## Optimal mass transportation in the Heisenberg group.(English)Zbl 1076.49023

Summary: We consider the problem of optimal transportation of absolutely continuous masses in the Heisenberg group $$H_n$$, in the case when the cost function is either the square of the Carnot-Carathéodory distance or the square of the Korányi norm. In both cases we show existence and uniqueness of an optimal transport map. In the former case the proof requires a delicate analysis of minimizing geodesics of the group and of the differentiability properties of the squared distance function. In the latter case the proof requires some fine properties of BV functions in the Heisenberg group.

### MSC:

 49Q20 Variational problems in a geometric measure-theoretic setting 43A80 Analysis on other specific Lie groups 49J45 Methods involving semicontinuity and convergence; relaxation 35H10 Hypoelliptic equations 53C17 Sub-Riemannian geometry
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