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Tesi etd-06052006-100000


Thesis type
Tesi di laurea specialistica
Author
Figalli, Alessio
email address
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
Commissione
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. <br>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<br>with $\sigma$-finite (n-1)-dimensional Hausdorff measure.<br>Moreover we prove that, in the case $c(x,y)=d^2(x,y)$, the transport map<br>is approximatively differentiable a.e. with respect to the volume measure, and we extend some results about concavity estimates and displacement convexity.<br>As a corollary of this existence-uniqueness result, we prove the equivalence between the notions of &#34;dispacement convexity&#34; and &#34;weak displacement convexity&#34; on non-compact Riemannian manifolds.<br>Finally we study the optimal transportation problem with distance-like costs, proving the existence of an optimal transport map on non-compact manifolds for<br>a large class of cost functions that includes the case $c(x,y)=d(x,y)$, under the only hypothesis that<br>the source measure is absolutely continuous with respect to the volume measure.<br>In particular, we assume compactness neither of the support of the source measure nor of that of the target measure.
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