Calculus of Variations and Geometric Measure Theory

A. Brancolini - B. Wirth

Equivalent formulations for the branched transport and urban planning problems

created by brancolin on 26 Sep 2017


Published Paper

Inserted: 26 sep 2017

Journal: J. Math. Pures Appl.
Volume: 106
Number: 4
Pages: 695-724
Year: 2016
Doi: 10.1016/j.matpur.2016.03.008

ArXiv: 1509.06698 PDF


We consider two variational models for transport networks, an urban planning and a branched transport model, in both of which there is a preference for networks that collect and transport lots of mass together rather than transporting all mass particles independently. The strength of this preference determines the ramification patterns and the degree of complexity of optimal networks. Traditionally, the models are formulated in very different ways, via cost functionals of the network in case of urban planning or via cost functionals of irrigation patterns or of mass fluxes in case of branched transport. We show here that actually both models can be described by all three types of formulations; in particular, the urban planning can be cast into a Eulerian (flux-based) or a Lagrangian (pattern-based) framework.

Keywords: Optimal transport, Branched transport, Wasserstein distance, Optimal Networks, urban planning, Irrigation