Regularity in Monge's mass transfer problem

Abstract

Abstract. In this paper, we study the regularity of optimal mappings in Monge's mass transfer problem. Using the approximation to Monge's cost function c(x, y) = |x − y| through the costs c ε (x, y) = ε 2 + |x − y| 2 , we consider the optimal mappings T ε for these costs, and we prove that the eigenvalues of the Jacobian matrix DT ε , which are all positive, are locally uniformly bounded. By an example we prove that T ε is in general not uniformly Lipschitz continuous as ε → 0, even if the mass distributions are positive and smooth, and the domains are c-convex. IntroductionThe Monge mass transfer problem consists in finding an optimal mapping from one mass distribution to another one such that the total cost is minimized among all measure preserving mappings. This problem was first proposed by Monge [27] and has been studied by many authors in the last two hundred years: among the main achievements in the 20th century we cite [21] and [16].In Monge's problem, the cost of moving a mass from point x to point y is proportional to the distance |x − y|, namely the cost function is given byThis is a natural cost function. In the last two decades, due to a range of applications, the optimal transportation for more general cost functions has been a subject of extensive studies. In order to present the framework more precisely, let Ω and Ω * be two bounded domains in the Euclidean space R n , and let f and g be two densities in Ω and Ω * respectively, satisfying the mass balance conditionLet c be a smooth cost function defined on Ω × Ω * .2010 Mathematics Subject Classification. 35J60, 35B65, 49Q20.

Lack of regularity of the transport density in the Monge problem

“…Inspired by [6,14], we will construct a family of counter-examples by, first, choosing which lines will be transport rays. Set γ > 0 and consider the following transport rays:…”

“…As in [6,14], it is not difficult to prove the existence of a Kantorovich potential u to assert that the rays (l a ) a∈(0,1) are, in fact, all the transport rays between f + and f − . This follows immediately from the fact that the unit vector of any transport ray l a is an irrotational vector field, which implies that there is a 1-Lipschitz function u such that (3.10) u(x) − u(y) = |x − y| ∀ x, y ∈ l a .…”

The Monge–Ampère equation and its link to optimal transportation

“…In spite of the fact that this latter cost does not satisfy any of the conditions (C0)-(C3), still existence of optimal transport maps can be proved (see for instance [6] for an account of the theory) and some regularity results have been obtained in [62]. Very recently, the strategy to approximate the Monge cost with c 3 has been used in [82] to deduce some estimates for the Monge problem by proving a priori bounds on the transport maps which are uniform as a → 0.…”

Untitled

“…Despite the fact that this latter cost does not satisfy any of the conditions (C0)-(C3), existence of optimal transport maps can still be proved (see for instance [6] for an account of the theory) and some regularity results have been obtained in [60]. Very recently, the strategy to approximate the Monge cost with c 3 has been used in [79] to deduce some estimates for the Monge problem by proving a priori bounds on the transport maps which are uniform as a → 0.…”