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A 2-dimensional shape optimization problem for tree branches
1. | Department of Mathematics, Penn State University, University Park, PA 16802, USA |
2. | Department of Mathematical Sciences, NTNU – Norwegian University of Science and Technology, NO-7491 Trondheim, Norway |
The paper is concerned with a shape optimization problem, where the functional to be maximized describes the total sunlight collected by a distribution of tree leaves, minus the cost for transporting water and nutrient from the base of trunk to all the leaves. In a 2-dimensional setting, the solution is proved to be unique and explicitly determined.
References:
[1] |
M. Bernot, V. Caselles and J.-M. Morel, Optimal Transportation Networks. Models and Theory, Springer Lecture Notes in Mathematics 1955, Berlin, 2009. |
[2] |
M. Bernot, V. Caselles and J.-M. Morel,
The structure of branched transportation networks, Calc. Var. Partial Differential Equations, 32 (2008), 279-317.
doi: 10.1007/s00526-007-0139-0. |
[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. Brancolini and B. Wirth,
Optimal energy scaling for micropatterns in transport networks, SIAM J. Math. Anal., 49 (2017), 311-359.
doi: 10.1137/15M1050227. |
[5] |
L. Brasco and F. Santambrogio,
An equivalent path functional formulation of branched transportation problems, Discrete Contin. Dyn. Syst., 29 (2011), 845-871.
doi: 10.3934/dcds.2011.29.845. |
[6] |
A. Bressan, S. T. Galtung, A. Reigstad and J. Ridder,
Competition models for plant stems, J. Differential Equations, 269 (2020), 1571-1611.
doi: 10.1016/j.jde.2020.01.013. |
[7] |
A. Bressan, M. Palladino and Q. Sun, Variational problems for tree roots and branches, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 7, 31 pp.
doi: 10.1007/s00526-019-1666-1. |
[8] |
A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, AIMS Series in Applied Mathematics, Springfield Mo. 2007. |
[9] |
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. |
[10] |
A. Bressan and Q. Sun, Weighted irrigation plans, submitted., |
[11] |
L. Cesari, Optimization - Theory and Applications, Springer-Verlag, 1983.
doi: 10.1007/978-1-4613-8165-5. |
[12] |
G. Devillanova and S. Solimini,
Some remarks on the fractal structure of irrigation balls, Adv. Nonlinear Stud., 19 (2019), 55-68.
doi: 10.1515/ans-2018-2035. |
[13] |
E. N. Gilbert,
Minimum cost communication networks., Bell System Tech. J., 46 (1967), 2209-2227.
doi: 10.1002/j.1538-7305.1967.tb04250.x. |
[14] |
F. Maddalena, S. Solimini and J.-M. Morel,
A variational model of irrigation patterns, Interfaces Free Bound., 5 (2003), 391-415.
doi: 10.4171/IFB/85. |
[15] |
J.-M. Morel and F. Santambrogio,
The regularity of optimal irrigation patterns, Arch. Ration. Mech. Anal., 195 (2010), 499-531.
doi: 10.1007/s00205-008-0210-9. |
[16] |
P. Pegon, F. Santambrogio and Q. Xia,
A fractal shape optimization problem in branched transport, J. Math. Pures Appl., 123 (2019), 244-269.
doi: 10.1016/j.matpur.2018.06.007. |
[17] |
F. Santambrogio,
Optimal channel networks, landscape function and branched transport, Interfaces Free Bound., 9 (2007), 149-169.
doi: 10.4171/IFB/160. |
[18] |
Q. Xia,
Optimal paths related to transport problems, Comm. Contemp. Math., 5 (2003), 251-279.
doi: 10.1142/S021919970300094X. |
[19] |
Q. Xia,
Interior regularity of optimal transport paths., Calc. Var. Partial Differential Equations, 20 (2004), 283-299.
doi: 10.1007/s00526-003-0237-6. |
[20] |
Q. Xia,
Motivations, ideas and applications of ramified optimal transportation, ESAIM Math. Model. Numer. Anal., 49 (2015), 1791-1832.
doi: 10.1051/m2an/2015028. |
show all references
References:
[1] |
M. Bernot, V. Caselles and J.-M. Morel, Optimal Transportation Networks. Models and Theory, Springer Lecture Notes in Mathematics 1955, Berlin, 2009. |
[2] |
M. Bernot, V. Caselles and J.-M. Morel,
The structure of branched transportation networks, Calc. Var. Partial Differential Equations, 32 (2008), 279-317.
doi: 10.1007/s00526-007-0139-0. |
[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. Brancolini and B. Wirth,
Optimal energy scaling for micropatterns in transport networks, SIAM J. Math. Anal., 49 (2017), 311-359.
doi: 10.1137/15M1050227. |
[5] |
L. Brasco and F. Santambrogio,
An equivalent path functional formulation of branched transportation problems, Discrete Contin. Dyn. Syst., 29 (2011), 845-871.
doi: 10.3934/dcds.2011.29.845. |
[6] |
A. Bressan, S. T. Galtung, A. Reigstad and J. Ridder,
Competition models for plant stems, J. Differential Equations, 269 (2020), 1571-1611.
doi: 10.1016/j.jde.2020.01.013. |
[7] |
A. Bressan, M. Palladino and Q. Sun, Variational problems for tree roots and branches, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 7, 31 pp.
doi: 10.1007/s00526-019-1666-1. |
[8] |
A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, AIMS Series in Applied Mathematics, Springfield Mo. 2007. |
[9] |
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. |
[10] |
A. Bressan and Q. Sun, Weighted irrigation plans, submitted., |
[11] |
L. Cesari, Optimization - Theory and Applications, Springer-Verlag, 1983.
doi: 10.1007/978-1-4613-8165-5. |
[12] |
G. Devillanova and S. Solimini,
Some remarks on the fractal structure of irrigation balls, Adv. Nonlinear Stud., 19 (2019), 55-68.
doi: 10.1515/ans-2018-2035. |
[13] |
E. N. Gilbert,
Minimum cost communication networks., Bell System Tech. J., 46 (1967), 2209-2227.
doi: 10.1002/j.1538-7305.1967.tb04250.x. |
[14] |
F. Maddalena, S. Solimini and J.-M. Morel,
A variational model of irrigation patterns, Interfaces Free Bound., 5 (2003), 391-415.
doi: 10.4171/IFB/85. |
[15] |
J.-M. Morel and F. Santambrogio,
The regularity of optimal irrigation patterns, Arch. Ration. Mech. Anal., 195 (2010), 499-531.
doi: 10.1007/s00205-008-0210-9. |
[16] |
P. Pegon, F. Santambrogio and Q. Xia,
A fractal shape optimization problem in branched transport, J. Math. Pures Appl., 123 (2019), 244-269.
doi: 10.1016/j.matpur.2018.06.007. |
[17] |
F. Santambrogio,
Optimal channel networks, landscape function and branched transport, Interfaces Free Bound., 9 (2007), 149-169.
doi: 10.4171/IFB/160. |
[18] |
Q. Xia,
Optimal paths related to transport problems, Comm. Contemp. Math., 5 (2003), 251-279.
doi: 10.1142/S021919970300094X. |
[19] |
Q. Xia,
Interior regularity of optimal transport paths., Calc. Var. Partial Differential Equations, 20 (2004), 283-299.
doi: 10.1007/s00526-003-0237-6. |
[20] |
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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