La soutenance de thèse de Zeinab Badreddine (McTAO, thèse sous la direction de L. Rifford et B. Bonnard) aura lieu le 4 décembre 2017 à 10:30 à Dijon, à l’Institut math. de Bourgogne (salle Baire).
Titre. Mass transportation in sub-Riemannian structures admitting singular minimizing geodesics
Abstract. This thesis is devoted to the study of mass transportation in sub-Riemannian geometry. More precisely, our aim is to extend previous results on the well-posedness of the Monge problem to cases of sub-Riemannian structures admitting singular minimizing geodesics. In the first part, we show that, in the case of rank-two analytic distributions in dimension four, we have existence and uniqueness of solutions for the sub-Riemannian quadratic cost, as soon as the distribution satisfies some growth condition. Our strategy to prove it, combines the technique used by Figalli-Rifford which is based on the regularity of the sub-Riemannian distance outside the diagonal in absence of singular minimizing curves, together with a localized contraction property for singular curves in the spirit of the previous work by Cavalletti and Huesmann. In the second part, we deal with regularity issues of the sub-Riemannian distance and we define a class of sub-Riemannian structures on Carnot groups, called h-ideal sub-Riemannian structures, on which the sub-Riemannian distance is h-semiconcave. Together with an assumption on the distribution, we prove heuristically the MCP property on Carnot groups. Anyway, we attempt to prove that MCP property defined on this class of Carnot groups is sufficient to apply the Cavalletti-Huesmann method to prove the well-posedness of the Monge problem.