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Mat. Sb., 2004, Volume 195, Number 9, Pages 57–74 (Mi msb845)  

This article is cited in 7 scientific papers (total in 7 papers)

Precise solutions of the one-dimensional Monge–Kantorovich problem

A. Yu. Plakhov

University of Aveiro

Abstract: The Monge–Kantorovich problem on finding a measure realizing the transportation of mass from $\mathbb R$ to $\mathbb R$ at minimum cost is considered. The initial and resulting distributions of mass are assumed to be the same and the cost of the transportation of the unit mass from a point $x$ to $y$ is expressed by an odd function $f(x+y)$ that is strictly concave on $\mathbb R_+$. It is shown that under certain assumptions about the distribution of the mass the optimal measure belongs to a certain family of measures depending on countably many parameters. This family is explicitly described: it depends only on the distribution of the mass, but not on $f$. Under an additional constraint on the distribution of the mass the number of the parameters is finite and the problem reduces to the minimization of a function of several variables. Examples of various distributions of the mass are considered.

DOI: https://doi.org/10.4213/sm845

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English version:
Sbornik: Mathematics, 2004, 195:9, 1291–1307

Bibliographic databases:

UDC: 517.98
MSC: Primary 49Q20; Secondary 46N10
Received: 11.11.2003

Citation: A. Yu. Plakhov, “Precise solutions of the one-dimensional Monge–Kantorovich problem”, Mat. Sb., 195:9 (2004), 57–74; Sb. Math., 195:9 (2004), 1291–1307

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. V. L. Levin, “Optimality conditions and exact solutions to the two-dimensional Monge–Kantorovich problem”, J. Math. Sci. (N. Y.), 133:4 (2006), 1456–1463  mathnet  crossref  mathscinet  zmath  elib  elib
    2. A. Yu. Plakhov, “Scattering in billiards and problems of Newtonian aerodynamics”, Russian Math. Surveys, 64:5 (2009), 873–938  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    3. Luis Caffarelli, Robert McCann, “Free boundaries in optimal transport and Monge-Ampère obstacle problems”, Ann of Math, 171:2 (2010), 673  crossref  mathscinet  zmath  isi
    4. Chiappori P.-A., McCann R.J., Nesheim L.P., “Hedonic price equilibria, stable matching, and optimal transport: equivalence, topology, and uniqueness”, Economic Theory, 42:2 (2010), 317–354  crossref  mathscinet  zmath  isi
    5. V. I. Bogachev, A. V. Kolesnikov, “The Monge–Kantorovich problem: achievements, connections, and perspectives”, Russian Math. Surveys, 67:5 (2012), 785–890  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    6. Plakhov A., “Newton's problem of minimal resistance under the single-impact assumption”, Nonlinearity, 29:2 (2016), 465–488  crossref  mathscinet  zmath  isi  scopus
    7. Plakhov A., Tchemisova T., “Problems of optimal transportation on the circle and their mechanical applications”, J. Differ. Equ., 262:3, 2 (2017), 2449–2492  crossref  mathscinet  zmath  isi  scopus
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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