Mass transportation in sub-Riemannian structures admitting singular minimizing geodesics

Abstract : This thesis is devoted to the study of the Monge transport problem for the quadratic cost in sub-Riemannian geometry and the essential conditions to obtain existence and uniqueness of solutions. These works consist in extending these results to the case of sub-Riemannian structures admitting singular minimizing geodesics. In a first part, we develop techniques inspired by works by Cavalletti and Huesmann in order to obtain significant results for structures of rank 2 in dimension 4. In a second part, we study analytical tools of the h-semiconcavity of the sub-Riemannian distance and we show how this type of regularity can lead to the well-posedness of the Monge problem in general cases.
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Zeinab Badreddine. Mass transportation in sub-Riemannian structures admitting singular minimizing geodesics. Differential Geometry [math.DG]. Université Bourgogne Franche-Comté, 2017. English. ⟨NNT : 2017UBFCK034⟩. ⟨tel-01675005v2⟩

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