# American Institute of Mathematical Sciences

September  2021, 16(3): 493-511. doi: 10.3934/nhm.2021014

## Irrigable measures for weighted irrigation plans

 Department of Mathematics, Penn State University, University Park, PA 16803, USA

Received  January 2020 Revised  April 2021 Published  September 2021 Early access  July 2021

A model of irrigation network, where lower branches must be thicker in order to support the weight of the higher ones, was recently introduced in [7]. This leads to a countable family of ODEs, describing the thickness of every branch, solved by backward induction. The present paper determines what kind of measures can be irrigated with a finite weighted cost. Indeed, the boundedness of the cost depends on the dimension of the support of the irrigated measure, and also on the asymptotic properties of the ODE which determines the thickness of branches.

Citation: Qing Sun. Irrigable measures for weighted irrigation plans. Networks and Heterogeneous Media, 2021, 16 (3) : 493-511. doi: 10.3934/nhm.2021014
##### References:
 [1] M. Bernot, V. Caselles and J. -M. Morel, Optimal Transportation Networks. Models and Theory, Lecture Notes in Mathematics, 1955, Springer, Berlin, 2009. [2] M. Bernot, V. Caselles and J.-M. Morel, Traffic plans, Publicacions Matemàtiques, 49 (2005), 417-451.  doi: 10.5565/PUBLMAT_49205_09. [3] A. Brancolini and S. Solimini, Fractal regularity results on optimal irrigation patterns, J. Math. Pures Appl., 102 (2014), 854-890.  doi: 10.1016/j.matpur.2014.02.008. [4] A. Bressan, M. Palladino and Q. Sun, Variational problems for tree roots and branches, Calc. Var. & Part. Diff. Equat., 59 (2020), Paper No. 7, 31 pp. doi: 10.1007/s00526-019-1666-1. [5] A. Bressan and F. Rampazzo, On differential systems with vector-valued impulsive controls, Boll. Un. Matematica Italiana B, 2 (1988), 641-656. [6] A. Bressan and Q. Sun, On the optimal shape of tree roots and branches, Math. Models & Methods Appl. Sci., 28 (2018), 2763-2801.  doi: 10.1142/S0218202518500604. [7] A. Bressan and Q. Sun, Weighted irrigation plans, submitted, arXiv: 1906.02232. [8] G. Devillanova and S. Solimini, On the dimension of an irrigable measure, Rend. Sem. Mat. Univ. Padova., 117 (2007), 1-49. [9] G. Devillanova and S. Solimini, Elementary properties of optimal irrigation patterns, Calc. Var. & Part. Diff. Equat., 28 (2007), 317-349.  doi: 10.1007/s00526-006-0046-9. [10] G. Devillanova and S. Solimini, Some remarks on the fractal structure of irrigation balls, Advanced Nonlinear Studies, 19 (2019), 55-68.  doi: 10.1515/ans-2018-2035. [11] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Revised Edition. CRC Press, 2015. [12] E. N. Gilbert, Minimum cost communication networks, Bell System Tech. J., 46 (1967), 2209-2227.  doi: 10.1002/j.1538-7305.1967.tb04250.x. [13] P. Hartman, Ordinary Differential Equations, , Second Edition, Birkhäuser, Boston, Mass., 1982. [14] F. Maddalena and S. Solimini, Synchronic and asynchronic descriptions of irrigation problems, Adv. Nonlinear Stud., 13 (2013), 583-623.  doi: 10.1515/ans-2013-0303. [15] F. Maddalena, G. Taglialatela and J.-M. Morel, A variational model of irrigation patterns, Interfaces Free Bound., 5 (2003), 391-415.  doi: 10.4171/IFB/85. [16] Q. Xia, Optimal paths related to transport problems, Comm. Contemp. Math., 5 (2003), 251-279.  doi: 10.1142/S021919970300094X. [17] Q. Xia, Motivations, ideas and applications of ramified optimal transportation, ESAIM Math. Model. Numer. Anal., 49 (2015), 1791-1832.  doi: 10.1051/m2an/2015028.

