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July  2011, 29(3): 845-871. doi: 10.3934/dcds.2011.29.845

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, Italy
2. 

Laboratoire de Mathématiques d'Orsay, UMR CNRS 8628, Faculté des Sciences, Université Paris-Sud XI, 91405 Orsay Cedex, France

Received  January 2010 Revised  May 2010 Published  November 2010

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: 90B2.
Citation: Lorenzo Brasco, Filippo Santambrogio. An equivalent path functional formulation of branched transportation problems. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 845-871. doi: 10.3934/dcds.2011.29.845
References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures," 2nd edition, Lectures in Mathematics ETH, Zürich, Birkhäuser Verlag, Basel, 2008.  Google Scholar

[2]

L. Ambrosio and P. Tilli, "Selected topics on 'Analysis in Metric Spaces'," Appunti dei corsi tenuti da docenti della Scuola, Scuola Normale Superiore, Pisa, 2000.  Google Scholar

[3]

M. Bernot, V. Caselles and J.-M. Morel, Traffic plans, Publ. Mat., 49 (2005), 417-451.  Google Scholar

[4]

M. Bernot, V. Caselles and J.-M. Morel, The structure of branched transportation networks, Calc. Var. Partial Differential Equations, 2 (2008), 279-371. doi: 10.1007/s00526-007-0139-0.  Google Scholar

[5]

M. Bernot, V. Caselles and J.-M. Morel, "Optimal Transportation Networks. Models and Theory," Lecture Notes in Mathemathics, 1955, Springer-Verlag, Berlin, 2009.  Google Scholar

[6]

M. Bernot and A. Figalli, Synchronized traffic plans and stability of optima, ESAIM Control Optim. Calc. Var., 14 (2008), 864-878. doi: 10.1051/cocv:2008012.  Google Scholar

[7]

S. Bianchini and A. Brancolini, Estimates on path functionals over Wasserstein spaces, SIAM J. Math. Anal., 42 (2010), 1179-1217. doi: 10.1137/100782693.  Google Scholar

[8]

A. Brancolini, G. Buttazzo and F. Santambrogio, Path functionals over Wasserstein spaces, J. Eur. Math. Soc., 8 (2006), 415-434. doi: 10.4171/JEMS/61.  Google Scholar

[9]

A. Brancolini and S. Solimini, On the Hölder regularity of the landscape function,, accepted on Interfaces Free Bound., ().   Google Scholar

[10]

L. Brasco, Curves of minimal action over metric spaces, Ann. Mat. Pura Appl., 189 (2010), 95-125. doi: 10.1007/s10231-009-0102-0.  Google Scholar

[11]

C. Dellacherie and P.-A. Meyer, "Probabilités et Potentiel" (French) [Probabilities and Potential], Chapitres I à IV. Édition entièrement refondue, Publications de l'Institut de Mathématique de l'Université de Strasbourg, No. XV, Actualités Scientifiques et Industrielles, 1372, Hermann, Paris, 1975.  Google Scholar

[12]

E. N. Gilbert and H. O. Pollak, Steiner minimal trees, SIAM J. Appl. Math., 16 (1968), 1-29. doi: 10.1137/0116001.  Google Scholar

[13]

S. Lisini, Characterization of absolutely continuous curves in Wasserstein spaces, Calc. Var. Partial Differential Equations, 28 (2007), 85-120. doi: 10.1007/s00526-006-0032-2.  Google Scholar

[14]

F. Maddalena, J.-M. Morel and S. Solimini, A variational model of irrigation patterns, Interfaces Free Bound., 5 (2003), 391-415. doi: 10.4171/IFB/85.  Google Scholar

[15]

F. Santambrogio, Optimal channel networks, landscape function and branched transport, Interfaces Free Bound., 9 (2007), 149-169.  Google Scholar

[16]

C. Villani, "Topics in Optimal Transportation," Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI, 2003.  Google Scholar

[17]

Q. Xia, Optimal paths related to transport problems, Commun. Contemp. Math., 5 (2003), 251-279. doi: 10.1142/S021919970300094X.  Google Scholar

show all references

References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures," 2nd edition, Lectures in Mathematics ETH, Zürich, Birkhäuser Verlag, Basel, 2008.  Google Scholar

[2]

L. Ambrosio and P. Tilli, "Selected topics on 'Analysis in Metric Spaces'," Appunti dei corsi tenuti da docenti della Scuola, Scuola Normale Superiore, Pisa, 2000.  Google Scholar

[3]

M. Bernot, V. Caselles and J.-M. Morel, Traffic plans, Publ. Mat., 49 (2005), 417-451.  Google Scholar

[4]

M. Bernot, V. Caselles and J.-M. Morel, The structure of branched transportation networks, Calc. Var. Partial Differential Equations, 2 (2008), 279-371. doi: 10.1007/s00526-007-0139-0.  Google Scholar

[5]

M. Bernot, V. Caselles and J.-M. Morel, "Optimal Transportation Networks. Models and Theory," Lecture Notes in Mathemathics, 1955, Springer-Verlag, Berlin, 2009.  Google Scholar

[6]

M. Bernot and A. Figalli, Synchronized traffic plans and stability of optima, ESAIM Control Optim. Calc. Var., 14 (2008), 864-878. doi: 10.1051/cocv:2008012.  Google Scholar

[7]

S. Bianchini and A. Brancolini, Estimates on path functionals over Wasserstein spaces, SIAM J. Math. Anal., 42 (2010), 1179-1217. doi: 10.1137/100782693.  Google Scholar

[8]

A. Brancolini, G. Buttazzo and F. Santambrogio, Path functionals over Wasserstein spaces, J. Eur. Math. Soc., 8 (2006), 415-434. doi: 10.4171/JEMS/61.  Google Scholar

[9]

A. Brancolini and S. Solimini, On the Hölder regularity of the landscape function,, accepted on Interfaces Free Bound., ().   Google Scholar

[10]

L. Brasco, Curves of minimal action over metric spaces, Ann. Mat. Pura Appl., 189 (2010), 95-125. doi: 10.1007/s10231-009-0102-0.  Google Scholar

[11]

C. Dellacherie and P.-A. Meyer, "Probabilités et Potentiel" (French) [Probabilities and Potential], Chapitres I à IV. Édition entièrement refondue, Publications de l'Institut de Mathématique de l'Université de Strasbourg, No. XV, Actualités Scientifiques et Industrielles, 1372, Hermann, Paris, 1975.  Google Scholar

[12]

E. N. Gilbert and H. O. Pollak, Steiner minimal trees, SIAM J. Appl. Math., 16 (1968), 1-29. doi: 10.1137/0116001.  Google Scholar

[13]

S. Lisini, Characterization of absolutely continuous curves in Wasserstein spaces, Calc. Var. Partial Differential Equations, 28 (2007), 85-120. doi: 10.1007/s00526-006-0032-2.  Google Scholar

[14]

F. Maddalena, J.-M. Morel and S. Solimini, A variational model of irrigation patterns, Interfaces Free Bound., 5 (2003), 391-415. doi: 10.4171/IFB/85.  Google Scholar

[15]

F. Santambrogio, Optimal channel networks, landscape function and branched transport, Interfaces Free Bound., 9 (2007), 149-169.  Google Scholar

[16]

C. Villani, "Topics in Optimal Transportation," Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI, 2003.  Google Scholar

[17]

Q. Xia, Optimal paths related to transport problems, Commun. Contemp. Math., 5 (2003), 251-279. doi: 10.1142/S021919970300094X.  Google Scholar

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