We consider the nonlinear optimal transportation problem of minimizing the cost functional C ∞ (λ) = λ-ess sup (x,y)∈Ω 2 |y − x| in the set of probability measures on Ω 2 having prescribed marginals. This corresponds to the question of characterizing the measures that realize the innite Wasserstein distance. We establish the existence of local solutions and characterize this class with the aid of an adequate version of cyclical monotonicity. Moreover, under natural assumptions, we show that local solutions are induced by transport maps.

We first consider the Monge problem in a convex bounded subset of R d. The cost is given by a general norm, and we prove the existence of an optimal transport map under the classical assumption that the first marginal is absolutely continuous with respect to the Lebesgue measure. In the final part of the paper we show how to extend this existence result to a general open subset of R d .

We study a class of optimal transport planning problems where the reference cost involves a non-linear function G(x, p) representing the transport cost between the Dirac measure δx and a target probability p. This allows to consider interesting models which favour multi-valued transport maps in contrast with the classical linear case ($G(x,p)=\int c(x,y)dp$) where finding single-valued optimal transport is a key issue. We present an existence result and a general duality principle which apply to many examples. Moreover, under a suitable subadditivity condition, we derive a Kantorovich–Rubinstein version of the dual problem allowing to show existence in some regular cases. We also consider the well studied case of Martingale transport and present some new perspectives for the existence of dual solutions in connection with Γ-convergence theory.

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

Abstract.In this paper, we prove that the L p approximants naturally associated to a supremal functional Γ-converge to it. This yields a lower semicontinuity result for supremal functionals whose supremand satisfy weak coercivity assumptions as well as a generalized Jensen inequality. The existence of minimizers for variational problems involving such functionals (together with a Dirichlet condition) then easily follows. In the scalar case we show the existence of at least one absolute minimizer (i.e. local solution) among these minimizers. We provide two different proofs of this fact relying on different assumptions and techniques.Mathematics Subject Classification. 49J45, 49J99.