*Accepted Paper*

**Inserted:** 7 apr 2012

**Last Updated:** 7 apr 2012

**Journal:** Milan J. Math.

**Year:** 2012

**Notes:**

Milan J. Math., (to appear), DOI: 10.1007*s00032-012-0175-x*

**Abstract:**

We present a survey on several mass transportation problems, in which a given mass dynamically moves from an initial configuration to a final one. The approach we consider is the one introduced by Benamou and Brenier in 5, where a suitable cost functional $F(\rho,v)$, depending on the density $\rho$ and on the velocity $v$ (which fulfill the continuity equation), has to be minimized. Acting on the functional $F$ various forms of mass transportation problems can be modeled, as for instance those presenting congestion effects, occurring in traffic simulations and in crowd motions, or concentration effects, which give rise to branched structures.

**Keywords:**
Optimal transport, Branched transport, continuity equation, functionals on measures, congested transport

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