Clop, AlbertCiti, GiovannaCircelli, MicheleUniversitat de Barcelona. Departament de Matemàtiques i Informàtica2024-12-192024-12-192024-07-03https://hdl.handle.net/2445/217201In this thesis we adapted the problem of continuous congested optimal transport to the Heisenberg group, equipped with a sub-Riemannian metric: we restricted the set of admissible paths to the horizontal curves. We obtained the existence of equilibrium configurations, known as Wardrop Equilibria, through the minimization of a convex functional, over a suitable set of measures on the horizontal curves. Moreover, such equilibria induce trans­ port plans that solve a Monge-Kantorovic problem associated with a cost, depending on the congestion itself, which we rigorously defined. We also proved the equivalence between this problem and a minimization problem defined over the set of p-summable horizontal vector fields with prescribed divergence. We showed that this new problem admits a dual formulation as a classical minimization problem of Calculus of Variations. In addition, even the Monge-Kantorovich problem associated with the sub-Riemannian distance turns out to be equivalent to a minimization problem over measures on horizontal curves. Passing through the notion of horizontal transport density, we proved that the Monge-Kantorovich problem can also be formulated as a minimization problem with a divergence-type constraint. Its dual formulation is the well-known Kantorovich duality theorem. In the end, we treated the continuous congested optimal transport problem with orthotropic cost function: we proved the Lipschitz regularity for solutions to a pseudo q-Laplacian-type equation arising from it.189 p.application/pdfengcc by (c) Circelli, Michele, 2024http://creativecommons.org/licenses/by/3.0/es/Varietats de RiemannAnells commutatiusRiemannian manifoldsCommutative ringsinfo:eu-repo/semantics/doctoralThesisinfo:eu-repo/semantics/openAccesshttp://hdl.handle.net/10803/692999