
- Previous Article
- NHM Home
- This Issue
-
Next Article
Emergent behaviors of Lohe Hermitian sphere particles under time-delayed interactions
Irrigable measures for weighted irrigation plans
Department of Mathematics, Penn State University, University Park, PA 16803, USA |
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 [
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. |
show all references
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. |




[1] |
Abdulrazzaq T. Abed, Azzam S. Y. Aladool. Applying particle swarm optimization based on Padé approximant to solve ordinary differential equation. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 321-337. doi: 10.3934/naco.2021008 |
[2] |
Sriram Nagaraj. Optimization and learning with nonlocal calculus. Foundations of Data Science, 2022 doi: 10.3934/fods.2022009 |
[3] |
Daniel G. Alfaro Vigo, Amaury C. Álvarez, Grigori Chapiro, Galina C. García, Carlos G. Moreira. Solving the inverse problem for an ordinary differential equation using conjugation. Journal of Computational Dynamics, 2020, 7 (2) : 183-208. doi: 10.3934/jcd.2020008 |
[4] |
Richard A. Norton, G. R. W. Quispel. Discrete gradient methods for preserving a first integral of an ordinary differential equation. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 1147-1170. doi: 10.3934/dcds.2014.34.1147 |
[5] |
Tomás Caraballo, Renato Colucci, Luca Guerrini. Bifurcation scenarios in an ordinary differential equation with constant and distributed delay: A case study. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2639-2655. doi: 10.3934/dcdsb.2018268 |
[6] |
Lukáš Adam, Jiří Outrata. On optimal control of a sweeping process coupled with an ordinary differential equation. Discrete and Continuous Dynamical Systems - B, 2014, 19 (9) : 2709-2738. doi: 10.3934/dcdsb.2014.19.2709 |
[7] |
Michael Herty, Veronika Sachers. Adjoint calculus for optimization of gas networks. Networks and Heterogeneous Media, 2007, 2 (4) : 733-750. doi: 10.3934/nhm.2007.2.733 |
[8] |
Ivar Ekeland. From Frank Ramsey to René Thom: A classical problem in the calculus of variations leading to an implicit differential equation. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 1101-1119. doi: 10.3934/dcds.2010.28.1101 |
[9] |
Lijuan Wang, Yashan Xu. Admissible controls and controllable sets for a linear time-varying ordinary differential equation. Mathematical Control and Related Fields, 2018, 8 (3&4) : 1001-1019. doi: 10.3934/mcrf.2018043 |
[10] |
Yuriy Golovaty, Anna Marciniak-Czochra, Mariya Ptashnyk. Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations. Communications on Pure and Applied Analysis, 2012, 11 (1) : 229-241. doi: 10.3934/cpaa.2012.11.229 |
[11] |
Qiong Meng, X. H. Tang. Multiple solutions of second-order ordinary differential equation via Morse theory. Communications on Pure and Applied Analysis, 2012, 11 (3) : 945-958. doi: 10.3934/cpaa.2012.11.945 |
[12] |
Jan-Hendrik Webert, Philip E. Gill, Sven-Joachim Kimmerle, Matthias Gerdts. A study of structure-exploiting SQP algorithms for an optimal control problem with coupled hyperbolic and ordinary differential equation constraints. Discrete and Continuous Dynamical Systems - S, 2018, 11 (6) : 1259-1282. doi: 10.3934/dcdss.2018071 |
[13] |
Nguyen Thi Hoai. Asymptotic approximation to a solution of a singularly perturbed linear-quadratic optimal control problem with second-order linear ordinary differential equation of state variable. Numerical Algebra, Control and Optimization, 2021, 11 (4) : 495-512. doi: 10.3934/naco.2020040 |
[14] |
Bernard Dacorogna, Alessandro Ferriero. Regularity and selecting principles for implicit ordinary differential equations. Discrete and Continuous Dynamical Systems - B, 2009, 11 (1) : 87-101. doi: 10.3934/dcdsb.2009.11.87 |
[15] |
Janosch Rieger. The Euler scheme for state constrained ordinary differential inclusions. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2729-2744. doi: 10.3934/dcdsb.2016070 |
[16] |
Zvi Artstein. Averaging of ordinary differential equations with slowly varying averages. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 353-365. doi: 10.3934/dcdsb.2010.14.353 |
[17] |
Tomasz Kapela, Piotr Zgliczyński. A Lohner-type algorithm for control systems and ordinary differential inclusions. Discrete and Continuous Dynamical Systems - B, 2009, 11 (2) : 365-385. doi: 10.3934/dcdsb.2009.11.365 |
[18] |
Stefano Maset. Conditioning and relative error propagation in linear autonomous ordinary differential equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2879-2909. doi: 10.3934/dcdsb.2018165 |
[19] |
W. Sarlet, G. E. Prince, M. Crampin. Generalized submersiveness of second-order ordinary differential equations. Journal of Geometric Mechanics, 2009, 1 (2) : 209-221. doi: 10.3934/jgm.2009.1.209 |
[20] |
Jacson Simsen, Mariza Stefanello Simsen, José Valero. Convergence of nonautonomous multivalued problems with large diffusion to ordinary differential inclusions. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2347-2368. doi: 10.3934/cpaa.2020102 |
2020 Impact Factor: 1.213
Tools
Metrics
Other articles
by authors
[Back to Top]