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

TL;DR

This paper compares two classical models of branched transport, providing a simplified proof of minimizer existence for the Lagrangian model and establishing the equivalence between Lagrangian and Eulerian formulations using energy formulas and flow decompositions.

Contribution

It offers a simplified proof of minimizer existence for the Lagrangian model and rigorously establishes the equivalence between Lagrangian and Eulerian branched transport models.

Findings

Proof of existence of minimizers for the Lagrangian model.

Rigorous connection between irrigation cost and Gilbert Energy.

Equivalence between Lagrangian and Eulerian models established.

Abstract

First we present two classical models of Branched Transport: the Lagrangian model introduced by Bernot, Caselles, Morel, Maddalena, Solimini, and the Eulerian model introduced by Xia. An emphasis is put on the Lagrangian model, 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 I$\alpha$ to the Gilbert Energy 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 by Santambrogio, to establish the equivalence between the Lagrangian and Eulerian models.