doi: 10.3934/dcds.2021167
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Stability of optimal traffic plans in the irrigation problem

1. 

EPFL SB, Station 8, CH-1015 Lausanne, Switzerland

2. 

Department of Mathematics, University of Maryland, 4176 Campus Dr, College Park, MD 20742, USA

3. 

Department of Decision Sciences and BIDSA, Bocconi University, Via Röntgen 1, 20136 Milano MI Italy

4. 

Dipartimento di Matematica, University of Trento, Via Sommarive, 14, 38123 Povo, Italy

5. 

CEREMADE, CNRS, UMR 7534, Université Paris-Dauphine, PSL University, INRIA Paris, Project team Mokaplan, 75016 Paris, France

6. 

DER de Mathématiques, ENS Paris-Saclay, Université Paris-Saclay, 91190 Gif-sur-Yvette, France

* Corresponding author: Antonio De Rosa

Received  May 2021 Revised  August 2021 Early access November 2021

Fund Project: The first author is partially supported by the Swiss National Science Foundation grant 200021_182565. The second author is partially supported by the NSF DMS Grant No. 1906451 and the NSF DMS Grant No. 2112311. The third author is partially supported by GNAMPA-INdAM

We prove the stability of optimal traffic plans in branched transport. In particular, we show that any limit of optimal traffic plans is optimal as well. This result goes beyond the Eulerian stability proved in [7], extending it to the Lagrangian framework.

Citation: Maria Colombo, Antonio De Rosa, Andrea Marchese, Paul Pegon, Antoine Prouff. Stability of optimal traffic plans in the irrigation problem. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021167
References:
[1] L. AmbrosioN. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, Oxford, 2000.   Google Scholar
[2]

M. BernotV. Caselles and J.-M. Morel, Traffic plans, Publ. Mat., 49 (2005), 417-451.  doi: 10.5565/PUBLMAT_49205_09.  Google Scholar

[3]

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

[4]

A. Brancolini and S. Solimini, Fractal regularity results on optimal irrigation patterns, J. Math. Pures Appl., 102 (2014), 854–890, URL https://www-sciencedirect-com-s.proxy.bu.dauphine.fr/science/article/pii/S0021782414000166. doi: 10.1016/j.matpur.2014.02.008.  Google Scholar

[5]

M. Colombo, A. De Rosa and A. Marchese, Improved stability of optimal traffic paths, Calc. Var. Partial Differential Equations, 57 (2018), 33pp. doi: 10.1007/s00526-017-1299-1.  Google Scholar

[6]

M. ColomboA. De Rosa and A. Marchese, Stability for the mailing problem, J. Math. Pures Appl., 128 (2019), 152-182.  doi: 10.1016/j.matpur.2019.01.020.  Google Scholar

[7]

M. ColomboA. De Rosa and A. Marchese, On the well-posedness of branched transportation, Comm. Pure Appl. Math., 74 (2021), 833-864.  doi: 10.1002/cpa.21919.  Google Scholar

[8]

M. ColomboA. De RosaA. Marchese and S. Stuvard, On the lower semicontinuous envelope of functionals defined on polyhedral chains, Nonlinear Anal., 163 (2017), 201-215.  doi: 10.1016/j.na.2017.08.002.  Google Scholar

[9]

G. Devillanova and S. Solimini, Elementary properties of optimal irrigation patterns, Calc. Var. Partial Differential Equations, 28 (2007), 317-349.  doi: 10.1007/s00526-006-0046-9.  Google Scholar

[10] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Revised edition, Textbooks in Mathematics, CRC Press, Boca Raton, FL, 2015.   Google Scholar
[11]

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.  Google Scholar

[12]

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

[13]

P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Fractals and rectifiability. Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995.  Google Scholar

[14]

E. Paolini and E. Stepanov, Optimal transportation networks as flat chains, Interfaces Free Bound., 8 (2006), 393-436.  doi: 10.4171/IFB/149.  Google Scholar

[15]

P. Pegon, Branched Transport and Fractal Structures, Theses, Université Paris-Saclay, 2017, URL https://tel.archives-ouvertes.fr/tel-01661457. Google Scholar

[16]

P. Pegon, On the lagrangian branched transport model and the equivalence with its eulerian formulation, Topological Optimization and Optimal Transport, Radon Ser. Comput. Appl. Math., 17 (2017), 281-303.   Google Scholar

[17]

