Calculus of Variations and Geometric Measure Theory
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P. Pegon

On the Lagrangian branched transport model and the equivalence with its Eulerian formulation

created by pegon on 21 Jul 2016


Accepted Paper

Inserted: 21 jul 2016
Last Updated: 21 jul 2016

Pages: 20
Year: 2016


We present the two classical models of Branched Transport: the Lagrangian model and its Eulerian counterpart, with an emphasis on the last one, for which we give a complete proof of existence of minimizers in a --hopefully-- simplified manner. We also treat in detail some $\sigma$-finiteness and rectifiability issues to yield rigorously the energy formula connecting the irrigation cost $\mathbf{I}_\alpha$ to the Gilbert Energy $\mathbf{E}_\alpha$. Our main purpose is to use this energy formula and exploit a Smirnov decomposition of vector flows, which was proved via the Dacorogna-Moser approach, to establish the equivalence between the Lagrangian and Eulerian models.

Keywords: Optimal transport, Branched transport, Rectifiability, Smirnov decomposition


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