*Preprint*

**Inserted:** 27 jul 2018

**Last Updated:** 10 oct 2018

**Pages:** 22

**Year:** 2018

**Abstract:**

In this paper we study general transportation problems in $\mathbb{R}^n$, in which $m$ different goods are moved simultaneously. The initial and final positions of the goods are represented by measures $\mu^-$, $\mu^+$ on $\mathbb{R}^n$ with values in $\mathbb{R}^m$. When the measures are finite atomic, a discrete transportation network is a measure $T$ on $\mathbb{R}^n$ with values in $\mathbb{R}^{n\times m}$ represented by an oriented graph $\mathcal{G}$ in $\mathbb{R}^n$ whose edges carry multiplicities in $\mathbb{R}^m$. The constraint is encoded in the relation ${\rm div}(T)=\mu^--\mu^+$. The cost of the discrete transportation $T$ is obtained integrating on $\mathcal{G}$ a general function $\mathcal{C}:\mathbb{R}^m\to\mathbb{R}$ of the multiplicity. When the initial data $\left( \mu^-,\mu^+\right)$ are arbitrary (possibly diffuse), the cost of a transportation network between them is computed by relaxation of the functional on graphs mentioned above. Our main result establishes the existence of cost-minimizing transportation networks for arbitrary data $\left( \mu^-,\mu^+\right)$. Furthermore, under additional assumptions on the cost integrand $\mathcal{C}$, we prove the existence of transportation networks with finite cost and the stability of the minimizers with respect to variations of the given data.

**Keywords:**
transportation network, branched transportation, Multi-material transport problem, Normal current

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