RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mat. Sb.: Year: Volume: Issue: Page: Find

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

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

Full text: PDF file (318 kB)
References: PDF file   HTML file

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

Bibliographic databases:

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

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
\Bibitem{Pla04} \by A.~Yu.~Plakhov \paper Precise solutions of the one-dimensional Monge--Kantorovich problem \jour Mat. Sb. \yr 2004 \vol 195 \issue 9 \pages 57--74 \mathnet{http://mi.mathnet.ru/msb845} \crossref{https://doi.org/10.4213/sm845} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2122369} \zmath{https://zbmath.org/?q=an:1080.49030} \transl \jour Sb. Math. \yr 2004 \vol 195 \issue 9 \pages 1291--1307 \crossref{https://doi.org/10.1070/SM2004v195n09ABEH000845} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000226336000004} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-12144283002} 

• http://mi.mathnet.ru/eng/msb845
• https://doi.org/10.4213/sm845
• http://mi.mathnet.ru/eng/msb/v195/i9/p57

 SHARE:

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
2. A. Yu. Plakhov, “Scattering in billiards and problems of Newtonian aerodynamics”, Russian Math. Surveys, 64:5 (2009), 873–938
3. Luis Caffarelli, Robert McCann, “Free boundaries in optimal transport and Monge-Ampère obstacle problems”, Ann of Math, 171:2 (2010), 673
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
5. V. I. Bogachev, A. V. Kolesnikov, “The Monge–Kantorovich problem: achievements, connections, and perspectives”, Russian Math. Surveys, 67:5 (2012), 785–890
6. Plakhov A., “Newton's problem of minimal resistance under the single-impact assumption”, Nonlinearity, 29:2 (2016), 465–488
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
•  Number of views: This page: 458 Full text: 213 References: 31 First page: 1