The Gilbert−Steiner problem is a mass transportation problem, where the cost of the transportation depends on the network used to move the mass and it is proportional to a certain power of the “flow”. In this paper, we introduce a new formulation of the problem, which turns it into the minimization of a convex functional in a class of currents with coefficients in a group. This framework allows us to define calibrations. We apply this technique to prove the optimality of a certain irrigation network in the separable Hilbert space , having countably many branching points and a continuous amount of endpoints.
DOI : 10.1051/cocv/2015028
Mots clés : Gilbert−Steiner problem, irrigation problem, calibrations, flatG-chains
@article{COCV_2016__22_2_543_0,
author = {Marchese, Andrea and Massaccesi, Annalisa},
title = {An optimal irrigation network with infinitely many branching points},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {543--561},
publisher = {EDP-Sciences},
volume = {22},
number = {2},
year = {2016},
doi = {10.1051/cocv/2015028},
zbl = {1343.49074},
mrnumber = {3491783},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2015028/}
}
TY - JOUR AU - Marchese, Andrea AU - Massaccesi, Annalisa TI - An optimal irrigation network with infinitely many branching points JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2016 SP - 543 EP - 561 VL - 22 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2015028/ DO - 10.1051/cocv/2015028 LA - en ID - COCV_2016__22_2_543_0 ER -
%0 Journal Article %A Marchese, Andrea %A Massaccesi, Annalisa %T An optimal irrigation network with infinitely many branching points %J ESAIM: Control, Optimisation and Calculus of Variations %D 2016 %P 543-561 %V 22 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2015028/ %R 10.1051/cocv/2015028 %G en %F COCV_2016__22_2_543_0
Marchese, Andrea; Massaccesi, Annalisa. An optimal irrigation network with infinitely many branching points. ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 2, pp. 543-561. doi : 10.1051/cocv/2015028. http://www.numdam.org/articles/10.1051/cocv/2015028/
and , Currents in metric spaces. Acta Math. 185 (2000) 1–80. | DOI | MR | Zbl
, and , Traffic plans. Publ. Mat. 49 (2005) 417–451. | DOI | MR | Zbl
, and , Are there infinite irrigation trees? J. Math. Fluid Mech. 8 (2006) 311–332. | DOI | MR | Zbl
M. Bernot, V. Caselles and J.-M. Morel,Optimal Transportation Networks, Models and theory. In vol. 1955 of Lect. Notes Math. Springer-Verlag, Berlin (2009). | MR | Zbl
F. Bethuel, A counterexample to the weak density of smooth maps between manifolds in Sobolev spaces. Preprint (2014). | arXiv | MR
and , Optimal design of gas pipeline networks. J. Oper. Res. Soc. 30 (1979) 1047–1060. | DOI | Zbl
and , Optimal networks for mass transportation problems. ESAIM: COCV 11 (2005) 88–101. | Numdam | MR | Zbl
, and , Path functionals over Wasserstein spaces. J. Eur. Math. Soc. 8 (2006) 415–434. | DOI | MR | Zbl
and , An equivalent path functional formulation of branched transportation problems. Discrete Contin. Dyn. Syst. 29 (2011) 845–871. | DOI | MR | Zbl
, and , Harmonic maps with defects. Comm. Math. Phys. 107 (1986) 649–705. | DOI | MR | Zbl
S. Conti, A. Garroni and A. Massaccesi, Lower semicontinuity and relaxation of functionals on one-dimensional currents with multiplicity in a lattice. Preprint (2013).
and , Rectifiable and flat chains in a metric space. Amer. J. Math. 134 (2012) 1–69. | DOI | MR | Zbl
, Least cost design of branched pipe network system. J. Environ. Eng. Division 100 (1974) 821–835. | DOI
H. Federer, Geometric measure theory. Die Grundl. Math. Wiss., Band 153. Springer-Verlag, New York Inc. (1969). | MR | Zbl
, Minimum cost communication networks. Bell Syst. Tech. J. 46 (1967) 2209–2227. | DOI
S.G. Krantz and H.R. Parks, Geometric Integration Theory. Cornerstones. Birkhäuser Boston Inc., Boston, MA (2008). | MR | Zbl
and , Transport distances and irrigation models. J. Convex Anal. 16 (2009) 121–152. | MR | Zbl
and , Synchronic and asynchronic descriptions of irrigation problems. Adv. Nonlin. Stud. 13 (2013) 583–623. | DOI | MR | Zbl
, and , A variational model of irrigation patterns. Interfaces Free Bound. 5 (2003) 391–415. | DOI | MR | Zbl
and , The Steiner tree problem revisited through rectifiable G-currents. Adv. Calc. Var. 9 (2016) 19–39. | MR | Zbl
, , and , An optimal bronchial tree may be dangerous. Nature. 427 (2004) 633–636. | DOI
and , The regularity of optimal irrigation patterns. Arch. Ration. Mech. Anal. 195 (2010) 499–531. | DOI | MR | Zbl
and , Optimal transportation networks as flat chains. Interfaces Free Bound. 8 (2006) 393–436. | DOI | MR | Zbl
and , Existence and regularity results for the Steiner problem. Calc. Var. Partial Differ. Equ. 46 (2013) 837–860. | DOI | MR | Zbl
, and , An example of an infinite Steiner tree connecting an uncountable set. Adv. Calc. Var. 8 (2015) 267–290. | DOI | MR | Zbl
and , The Steiner problem for infinitely many points. Rend. Semin. Mat. Univ. Padova 124 (2010) 43–56. | DOI | Numdam | MR | Zbl
L. Simon, Lectures on Geometric Measure Theory. In vol. 3 of Proc. of the Centre for Mathematical Analysis. Australian National University Centre for Mathematical Analysis, Canberra (1983). | MR | Zbl
, Optimization model of transport currents. Problems in mathematical analysis. J. Math. Sci. (N. Y.) 135 (2006) 3457–3484. | DOI | MR | Zbl
, and , A general model for the origin of allometric scaling laws in biology. Science 276 (1997) 122–126. | DOI
, Rectifiability of flat chains. Ann. Math.150 (1999) 165–184. | DOI | MR | Zbl
, Optimal paths related to transport problems. Commun. Contemp. Math. 5 (2003) 251–279. | DOI | MR | Zbl
, Interior regularity of optimal transport paths. Calc. Var. Partial Differ. Equ. 20 (2004) 283–299. | DOI | MR | Zbl
, The formation of a tree leaf. ESAIM: COCV 13 (2007) 359–377. | Numdam | MR | Zbl
, Minimum concave cost flows in certain networks. Manag. Sci. 14 (1968) 429–450. | DOI | MR | Zbl
Cité par Sources :
