n+1 and its n-dimensional submanifold X₀ with the first marginal of the form align* X f (x) \, d (x) =X f (x) (x) \, dL^n+1 (x) +ₗ䃐 f (x₀) \, d S (x₀), \ \ f Cb (X). align* Here, S is a measure on X₀ such that S L^n on each coordinate chart of X₀. This can be seen as a two-layers problem as the measure charges both n- and n+1-dimensional subsets.