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##### References:
 [1] M. Bernot, V. Caselles and J. -M. Morel, Optimal Transportation Networks. Models and Theory, Lecture Notes in Mathematics, 1955, Springer, Berlin, 2009. [2] M. Bernot, V. Caselles and J.-M. Morel, Traffic plans, Publicacions Matemàtiques, 49 (2005), 417-451.  doi: 10.5565/PUBLMAT_49205_09. [3] A. Brancolini and S. Solimini, Fractal regularity results on optimal irrigation patterns, J. Math. Pures Appl., 102 (2014), 854-890.  doi: 10.1016/j.matpur.2014.02.008. [4] A. Bressan, M. Palladino and Q. Sun, Variational problems for tree roots and branches, Calc. Var. & Part. Diff. Equat., 59 (2020), Paper No. 7, 31 pp. doi: 10.1007/s00526-019-1666-1. [5] A. Bressan and F. Rampazzo, On differential systems with vector-valued impulsive controls, Boll. Un. Matematica Italiana B, 2 (1988), 641-656. [6] A. Bressan and Q. Sun, On the optimal shape of tree roots and branches, Math. Models & Methods Appl. Sci., 28 (2018), 2763-2801.  doi: 10.1142/S0218202518500604. [7] A. Bressan and Q. Sun, Weighted irrigation plans, submitted, arXiv: 1906.02232. [8] G. Devillanova and S. Solimini, On the dimension of an irrigable measure, Rend. Sem. Mat. Univ. Padova., 117 (2007), 1-49. [9] G. Devillanova and S. Solimini, Elementary properties of optimal irrigation patterns, Calc. Var. & Part. Diff. Equat., 28 (2007), 317-349.  doi: 10.1007/s00526-006-0046-9. [10] G. Devillanova and S. Solimini, Some remarks on the fractal structure of irrigation balls, Advanced Nonlinear Studies, 19 (2019), 55-68.  doi: 10.1515/ans-2018-2035. [11] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Revised Edition. CRC Press, 2015. [12] E. N. Gilbert, Minimum cost communication networks, Bell System Tech. J., 46 (1967), 2209-2227.  doi: 10.1002/j.1538-7305.1967.tb04250.x. [13] P. Hartman, Ordinary Differential Equations, , Second Edition, Birkhäuser, Boston, Mass., 1982. [14] F. Maddalena and S. Solimini, Synchronic and asynchronic descriptions of irrigation problems, Adv. Nonlinear Stud., 13 (2013), 583-623.  doi: 10.1515/ans-2013-0303. [15] F. Maddalena, G. Taglialatela and J.-M. Morel, A variational model of irrigation patterns, Interfaces Free Bound., 5 (2003), 391-415.  doi: 10.4171/IFB/85. [16] Q. Xia, Optimal paths related to transport problems, Comm. Contemp. Math., 5 (2003), 251-279.  doi: 10.1142/S021919970300094X. [17] Q. Xia, Motivations, ideas and applications of ramified optimal transportation, ESAIM Math. Model. Numer. Anal., 49 (2015), 1791-1832.  doi: 10.1051/m2an/2015028.
Left: A free standing tree with 5 branches. In this example, $\mathcal{O}(1) = \{2,3 \},\mathcal{O}(3) = \{4,5\}, \mathcal{O}(2) = \mathcal{O}(4) = \mathcal{O}(5) = \emptyset$. Right: On each branch, the weight decreases as one moves from the lower portion to the tip
Left: Two finite truncation plans, showing three maximal $\varepsilon$-good paths (thick lines) and six maximal $\varepsilon'$-good paths (thin lines), for $0<\varepsilon'<\varepsilon$. Right: The three maximal $\varepsilon$-good paths can be partitioned into five elementary branches, by the Path Splitting Algorithm
Left: The dyadic approxmiated measure $\mu_1$ is supported on the four centers $x_1^1,\ldots,x^1_4$ of small cubes. Right: Dyadic approximated measures corresponding to a family of partitions into dyadic cubes in $\bf{R}^2$
The dyadic irrigation plans in $\bf{R}^2$. Left: The dyadic irrigation plan $\chi_1$. The multiplicity on each branch equals to the mass on the terminal point. Right: The dyadic irrigation plan $\chi_2$. The particles are first transported to the 4 centers in $\mathcal{P}_1$, then on each center in $\mathcal{P}_1$, the particles are transported to the neighboring 4 centers in $\mathcal{P}_2$
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