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Sibirsk. Mat. Zh., 2018, Volume 59, Number 6, Pages 1370–1374 (Mi smj3049)  

The Monge problem of “piles and holes” on the torus and the problem of small denominators

V. V. Kozlov

Steklov Institute of Mathematics, Moscow, Russia

Abstract: We discuss the problem of existence of a smooth endomorphism of a closed $n$-dimensional manifold carrying a differential $n$-form into a prescribed volume form. Of course, we assume that the integrals of these forms over the whole manifold are equal. The solution of this problem for the $n$-dimensional torus reduces to the problem of small denominators well known in analysis.

Keywords: Monge–Kantorovich problem, smooth endomorphisms, small denominators.

Funding Agency Grant Number
Russian Science Foundation 14-50-00005
The author was supported by the Russian Foundation for Basic Research (Grant 14-50-00005).


DOI: https://doi.org/10.17377/smzh.2018.59.611

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English version:
Siberian Mathematical Journal, 2018, 59:6, 1090–1093

Bibliographic databases:

Document Type: Article
UDC: 519.2+517.9
MSC: 35R30
Received: 07.06.2018

Citation: V. V. Kozlov, “The Monge problem of “piles and holes” on the torus and the problem of small denominators”, Sibirsk. Mat. Zh., 59:6 (2018), 1370–1374; Siberian Math. J., 59:6 (2018), 1090–1093

Citation in format AMSBIB
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\paper The Monge problem of ``piles and holes'' on the torus and the problem of small denominators
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\yr 2018
\vol 59
\issue 6
\pages 1370--1374
\mathnet{http://mi.mathnet.ru/smj3049}
\crossref{https://doi.org/10.17377/smzh.2018.59.611}
\transl
\jour Siberian Math. J.
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\pages 1090--1093
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