Related papers: Partial $W^{2,p}$ regularity for optimal transport…

The optimal transport problem with quadratic regularization is useful when sparse couplings are desired. The density of the optimal coupling is described by two functions called potentials; equivalently, potentials can be defined as a…

We prove that, for general cost functions on $\mathbb{R}^n$, or for the cost $d^2/2$ on a Riemannian manifold, optimal transport maps between smooth densities are always smooth outside a closed singular set of measure zero.

We prove a sharp global $W^{2,\,p}$ estimate for potentials of optimal transport maps that take a certain class of non-convex planar domains to convex ones.

For $0<p<+\infty$, we prove a global $W^{2,p}$-estimate for potentials of optimal transport maps between convex domains in the plane. Among the tools developed for that purpose are obliqueness in general convex domains and estimates for the…

In this paper, we investigate the optimal transport problem when the source is a non-convex polygonal domain in $\mathbb{R}^2$. We show a global $W^{2,1+\epsilon}$ estimate for potentials of optimal transport. Our method applies to a more…

We consider maps $T$ solving the optimal transport problem with a cost $c(x-y)$ modeled on the $p$-cost. For H\"older continuous marginals, we prove a $C^{1,\alpha}$-partial regularity result for $T $in the set $\{|T(x)-x|>0\}$.

This note concerns the relationship between conditions on cost functions and domains and the convexity properties of potentials in optimal transportation and the continuity of the associated optimal mappings. In particular, we prove that if…

We prove a pointwise $C^{2,\,\alpha}$ estimate for the potential of the optimal transport map in the case that the densities are only close to constant in a certain $L^p$ sense.

The classical problem of optimal transportation can be formulated as a linear optimization problem on a convex domain: among all joint measures with fixed marginals find the optimal one, where optimality is measured against a cost function.…

We develop an $\e$-regularity theory at the boundary for a general class of Monge-Amp\`ere type equations arising in optimal transportation. As a corollary we deduce that optimal transport maps between H\"older densities supported on $C^2$…

In this paper, we establish a global regularity result for the optimal transport problem with the quadratic cost, where the domains may not be convex. This result is obtained by a perturbation argument, using a recent global regularity of…

We consider the problem of finding an optimal transport plan between an absolutely continuous measure $\mu$ on $\mathcal{X} \subset \mathbb{R}^d$ and a finitely supported measure $\nu$ on $\mathbb{R}^d$ when the transport cost is the…

We establish novel quantitative stability results for optimal transport problems with respect to perturbations in the target measure. We provide explicit bounds on the stability of optimal transport potentials and maps, which are relevant…

We identify a condition for regularity of optimal transport maps that requires only three derivatives of the cost function, for measures given by densities that are only bounded above and below. This new condition is equivalent to the weak…

We consider the problem of optimal transportation with general cost between a empirical measure and a general target probability on R d , with d $\ge$ 1. We extend results in [19] and prove asymptotic stability of both optimal transport…

In this short note, we show that given a cost function $c$, any coupling $\pi$ of two probability measures where the second is a discrete measure can be associated to a certain bipartite graph containing a perfect matching, based on the…

We consider regularised quadratic optimal transport with subquadratic polynomial or entropic regularisation. In both cases, we prove interior Lipschitz-estimates on a transport-like map and interior gradient Lipschitz-estimates on the…

We give natural topological conditions on the support of the target measure under which solutions to the optimal transport problem with cost function satisfying the (weak) Ma, Trudinger, and Wang condition cannot have any isolated singular…

We prove that for two-marginal optimal transport with Coulomb cost, the optimal map is a $C^{1,\alpha}$ diffeomorphism outside a closed set of Lebesgue measure zero provided the marginals are $\alpha$-H\"older continuous and bounded away…

We prove existence of an optimal transport map in the Monge-Kantorovich problem associated to a cost $c(x,y)$ which is not finite everywhere, but coincides with $|x-y|^2$ if the displacement $y-x$ belongs to a given convex set $C$ and it is…