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 |
Files in This Item:
File | Description | Size | Format | |
---|---|---|---|---|
MC_PhD_THESIS.pdf | 1.42 MB | Adobe PDF | View/Open |
This item is licensed under a
Creative Commons License