A multi-material transport problem with arbitrary marginals

Andrea Marchese, Annalisa Massaccesi, Salvatore Stuvard, Riccardo Tione

Published 2018-07-28Version 1

In this paper we study general transportation problems in $\mathbb{R}^n$, in which $m$ different goods are moved simultaneously. The initial and final displacements 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 very general function $\mathcal{C}:\mathbb{R}^m\to\mathbb{R}$ of the multiplicity. The proof of the existence of minimizers for arbitrary (possibly diffuse) data $(\mu^-,\mu^+)$ requires an explicit formula for the relaxation, on arbitrary transportation networks, of the functional on graphs defined above. Under additional assumptions on $\mathcal{C}$, we prove the existence of transportation networks with finite cost and the stability of the minimizers with respect to variations of the given data. The proofs of the main results of the paper require notions from the theory of currents with coefficients in a group. In the process, we give details of the proof of a useful result concerning the relaxation of general functionals defined on polyhedral chains with coefficients in groups.