April  2014, 34(4): 1683-1700. doi: 10.3934/dcds.2014.34.1683

On landscape functions associated with transport paths

1. 

University of California at Davis, Department of Mathematics, One Shields Ave, Davis,CA, 95616, United States

Received  October 2012 Revised  March 2013 Published  October 2013

In this paper, we introduce a multiple-sources version of the landscape function which was originally introduced by Santambrogio in [10]. More precisely, we study landscape functions associated with a transport path between two atomic measures of equal mass. We also study p-harmonic functions on a directed graph for nonpositive $p$. We show an equivalence relation between landscape functions associated with an $\alpha $-transport path and $ p$-harmonic functions on the underlying graph of the transport path for $ p=\alpha /(\alpha -1)$, which is the conjugate of $\alpha $. Furthermore, we prove the Lipschitz continuity of a landscape function associated with an optimal transport path on each of its connected components.
Citation: Qinglan Xia. On landscape functions associated with transport paths. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1683-1700. doi: 10.3934/dcds.2014.34.1683
References:
[1]

R. B. Bapat, "Graphs and Matrices,", Universitext. Springer, (2010).  doi: 10.1007/978-1-84882-981-7.  Google Scholar

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M. Bernot, V. Caselles and J.-M. Morel, Traffic plans,, Publicacions Matematiques, 49 (2005), 417.  doi: 10.5565/PUBLMAT_49205_09.  Google Scholar

[3]

M. Bernot, V. Caselles and J. M. Morel, "Optimal Transportation Networks: Models and Theory,", Lecture Notes in Mathematics, 1955 (2009).   Google Scholar

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A. Brancolini, G. Buttazzo and F. Santambrogio, Path functions over Wasserstein spaces,, Journal of the European Mathematical Society, 8 (2006), 415.  doi: 10.4171/JEMS/61.  Google Scholar

[5]

A. Brancolini and S. Solimini, On the Hölder regularity of the landscape function,, Interfaces and Free Boundaries, 13 (2011), 191.  doi: 10.4171/IFB/254.  Google Scholar

[6]

L. Brasco, G. Buttazzo and F. Santambrogio, A Benamou-Brenier approach to branched transport,, SIAM J. Math. Anal., 43 (2011), 1023.  doi: 10.1137/10079286X.  Google Scholar

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G. Devillanova and S. Solimini, On the dimension of an irrigable measure,, Rend. Semin. Mat. Univ. Padova, 117 (2007), 1.   Google Scholar

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

[9]

J. M. Morel and F. Santambrogio, The regularity of optimal irrigation patterns,, Arch. Mat. Rat. Mech., 195 (2010), 499.  doi: 10.1007/s00205-008-0210-9.  Google Scholar

[10]

F. Santambrogio, Optimal channel networks, landscape function and branched transport,, Interfaces and Free Boundaries, 9 (2007), 149.  doi: 10.4171/IFB/160.  Google Scholar

[11]

C. Villani, "Topics in Optimal Transportation,", Graduate Studies in Mathematics, 58 (2003).  doi: 10.1007/b12016.  Google Scholar

[12]

Q. Xia, Optimal paths related to transport problems,, Communications in Contemporary Mathematics, 5 (2003), 251.  doi: 10.1142/S021919970300094X.  Google Scholar

[13]

Q. Xia, Interior regularity of optimal transport paths,, Calculus of Variations and Partial Differential Equations, 20 (2004), 283.  doi: 10.1007/s00526-003-0237-6.  Google Scholar

[14]

Q. Xia, The formation of a tree leaf,, ESAIM Control, 13 (2007), 359.  doi: 10.1051/cocv:2007016.  Google Scholar

[15]

Q. Xia, The geodesic problem in quasimetric spaces,, Journal of Geometric Analysis, 19 (2009), 452.  doi: 10.1007/s12220-008-9065-4.  Google Scholar

[16]

Q. Xia, Boundary regularity of optimal transport paths,, Advances in Calculus of Variations, 4 (2011), 153.  doi: 10.1515/ACV.2010.024.  Google Scholar

[17]

Q. Xia, Ramified optimal transportation in geodesic metric spaces,, Advances in Calculus of Variations, 4 (2011), 277.  doi: 10.1515/ACV.2011.002.  Google Scholar

[18]

Q. Xia, Numerical simulation of optimal transport paths,, in, 1 (2010), 521.  doi: 10.1109/ICCMS.2010.30.  Google Scholar

[19]

Q. Xia and A. Vershynina, On the transport dimension of measures,, SIAM Journal on Mathematical Analysis, 41 (2010), 2407.  doi: 10.1137/090770205.  Google Scholar

