On Monge–Ampère type equations arising in optimal transportation problems

We prove some asymptotic interior regularity results for potential functions of optimal transportation problems with power costs. We show that our problems are equivalent to optimal transportation problems whose cost functions are sufficiently small perturbations of the quadratic cost but they do not satisfy the well known condition (A.3) guaranteeing regularity. The proof consists in a perturbation argument from the standard MongeAmpère equation in order to obtain interior Hölder estimates for second derivatives of potentials, and a careful understanding of why we might fail to have an Alexandroff weak solution when restricted to subdomains. In particular, we provide some quantitative estimates along the way on how the equation degenerates near the boundary.

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“…Let us give here the necessary definitions for general strictly convex cost function (cf. Gutiérrez-Nguyen [24]). …”

Section: A Review Of Convex Costsmentioning

confidence: 99%

“…The last integral above was computed in Gutiérrez-Nguyen [24] (equations (67) and (68)), and the proposition follows from these estimates.…”

Section: An Alexandroff Type Estimatementioning

confidence: 99%

A Perturbation Argument for a Monge–Ampère Type Equation Arising in Optimal Transportation

Caffarelli

González

Nguyen

2014

Arch Rational Mech Anal

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“…Since v ∈ C(Ω), v is continuous up to the boundary on Ω. By the Aleksandrov's maximum principle [37,Proposition 6.15] applied to z on the convex set Ω ⊂ Ω where z < 0.…”

Section: By (44)mentioning

confidence: 99%

Smooth approximations of the Aleksandrov solution of the Monge-Ampère equation

Awanou

2015

Communications in Mathematical Sciences

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A discrete analogue of the Dirichlet problem of the Aleksandrov theory of the Monge-Ampère equation is derived in this paper. The discrete solution is not required to be convex, but only discrete convex in the sense of Oberman. We prove that the uniform limit on compact subsets of discrete convex functions which are uniformly bounded and which interpolate the Dirichlet boundary data is a continuous convex function which satisfies the boundary condition strongly. The domain of the solution needs not be uniformly convex. We obtain the first proof of convergence of a wide stencil finite difference scheme to the Aleksandrov solution of the elliptic Monge-Ampère equation when the right hand side is a sum of Dirac masses. The discrete scheme we analyze for the Dirichlet problem, when coupled with a discretization of the second boundary condition, can be used to get a good initial guess for geometric methods solving optimal transport between two measures.

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“…This problem was considered in [GN07] for general cost functions, but in our case we need to have a more precise characterization of the solution; see [GN07,Theorem 6.7].…”

Section: Homogeneous Dirichlet Problemmentioning

confidence: 99%

“…Consequently, from the results of Loeper [Loe09, Proposition 2.11 and Theorem 3.1], the set F u (x) defined in Definition 2.1 is in general not connected. We refer to the paper [GN07] for results on Monge-Ampère type equations arising in optimal mass transportation for general cost functions and properties of the subdifferential F u . Optimal mass transportation has recently become a very active area of research; we mention, in particular, the fundamental work of Ma, Trudinger and Wang [MTW05], for smooth cost functions.…”

Section: Introductionmentioning

confidence: 99%