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

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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 given 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 use either mass fluxes (vector-valued measures, Eulerian formulation) or patterns (probabilities on the space of particle paths, Lagrangian formulation). In the branched transport problem the transportation cost is a fractional power of the transported mass. In this paper we instead analyse the much more general class of transport problems in which the transportation cost is merely a nonnegative increasing and subadditive function (in a certain sense this is the broadest possible generalization of branched transport). In particular, we address the problem of the equivalence of the above-mentioned formulations in this wider context. However, the newly-introduced class of transportation costs lacks strict concavity which complicates the analysis considerably. New ideas are required, in particular, it turns out convenient to state the problem via 1-currents. Our analysis also includes the well-posedness, some network properties, as well as a metrization and a length space property of the model cost, which were previously only known for branched transport. Some already existing arguments in that field are given a more concise and simpler form.

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Acknowledgements

This work was supported by the Deutsche Forschungsgemeinschaft (DFG), Cells-in-Motion Cluster of Excellence (EXC 1003-CiM), University of Münster, Germany. B.W.’s research was supported by the Alfried Krupp Prize for Young University Teachers awarded by the Alfried Krupp von Bohlen und Halbach-Stiftung.

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  1. Institute for Numerical and Applied Mathematics, University of Münster, Einsteinstraße 62, 48149, Münster, Germany

    Alessio Brancolini & Benedikt Wirth

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  1. Alessio Brancolini
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  2. Benedikt Wirth
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Correspondence to Alessio Brancolini.

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Communicated by L. Ambrosio.

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Brancolini, A., Wirth, B. General transport problems with branched minimizers as functionals of 1-currents with prescribed boundary. Calc. Var. 57, 82 (2018). https://doi.org/10.1007/s00526-018-1364-4

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