[20]

Q. Xia and D. Unger, Diffusion-limited aggregation driven by optimal transportation,, Fractals, 18 (2010), 247.  doi: 10.1142/S0218348X10004877.  Google Scholar

[21]

Q. Xia and S. Xu, On the Ramified Optimal Allocation Problem,, Networks and Heterogeneous Media, 8 (2013), 591.  doi: 10.3934/nhm.2013.8.591.  Google Scholar

show all references

References:
[1]

R. B. Bapat, "Graphs and Matrices,", Universitext. Springer, (2010).  doi: 10.1007/978-1-84882-981-7.  Google Scholar

[2]

M. Bernot, V. Caselles and J.-M. Morel, Traffic plans,, Publicacions Matematiques, 49 (2005), 417.  doi: 10.5565/PUBLMAT_49205_09.  Google Scholar

[3]

M. Bernot, V. Caselles and J. M. Morel, "Optimal Transportation Networks: Models and Theory,", Lecture Notes in Mathematics, 1955 (2009).   Google Scholar

[4]

A. Brancolini, G. Buttazzo and F. Santambrogio, Path functions over Wasserstein spaces,, Journal of the European Mathematical Society, 8 (2006), 415.  doi: 10.4171/JEMS/61.  Google Scholar

[5]

A. Brancolini and S. Solimini, On the Hölder regularity of the landscape function,, Interfaces and Free Boundaries, 13 (2011), 191.  doi: 10.4171/IFB/254.  Google Scholar

[6]

L. Brasco, G. Buttazzo and F. Santambrogio, A Benamou-Brenier approach to branched transport,, SIAM J. Math. Anal., 43 (2011), 1023.  doi: 10.1137/10079286X.  Google Scholar

[7]

G. Devillanova and S. Solimini, On the dimension of an irrigable measure,, Rend. Semin. Mat. Univ. Padova, 117 (2007), 1.   Google Scholar

[8]

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

[9]

J. M. Morel and F. Santambrogio, The regularity of optimal irrigation patterns,, Arch. Mat. Rat. Mech., 195 (2010), 499.  doi: 10.1007/s00205-008-0210-9.  Google Scholar

[10]

F. Santambrogio, Optimal channel networks, landscape function and branched transport,, Interfaces and Free Boundaries, 9 (2007), 149.  doi: 10.4171/IFB/160.  Google Scholar

[11]

C. Villani, "Topics in Optimal Transportation,", Graduate Studies in Mathematics, 58 (2003).  doi: 10.1007/b12016.  Google Scholar

[12]

Q. Xia, Optimal paths related to transport problems,, Communications in Contemporary Mathematics, 5 (2003), 251.  doi: 10.1142/S021919970300094X.  Google Scholar

[13]

Q. Xia, Interior regularity of optimal transport paths,, Calculus of Variations and Partial Differential Equations, 20 (2004), 283.  doi: 10.1007/s00526-003-0237-6.  Google Scholar

[14]

Q. Xia, The formation of a tree leaf,, ESAIM Control, 13 (2007), 359.  doi: 10.1051/cocv:2007016.  Google Scholar

[15]

Q. Xia, The geodesic problem in quasimetric spaces,, Journal of Geometric Analysis, 19 (2009), 452.  doi: 10.1007/s12220-008-9065-4.  Google Scholar

[16]

Q. Xia, Boundary regularity of optimal transport paths,, Advances in Calculus of Variations, 4 (2011), 153.  doi: 10.1515/ACV.2010.024.  Google Scholar

[17]

Q. Xia, Ramified optimal transportation in geodesic metric spaces,, Advances in Calculus of Variations, 4 (2011), 277.  doi: 10.1515/ACV.2011.002.  Google Scholar

[18]

Q. Xia, Numerical simulation of optimal transport paths,, in, 1 (2010), 521.  doi: 10.1109/ICCMS.2010.30.  Google Scholar

[19]

Q. Xia and A. Vershynina, On the transport dimension of measures,, SIAM Journal on Mathematical Analysis, 41 (2010), 2407.  doi: 10.1137/090770205.  Google Scholar

[20]

Q. Xia and D. Unger, Diffusion-limited aggregation driven by optimal transportation,, Fractals, 18 (2010), 247.  doi: 10.1142/S0218348X10004877.  Google Scholar

[21]

Q. Xia and S. Xu, On the Ramified Optimal Allocation Problem,, Networks and Heterogeneous Media, 8 (2013), 591.  doi: 10.3934/nhm.2013.8.591.  Google Scholar

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