Regularity of Potential Functions of the Optimal Transportation Problem

By Xinwen Ma, Neil S. Trudinger, Xu-jia Wang. 2005

439 citations recorded by the source corpus.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ff7fc66c58ee2fa416ee85f8204da6053d0b31a7

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Nearly Round Spheres Look Convex (2012, methodology): “The focalization problem was the motivation for several progress: a “prob-abilistic” perturbation lemma for paths crossing the tangent cut locus [17], and more importantly the introduction of the extended Ma-Trudinger-Wang tensor by Figalli and Rifford [14].”
Partial regularity for optimal transport maps (2012, methodology): “A. FIGALLI, Regularity of optimal transport maps [after Ma-Trudinger-Wang and Loeper].”
Unidimensional and Evolution Methods for Optimal Transportation (2013, background): “Trudinger, and Xu-Jia Wang [41]; a more speci c result, on products of spheres, has been recently proved by Alessio Figalli, Young-Heon Kim, and Robert J.”
On the inverse implication of Brenier-Mccann theorems and the structure of (P 2 (M),W 2 ) (2011, background): “A sufficient condition for this regularity is the satisfaction of the MaTrudinger-Wang condition (see [22]).”
Hedonic price equilibria, stable matching, and optimal transport: equivalence, topology, and uniqueness (2007, background): “Even in the semiconvex-buyer setting Dom Dq has full μ-measure, because q is locally semiconvex, hence differentiable except on a countably rectifiable hypersurface, to which μ0 assigns zero mass; see McCann (1995) and Gangbo and McCann (1996) or Theorems 10.”
Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling (2015, background): “Surprisingly, at the beginning of the present century, Ma-TrudingerWang [219] found out the key for the regularity under different costs.”
Multi-marginal optimal transport and multi-agent matching problems: uniqueness and structure of solutions (2012, methodology): “It also served as an early example of a cost function satisfying (A3w), the crucial condition introduced by Ma, Trudinger and Wang governing the regularity of optimal maps in two marginal problems [22].”
Hölder Continuity and Injectivity of Optimal Maps (2011, methodology): “Their condition relaxes the hypothesis (A3) (=(B3)s) proposed earlier with Ma [45].”
Regularity of optimal transport in curved geometry: The nonfocal case (2010, background): “) Then one may conclude, using arguments similar to those in [22], that ψ is C if f and g are.”
Continuity, curvature, and the general covariance of optimal transportation (2007, methodology): “Let us now exploit the geodesic structure which the pseudo-metric h induces on N ⊂ M × M̄ to recover Ma, Trudinger & Wang’s notions concerning c-convex domains [34] in our setting.”
Hölder regularity of optimal mappings in optimal transportation (2009, methodology): “From Theorem 2 we infer that if u is a generalized solution of Eq. (1.1) with f ≡ 0, then u ∈ C 1,1 (�) . This result was proved in [ 18 ].”
Regularity of Optimal Maps on the Sphere: the Quadratic Cost and the Reflector Antenna (2013, methodology): “Concerning classical regularity, Ma, Trudinger and Wang [17] proved the following: Theorem 3.7 ([17]) Let c be a C4(Ω × Ω′) cost function that satisfies assumptions A1, A2, As on (Ω × Ω′), Ω′ being c-convex with respect to Ω.”
Second order estimates for Hessian type fully nonlinear elliptic equations on Riemannian manifolds (2014, background): “In [39] Ma, Trudinger and Wang introduced the following condition to establish interior regularity for optimal transports: there exists c0 > 0 such that (1.”
NECESSARY AND SUFFICIENT CONDITIONS FOR CONTINUITY OF OPTIMAL TRANSPORT MAPS ON RIEMANNIAN MANIFOLDS (2011, methodology): “As before, if the last inequality in (2.1) is strict unless ξ = 0 or η = 0, then M is said to satisfy the extended strict Ma–Trudinger–Wang condition (MTW + ).”
Free boundaries in optimal transport and Monge-Ampere obstacle problems (2010, background): “x0; y/ is a familiar criterion from Gangbo [40], Carlier [24], and Ma, Trudinger and Wang [56] for uniqueness of total transfer; it follows from strict convexity in Caffarelli [18] and Gangbo and McCann [41].”