Continuity and Estimates for Multimarginal Optimal Transportation Problems with Singular Costs

Abstract

We consider some repulsive multimarginal optimal transportation problems which include, as a particular case, the Coulomb cost. We prove a regularity property of the minimizers (optimal transportation plan) from which we deduce existence and some basic regularity of a maximizer for the dual problem (Kantorovich potential). This is then applied to obtain some estimates of the cost and to the study of continuity properties.This problem is called a multimarginal optimal transportation problem and elements of Π(ρ) are called transportation plans for ρ. Some general results about multimarginal optimal transportation problems are available in [3,17,21,22,23]. Results for particular cost functions are available, for example in [11] for the quadratic cost, with some generalization in [15], and in [4] for the determinant cost function.Optimization problems for the cost function C(ρ) in (1.1) intervene in the socalled Density Functional Theory (DFT), we refer to [16,18] for the basic theory of DFT and to [13,14,24,25,26] for recent development which are of interest for us. Some new applications are emerging for example in [12]. In the particular case of the Coulomb cost there are also many other open questions related to the applications. Recent results on the topic are contained in [2,5,6,7,10,20] and some of them will

Duality theory for multi-marginal optimal transport with repulsive costs in metric spaces

“…In Section 4, we extend the duality results of [18,9,2] for a class of unbounded cost functions (Theorem 4.1) and in Section 5 we obtain regularity results of Kantorovich potentials (Theorem 5.2) as well as continuity of the cost functional as a function of the marginal ρ.…”

“…By using arguments from physics, Seidl suggested that, at least in the case when ρ is radially symmetric, a minimizer γ in (1.1) exists and is concentrated on the graph of a map T : R 3 → R 3 , T ♯ ρ = ρ, and its iterates, i.e. γ = (Id, T, T (2) , . .…”

“…More recently, De Pascale [9] and Buttazzo, Champion and De Pascale [2] extended the duality theory for a class of repulsive cost functions c : R dN → R ∪ {+∞} which are bounded from below, allowing, for instance, the inclusion of the Coulomb (s = 1) and Riesz cost functions…”

“…The main contribution of this paper is to extend the duality theory for logarithmic costs. Some of our proofs are based on arguments present in [2]. One ingredient to tackle the problem of costs that are not bounded from below is to consider, for R ∈ ]0, ∞[, the truncated cost functions…”

Multi-marginal entropy-transport with repulsive cost

“…Now we may consider two, a priori different, entropy-regularized MOT problems: the one introduced in (2.3) 4) and the problem with the reference measure chosen to be (ρm) ⊗N…”