An equivalent path functional formulation of branched transportation problems

Lorenzo Brasco; Filippo Santambrogio

doi:10.3934/dcds.2011.29.845

Article Contents

2011, Volume 29, Issue 3: 845-871. Doi: 10.3934/dcds.2011.29.845

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An equivalent path functional formulation of branched transportation problems

Lorenzo Brasco^1, and
Filippo Santambrogio^2,

1.
Dipartimento di Matematica e Applicazioni, Università di Napoli "Federico II", Complesso di Monte S. Angelo, Via Cintia, 80126 Napoli
2.
Laboratoire de Mathématiques d'Orsay, UMR CNRS 8628, Faculté des Sciences, Université Paris-Sud XI, 91405 Orsay Cedex

Received: December 31, 2009

Revised: April 30, 2010

Published: August 2011

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Abstract

We consider two models for branched transport: the one introduced in Bernot et al. (Publ Mat 49:417-451, 2005), which makes use of a functional defined on measures over the space of Lipschitz paths, and the path functional model presented in Brancolini et al. (J Eur Math Soc 8:415-434, 2006), where one minimizes some suitable action functional defined over the space of measure-valued Lipschitz curves, getting sort of a Riemannian metric on the space of probabilities, favouring atomic measures, with a cost depending on the masses of each of their atoms. We prove that modifying the latter model according to Brasco (Ann Mat Pura Appl 189:95-125, 2010), then the two models turn out to be equivalent.

Keywords:

Branched transport; irrigation patterns; path functionals.

Mathematics Subject Classification: Primary: 49Q20; Secondary: 90B20.

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