Uniqueness and transport density in Monge's mass transportation problem

By Mikhail Feldman, Robert J. McCann. 2002

85 citations recorded by the source corpus.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/1247f956fc26acb7f22e99852fc36ff27db04fcb

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Integral Estimates for Transport Densities (2004, methodology): “Introduction This paper provides some partial differential equation (PDE) methods that improve upon the L estimates of the ‘transport densities’ in certain Monge–Kantorovich mass transfer problems, as derived in an earlier paper by the first and third authors [4], and in some cases also in [7].”
Optimal transport maps in Monge-Kantorovich problem (2003, background): “One can also show ([2], [13], [16], [14]) that CT is unique (unlike 7) if either p or v are absolutely continuous.”
Optimal transportation, topology and uniqueness (2010, unknown): “Uniqueness fails in this context [54] [45].”
Existence and stability results in the L 1 theory of optimal transportation (2003, background): “The measure σ := aL n , the so-called transport density, plays an important rˆ ole in the theory: its uniqueness and its regularity are discussed, under more general assumptions on µ and ν, in [ 27 ], [20], [21].”
Constructing optimal maps for Monge's transport problem as a limit of strictly convex costs (2001, background): “Lecture notes subsequently released by Ambrosio [2] contain an excellent summary of progress on Monge’s problem cast into the framework of geometric measure theory; they include a derivation for existence and uniqueness of a transport density corresponding to f2 L1(Rn) which parallels certain result”
Sharp summability for Monge Transport density via Interpolation (2004, background): “It has been proved (see [1, 8]) that there always exist (in this setting) optimal transports and in particular there are invertible optimal transports whose inverse is also an optimal transport for −f .”
Free boundaries in optimal transport and Monge-Ampere obstacle problems (2010, background): “Monge’s cost c(x,y) = |x − y| in particular has attracted recent attention from Ambrosio [5] & Pratelli [8], Ambrosio, Kircheim & Pratelli [7], Caffarelli, Feldman & McCann [19], DePascale, Evans & Pratelli [30], Feldman & McCann [36] [35], and Trudinger & Wang [76] following the work of Evans & Gan”
Monge's transport problem on a Riemannian manifold (2001, methodology): “Thus for the proof of Theorem 28 we follow the argument of [11, Section 3], skipping Lemma 3.1 of [ 11, Section 3 ] since this lemma considers the case when x1, s(x1), x2, s(x2 )a re not in one line.”
Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling (2015, background): “The first result – first proven in [162] – will concern L(1) summability (i.”
Regularity properties for Monge transport density and for solutions of some shape optimization problem (2002, methodology): “Note. After the completion of this work we heard about a related work by Feldman and McCann,[ 12 ].”