## Tesi etd-06052003-174133 |

Thesis type

Tesi di laurea vecchio ordinamento

Author

Santambrogio, Filippo

email address

f.santambrogio@sns.it

URN

etd-06052003-174133

Title

Misure ottime per costi di trasporto e funzionali locali

Struttura

SCIENZE MATEMATICHE, FISICHE E NATURALI

Corso di studi

MATEMATICA

Supervisors

**relatore**Prof. Buttazzo, Giuseppe

Parole chiave

- nonconvex functionals over measures
- urban planning
- mass transportation
- c-concavity
- urban planning
- mass transportation
- nonconvex functionals over measures
- duality formula

Data inizio appello

03/07/2003;

ConsultabilitÃ

Completa

Riassunto analitico

We consider minimization problems for functionals defined on couples of probability measures on a space $Omega$ (which could be an abstract metric space or a subset of the Euclidean space), built by summing up three terms. Two of them are local lower semicontinuous functionals, each concerning just one of the measures, while the third is given by the Monge-Kantorovich optimal cost of transport between them (precisely it is the p-th Wasserstein distance to the power of p). The two local functionals, coming from the theory developed by G.BouchittÃ© and G.Buttazzo, have opposite behaviours and force the first measure to be spread all over $Omega$ and the other one to be a concentrated atomic measure. We give existence results in the case of a compact metric space and, in the Euclidean space, in the case of a bounded convex set or of the whole space itself. We also show necessary optimality conditions, which imply some regularity for the Lebesgue-density of the spread measure (essentially Lipschitz-type) but also, in some cases, properties identifying in a quite precise way the optimal couples. This model can be applied in several decisional problems but in particular it arises from urban planning questions: in this case the spread measure stands for the distribution of the population in a city and the concentrated one for public services and offices.

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