## Thesis etd-09232010-161155 |

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Thesis type

Tesi di dottorato di ricerca

Author

BRASCO, LORENZO

URN

etd-09232010-161155

Thesis title

Geodesics and PDE methods in transport models

Academic discipline

MAT/05

Course of study

MATEMATICA

Supervisors

**tutor**Prof. Buttazzo, Giuseppe

**relatore**Prof. Carlier, Guillaume

**correlatore**Prof. Santambrogio, Filippo

Keywords

- analysis in metric spaces
- branched transport
- continuity equation
- degenerate elliptic equations
- Di Perna-Lions
- nonconvex variational problems
- optimal transport
- traffic congestion
- Wardrop equilibrium

Graduation session start date

11/10/2010

Availability

Full

Summary

This thesis is devoted to to the study of optimal transport problems, alternative to the so called Monge-Kantorovich one: they naturally arise in some real world applications, like in the design of optimal transportation networks or in urban traffic modeling. More precisely, we consider problems where the transport cost has a nonlinear dependence on the mass: typically in this type of problems, to move a mass $m$ for a distance $\ell$ costs $\varphi(m) \ell$, where $\varphi$ is a given function, thus giving rise to a total cost of the type $\sum\varphi(m) \ell$.

Two interesting cases are widely addressed in this work: the case where $\varphi$ is subadditive (branched transport), so that masses have the interest to travel together in order to lower the total cost; the case of $\varphi$ superadditive (congested transport), where on the contrary the mass tends to be as widespread as possible.

In the case of branched transport, we introduce two new dynamical models: in the first one, the transport is described through the employ of curves of probability measures minimizing a length-type functional (with a weight function penalizing non atomic measures). On the other hand, the second model is much more in the spirit of the celebrated Benamou-Brenier formulation for Wasserstein distances: in particular, the transport is described by means of pairs ``curve of measures--velocity vector field'', satisfying the continuity equation and minimizing a suitable dynamical energy (which is a non convex one, actually). For both models we prove existence of minimal configurations and equivalence with other modelizations existing in literature.

Concerning the case of congested transport, we review in great details two already existing models, proving their equivalence: while the first one can be viewed as a Lagrangian approach to the problem and it has some interesting links with traffic equilibrium issues, the second one is a divergence-constrained convex optimization problem.

The proof of this equivalence represents the central core of the second part of the work and contains various points of interest: among them, the DiPerna-Lions theory of flows of weakly differentiable vector fields, the Dacorogna-Moser construction for transport maps and, above all, some regularity estimates (that we derive here) for a very degenerate elliptic equation, that seems to be quite unexplored.

Two interesting cases are widely addressed in this work: the case where $\varphi$ is subadditive (branched transport), so that masses have the interest to travel together in order to lower the total cost; the case of $\varphi$ superadditive (congested transport), where on the contrary the mass tends to be as widespread as possible.

In the case of branched transport, we introduce two new dynamical models: in the first one, the transport is described through the employ of curves of probability measures minimizing a length-type functional (with a weight function penalizing non atomic measures). On the other hand, the second model is much more in the spirit of the celebrated Benamou-Brenier formulation for Wasserstein distances: in particular, the transport is described by means of pairs ``curve of measures--velocity vector field'', satisfying the continuity equation and minimizing a suitable dynamical energy (which is a non convex one, actually). For both models we prove existence of minimal configurations and equivalence with other modelizations existing in literature.

Concerning the case of congested transport, we review in great details two already existing models, proving their equivalence: while the first one can be viewed as a Lagrangian approach to the problem and it has some interesting links with traffic equilibrium issues, the second one is a divergence-constrained convex optimization problem.

The proof of this equivalence represents the central core of the second part of the work and contains various points of interest: among them, the DiPerna-Lions theory of flows of weakly differentiable vector fields, the Dacorogna-Moser construction for transport maps and, above all, some regularity estimates (that we derive here) for a very degenerate elliptic equation, that seems to be quite unexplored.

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