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Zbl 1065.43012
Rigot, Séverine
Optimal measure transport in the Heisenberg group. (Transport optimal de mesure dans le groupe de Heisenberg.) (French)
[A] Seminar on spectral theory and geometry. 2003--2004. St. Martin d'Hères: Université de Grenoble I, Institut Fourier. Séminaire de Théorie Spectrale et Géométrie 22, 9-23 (2004).

The author presents some results concerning the optimal measure transport problem in the Heisenberg group $H^n$. The problem is to find, given two measures $\mu$ and $\nu$, a minimizer of $$ \psi\to \int_{H^n} c(x,\psi(x))d\mu(x) $$ among all functions $\psi$ such that $\psi_\# \mu=\nu$. Here the cost function $c$ is given by $c(x,y)=d(x,y)^2/2$, with $d$ the geodesic distance. The author gives some differentiability properties of the distance function and of $c$-concave functions, and the main result presented is the existence of a unique optimal plane transport under the assumption that $\mu<< {\cal L}^{2n+1}$ and $$ \int _{H^n} d(0,x)^2d\mu(x)+\int _{H^n} d(0,x)^2d\nu(x) <+\infty. $$ The proofs of these results are mainly contained in the papers [{\it L.~Ambrosio} and {\it S.~Rigot}, J. Funct. Anal. 208, 261--301 (2004; Zbl 1076.49023)] and [{\it S.~Rigot}, ``Mass transportation in groups of type $H$'', preprint].
[Michele Miranda (Lecce)]

MSC 2000:: *43A80 Analysis on other specific Lie groups
49Q20 Variational problems in geometric measure-theoretic setting
53C17 Sub-Riemannian geometry

Keywords: optimal transport; Heisenberg group

Citations: Zbl 1076.49023

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	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

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