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 prescribed 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 \({\mathrm{div}}(T)=\mu ^- -\mu ^+\). The cost of the discrete transportation T is obtained integrating on \({\mathcal {G}}\) a general function \({\mathcal {C}}:{\mathbb {R}}^m\rightarrow {\mathbb {R}}\) of the multiplicity. When the initial data \(\left( \mu ^-,\mu ^+\right) \) are arbitrary (possibly diffuse) measures, 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. Finally, we provide an explicit integral representation formula for the cost of rectifiable transportation networks, and we characterize the costs such that every transportation network with finite cost is rectifiable.

在本文中，我们研究了\（{\ mathbb {R}} ^ n \）中的一般运输问题，其中m个不同的货物同时移动。货物的初始和最终位置由\（{\ mathbb {R}} ^ n \）上的度量\（\ mu ^-\），\（\ mu ^ + \）来指定，值的值为\（{\ mathbb {R}} ^ m \）。当度量是有限原子时，离散运输网络是\（{\ mathbb {R}} ^ n \）上的度量T，其值在\（{\ mathbb {R}} ^ {n \ times m} \）中由\（{\ mathbb {R}} ^ n \）中的定向图\（{\ mathcal {G}} \）表示，其边缘带有多重性\（{\ mathbb {R}} ^ m \）。约束被编码为关系\（{\ mathrm {div}}（T）= \ mu ^--\ mu ^ + \）。离散运输T的成本是在\（{\ mathcal {G}} \）上集成通用函数\（{\ mathcal {C}}：{\ mathbb {R}} ^ m \ rightarrow {\ mathbb { R}} \）的多重性。当初始数据\（\ left（\ mu ^-，\ mu ^ + \ right）\）是任意（可能是分散的）度量时，它们之间的运输网络成本是通过上述图上的函数松弛来计算的。我们的主要结果为任意数据\（\ left（\ mu ^-，\ mu ^ + \ right）\）建立了成本最小的运输网络。。此外，在关于成本被积\（{\ mathcal {C}} \）的附加假设下，我们证明了有限成本的运输网络的存在以及最小化器在给定数据变化方面的稳定性。最后，我们为可纠正的运输网络的成本提供了一个明确的积分表示公式，并对这些成本进行了表征，以使每个具有有限成本的运输网络都可以纠正。