# Tesi etd-06052006-100000

Thesis type
Tesi di laurea specialistica
Author
Figalli, Alessio
a.figalli@sns.it
URN
etd-06052006-100000
Title
Trasporto ottimale su varietà non compatte
Struttura
SCIENZE MATEMATICHE, FISICHE E NATURALI
Corso di studi
MATEMATICA
Supervisors
relatore Ambrosio, Luigi
relatore Prof. Alberti, Giovanni
Parole chiave
• Tonelli Lagrangians
• Mané potential
• displacement convexity
• approximate differential
• mass transportation
• non-compact manifolds
Data inizio appello
23/06/2006;
Consultabilità
Completa
Riassunto analitico
We consider the optimal transportation problem on non-compact manifolds.
We yield existence and uniqueness of a unique transport map in the case of cost functions induced by a $C^2$-Lagrangian, provided that the source measure vanishes on sets
with $\sigma$-finite (n-1)-dimensional Hausdorff measure.
Moreover we prove that, in the case $c(x,y)=d^2(x,y)$, the transport map
is approximatively differentiable a.e. with respect to the volume measure, and we extend some results about concavity estimates and displacement convexity.
As a corollary of this existence-uniqueness result, we prove the equivalence between the notions of "dispacement convexity" and "weak displacement convexity" on non-compact Riemannian manifolds.
Finally we study the optimal transportation problem with distance-like costs, proving the existence of an optimal transport map on non-compact manifolds for
a large class of cost functions that includes the case $c(x,y)=d(x,y)$, under the only hypothesis that
the source measure is absolutely continuous with respect to the volume measure.
In particular, we assume compactness neither of the support of the source measure nor of that of the target measure.
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