# General transport problems with branched minimizers as functionals of 1-currents with prescribed boundary

## TL;DR

This paper extends the branched transport model to more general cost functions, establishing equivalence between formulations and analyzing properties like well-posedness and network structure.

## Contribution

It generalizes the classical branched transport model to subadditive costs and introduces a 1-current framework for analysis, simplifying and broadening previous approaches.

## Key findings

- Established equivalence between flux and pattern formulations.
- Proved well-posedness and metric properties of the generalized model.
- Analyzed network structure and properties under new cost functions.

## Abstract

A prominent model for transportation networks is branched transport, which seeks the optimal transportation scheme to move material from a given initial to a final distribution. The cost of the scheme encodes a higher transport efficiency the more mass is moved together, which automatically leads to optimal transportation networks with a hierarchical branching structure. The two major existing model formulations, either using mass fluxes (vector-valued measures) or patterns (probabilities on the space of particle paths), are rather different. Once their equivalence was established, the analysis of optimal networks could rest on both. The transportation cost of classical branched transport is a fractional power of the transported mass, and several model properties and proof techniques build on its strict concavity. We generalize the model and its analysis to the most general class of reasonable transportation costs, essentially increasing, subadditive functions. This requires several modifications or new approaches. In particular, for the equivalence between mass flux and pattern formulation it turns out advantageous to resort to a description via 1-currents, an intuition which already Xia exploited. In addition, some already existing arguments are given a more concise and perhaps simpler form. The analysis includes the well-posedness, a metrization and a length space property of the model cost, the equivalence between the different model formulations, as well as a few network properties.