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,Yⁿ⁺¹ let (X,BX,μ) and (Y,BY,ν) be two Borel probability spaces, c:X× Y be a cost function, and consider the problem {align*}{MKP}{MKPEQ} inf\{∫X× Y c(x,y)\,dλ\ :\ λ ∈Π(μ,ν) \}. {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=X̄×\{x̄\}, Y=ₖ₌₁^K (Ȳₖ× \{ȳ_k\}), {align*} where X̄, Ȳₖ⊆ Rⁿ for k∈ \1,…,K\, x̄∈ R, and ̄₁,..., ȳK\⊆ R. Here, we assume that μ|X̄ⁿ (the Lebesgue measure on Rⁿ), but μ is singular w.r.t. Lⁿ⁺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⊆ Rⁿ⁺¹ and its n-dimensional submanifold X₀ with the first marginal of the form {align*} ∫_X f(x)\,dμ(x)=∫_X f(x)α(x)\,dLⁿ⁺¹(x)+∫X_0 f(x_0)\,d S(x_0),\ \ ∀ f∈ C_b(X). {align*} Here, S is a measure on X₀ such that S Lⁿ on each coordinate chart of X₀. This can be seen as a two-layers problem as the measure μ charges both n- and $n+1$-dimensional subsets.