Article Contents
Article Contents

# On landscape functions associated with transport paths

• 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.
Mathematics Subject Classification: Primary: 90B06, 49Q20.

 Citation:

•  [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.