Please use this identifier to cite or link to this item: https://hdl.handle.net/2445/217201
Title: Congested Optimal Transport in the Heisenberg Group
Author: Circelli, Michele
Director/Tutor: Clop, Albert
Citi, Giovanna
Keywords: Varietats de Riemann
Anells commutatius
Riemannian manifolds
Commutative rings
Issue Date: 3-Jul-2024
Publisher: Universitat de Barcelona
Abstract: In 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.
URI: https://hdl.handle.net/2445/217201
Appears in Collections:Tesis Doctorals - Departament - Matemàtiques i Informàtica

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