F. Santambrogio, A Dacorogna-Moser approach to flow decomposition and minimal flow problems, Congrès SMAI, ESAIM Proc. Surveys, EDP Sci., Les Ulis, 45 (2014), 265–274. doi: 10.1051/proc/201445027.  Google Scholar

[18]

L. Simon, Lectures on Geometric Measure Theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, 3. Australian National University, Centre for Mathematical Analysis, Canberra, 1983.  Google Scholar

[19]

S. K. Smirnov, Decomposition of solenoidal vector charges into elementary solenoids, and the structure of normal one-dimensional flows, Algebra i Analiz, 5 (1993), 206-238.   Google Scholar

[20]

Q. Xia, Optimal paths related to transport problems, Commun. Contemp. Math., 5 (2003), 251-279.  doi: 10.1142/S021919970300094X.  Google Scholar

show all references

References:
[1] L. AmbrosioN. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, Oxford, 2000.   Google Scholar
[2]

M. BernotV. Caselles and J.-M. Morel, Traffic plans, Publ. Mat., 49 (2005), 417-451.  doi: 10.5565/PUBLMAT_49205_09.  Google Scholar

[3]

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

[4]

A. Brancolini and S. Solimini, Fractal regularity results on optimal irrigation patterns, J. Math. Pures Appl., 102 (2014), 854–890, URL https://www-sciencedirect-com-s.proxy.bu.dauphine.fr/science/article/pii/S0021782414000166. doi: 10.1016/j.matpur.2014.02.008.  Google Scholar

[5]

M. Colombo, A. De Rosa and A. Marchese, Improved stability of optimal traffic paths, Calc. Var. Partial Differential Equations, 57 (2018), 33pp. doi: 10.1007/s00526-017-1299-1.  Google Scholar

[6]

M. ColomboA. De Rosa and A. Marchese, Stability for the mailing problem, J. Math. Pures Appl., 128 (2019), 152-182.  doi: 10.1016/j.matpur.2019.01.020.  Google Scholar

[7]

M. ColomboA. De Rosa and A. Marchese, On the well-posedness of branched transportation, Comm. Pure Appl. Math., 74 (2021), 833-864.  doi: 10.1002/cpa.21919.  Google Scholar

[8]

M. ColomboA. De RosaA. Marchese and S. Stuvard, On the lower semicontinuous envelope of functionals defined on polyhedral chains, Nonlinear Anal., 163 (2017), 201-215.  doi: 10.1016/j.na.2017.08.002.  Google Scholar

[9]

G. Devillanova and S. Solimini, Elementary properties of optimal irrigation patterns, Calc. Var. Partial Differential Equations, 28 (2007), 317-349.  doi: 10.1007/s00526-006-0046-9.  Google Scholar

[10] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Revised edition, Textbooks in Mathematics, CRC Press, Boca Raton, FL, 2015.   Google Scholar
[11]

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.  Google Scholar

[12]

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

[13]

P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Fractals and rectifiability. Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995.  Google Scholar

[14]

E. Paolini and E. Stepanov, Optimal transportation networks as flat chains, Interfaces Free Bound., 8 (2006), 393-436.  doi: 10.4171/IFB/149.  Google Scholar

[15]

P. Pegon, Branched Transport and Fractal Structures, Theses, Université Paris-Saclay, 2017, URL https://tel.archives-ouvertes.fr/tel-01661457. Google Scholar

[16]

P. Pegon, On the lagrangian branched transport model and the equivalence with its eulerian formulation, Topological Optimization and Optimal Transport, Radon Ser. Comput. Appl. Math., 17 (2017), 281-303.   Google Scholar

[17]

F. Santambrogio, A Dacorogna-Moser approach to flow decomposition and minimal flow problems, Congrès SMAI, ESAIM Proc. Surveys, EDP Sci., Les Ulis, 45 (2014), 265–274. doi: 10.1051/proc/201445027.  Google Scholar

[18]

L. Simon, Lectures on Geometric Measure Theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, 3. Australian National University, Centre for Mathematical Analysis, Canberra, 1983.  Google Scholar

[19]

S. K. Smirnov, Decomposition of solenoidal vector charges into elementary solenoids, and the structure of normal one-dimensional flows, Algebra i Analiz, 5 (1993), 206-238.   Google Scholar

[20]

Q. Xia, Optimal paths related to transport problems, Commun. Contemp. Math., 5 (2003), 251-279.  doi: 10.1142/S021919970300094X.  Google Scholar

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