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 and Continuous Dynamical Systems, 2014, 34 (4) : 1683-1700. doi: 10.3934/dcds.2014.34.1683
References:
[1]

R. B. Bapat, "Graphs and Matrices," Universitext. Springer, London; Hindustan Book Agency, New Delhi, 2010. x+171 pp. doi: 10.1007/978-1-84882-981-7.

[2]

M. Bernot, V. Caselles and J.-M. Morel, Traffic plans, Publicacions Matematiques, 49 (2005), 417-451. doi: 10.5565/PUBLMAT_49205_09.

[3]

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

[4]

A. Brancolini, G. Buttazzo and F. Santambrogio, Path functions over Wasserstein spaces, Journal of the European Mathematical Society, 8 (2006), 415-434. doi: 10.4171/JEMS/61.

[5]

A. Brancolini and S. Solimini, On the Hölder regularity of the landscape function, Interfaces and Free Boundaries, 13 (2011), 191-222. doi: 10.4171/IFB/254.

[6]

L. Brasco, G. Buttazzo and F. Santambrogio, A Benamou-Brenier approach to branched transport, SIAM J. Math. Anal., 43 (2011), 1023-1040. doi: 10.1137/10079286X.

[7]

G. Devillanova and S. Solimini, On the dimension of an irrigable measure, Rend. Semin. Mat. Univ. Padova, 117 (2007), 1-49.

[8]

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

[9]

J. M. Morel and F. Santambrogio, The regularity of optimal irrigation patterns, Arch. Mat. Rat. Mech., 195 (2010), 499-531. doi: 10.1007/s00205-008-0210-9.

[10]

F. Santambrogio, Optimal channel networks, landscape function and branched transport, Interfaces and Free Boundaries, 9 (2007), 149-169. doi: 10.4171/IFB/160.

[11]

C. Villani, "Topics in Optimal Transportation," Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI, 2003. doi: 10.1007/b12016.

[12]

Q. Xia, Optimal paths related to transport problems, Communications in Contemporary Mathematics, 5 (2003), 251-279. doi: 10.1142/S021919970300094X.

[13]

Q. Xia, Interior regularity of optimal transport paths, Calculus of Variations and Partial Differential Equations, 20 (2004), 283-299. doi: 10.1007/s00526-003-0237-6.

[14]

Q. Xia, The formation of a tree leaf, ESAIM Control, Optimisation and Calculus of Variations, 13 (2007), 359-377 (electronic). doi: 10.1051/cocv:2007016.

[15]

Q. Xia, The geodesic problem in quasimetric spaces, Journal of Geometric Analysis, 19 (2009), 452-479. doi: 10.1007/s12220-008-9065-4.

[16]

Q. Xia, Boundary regularity of optimal transport paths, Advances in Calculus of Variations, 4 (2011), 153-174. doi: 10.1515/ACV.2010.024.

[17]

Q. Xia, Ramified optimal transportation in geodesic metric spaces, Advances in Calculus of Variations, 4 (2011), 277-307. doi: 10.1515/ACV.2011.002.

[18]

Q. Xia, Numerical simulation of optimal transport paths, in "The Second International Conference on Computer Modeling and Simulation," 1, IEEE, (2010), 521-525. doi: 10.1109/ICCMS.2010.30.

[19]

Q. Xia and A. Vershynina, On the transport dimension of measures, SIAM Journal on Mathematical Analysis, 41 (2010), 2407-2430. doi: 10.1137/090770205.

[20]

Q. Xia and D. Unger, Diffusion-limited aggregation driven by optimal transportation, Fractals, 18 (2010), 247-253. doi: 10.1142/S0218348X10004877.

[21]

Q. Xia and S. Xu, On the Ramified Optimal Allocation Problem, Networks and Heterogeneous Media, 8 (2013), 591-624. doi: 10.3934/nhm.2013.8.591.

show all references

References:
[1]

R. B. Bapat, "Graphs and Matrices," Universitext. Springer, London; Hindustan Book Agency, New Delhi, 2010. x+171 pp. doi: 10.1007/978-1-84882-981-7.

[2]

M. Bernot, V. Caselles and J.-M. Morel, Traffic plans, Publicacions Matematiques, 49 (2005), 417-451. doi: 10.5565/PUBLMAT_49205_09.

[3]

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

[4]

A. Brancolini, G. Buttazzo and F. Santambrogio, Path functions over Wasserstein spaces, Journal of the European Mathematical Society, 8 (2006), 415-434. doi: 10.4171/JEMS/61.

[5]

A. Brancolini and S. Solimini, On the Hölder regularity of the landscape function, Interfaces and Free Boundaries, 13 (2011), 191-222. doi: 10.4171/IFB/254.

[6]

L. Brasco, G. Buttazzo and F. Santambrogio, A Benamou-Brenier approach to branched transport, SIAM J. Math. Anal., 43 (2011), 1023-1040. doi: 10.1137/10079286X.

[7]

G. Devillanova and S. Solimini, On the dimension of an irrigable measure, Rend. Semin. Mat. Univ. Padova, 117 (2007), 1-49.

[8]

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

[9]

J. M. Morel and F. Santambrogio, The regularity of optimal irrigation patterns, Arch. Mat. Rat. Mech., 195 (2010), 499-531. doi: 10.1007/s00205-008-0210-9.

[10]

F. Santambrogio, Optimal channel networks, landscape function and branched transport, Interfaces and Free Boundaries, 9 (2007), 149-169. doi: 10.4171/IFB/160.

[11]

C. Villani, "Topics in Optimal Transportation," Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI, 2003. doi: 10.1007/b12016.

[12]

Q. Xia, Optimal paths related to transport problems, Communications in Contemporary Mathematics, 5 (2003), 251-279. doi: 10.1142/S021919970300094X.

[13]

Q. Xia, Interior regularity of optimal transport paths, Calculus of Variations and Partial Differential Equations, 20 (2004), 283-299. doi: 10.1007/s00526-003-0237-6.

[14]

Q. Xia, The formation of a tree leaf, ESAIM Control, Optimisation and Calculus of Variations, 13 (2007), 359-377 (electronic). doi: 10.1051/cocv:2007016.

[15]

Q. Xia, The geodesic problem in quasimetric spaces, Journal of Geometric Analysis, 19 (2009), 452-479. doi: 10.1007/s12220-008-9065-4.

[16]

Q. Xia, Boundary regularity of optimal transport paths, Advances in Calculus of Variations, 4 (2011), 153-174. doi: 10.1515/ACV.2010.024.

[17]

Q. Xia, Ramified optimal transportation in geodesic metric spaces, Advances in Calculus of Variations, 4 (2011), 277-307. doi: 10.1515/ACV.2011.002.

[18]

Q. Xia, Numerical simulation of optimal transport paths, in "The Second International Conference on Computer Modeling and Simulation," 1, IEEE, (2010), 521-525. doi: 10.1109/ICCMS.2010.30.

[19]

Q. Xia and A. Vershynina, On the transport dimension of measures, SIAM Journal on Mathematical Analysis, 41 (2010), 2407-2430. doi: 10.1137/090770205.

[20]

Q. Xia and D. Unger, Diffusion-limited aggregation driven by optimal transportation, Fractals, 18 (2010), 247-253. doi: 10.1142/S0218348X10004877.

[21]

Q. Xia and S. Xu, On the Ramified Optimal Allocation Problem, Networks and Heterogeneous Media, 8 (2013), 591-624. doi: 10.3934/nhm.2013.8.591.

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