Models for branched networks are often expressed as the minimization of an energy over vector measures concentrated on -dimensional rectifiable sets with a divergence constraint. We study a Modica–Mortola type approximation , introduced by Edouard Oudet and Filippo Santambrogio, which is defined over vector measures. These energies induce some pseudo-distances between functions obtained through the minimization problem : . We prove some uniform estimates on these pseudo-distances which allow us to establish a -convergence result for these energies with a divergence constraint.
Accepté le :
DOI : 10.1051/cocv/2015049
Mots clés : Branched transportation networks, Γ-convergence, phase field models
@article{COCV_2017__23_1_309_0, author = {Monteil, Antonin}, title = {Uniform estimates for a {Modica{\textendash}Mortola} type approximation of branched transportation}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {309--335}, publisher = {EDP-Sciences}, volume = {23}, number = {1}, year = {2017}, doi = {10.1051/cocv/2015049}, mrnumber = {3601026}, zbl = {1385.49006}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2015049/} }
TY - JOUR AU - Monteil, Antonin TI - Uniform estimates for a Modica–Mortola type approximation of branched transportation JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2017 SP - 309 EP - 335 VL - 23 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2015049/ DO - 10.1051/cocv/2015049 LA - en ID - COCV_2017__23_1_309_0 ER -
%0 Journal Article %A Monteil, Antonin %T Uniform estimates for a Modica–Mortola type approximation of branched transportation %J ESAIM: Control, Optimisation and Calculus of Variations %D 2017 %P 309-335 %V 23 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2015049/ %R 10.1051/cocv/2015049 %G en %F COCV_2017__23_1_309_0
Monteil, Antonin. Uniform estimates for a Modica–Mortola type approximation of branched transportation. ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 1, pp. 309-335. doi : 10.1051/cocv/2015049. http://www.numdam.org/articles/10.1051/cocv/2015049/
T. plans. Publ. Mat. 49 (2005) 417–451. | DOI | MR | Zbl
, and ,The structure of branched transportation networks. Calc. Var. Partial Differ. Eq. 32 (2008) 279–317. | DOI | MR | Zbl
, and ,M. Bernot, V. Caselles and J.-M. Morel, Optimal transportation networks: models and theory. Vol. 1955. Springer Science & Business Media (2009). | MR | Zbl
F. Bethuel, A counterexample to the weak density of smooth maps between manifolds in Sobolev spaces. Preprint (2014). | arXiv | MR
G. Bouchitté and P. Seppecher, Cahn and Hilliard fluid on an oscillating boundary. In Motion by mean curvature and related topics (Trento, 1992). De Gruyter, Berlin (1994) 23–42. | MR | Zbl
Transitions de phases avec un potentiel dégénéré à l’infini, application à l’équilibre de petites gouttes. C. R. Acad. Sci. Paris Sér. I Math. 323 (1996) 1103–1108. | MR | Zbl
, and ,On the equation and application to control of phases. J. Amer. Math. Soc. 16 (2003) 393–426. | DOI | MR | Zbl
and ,A. Braides, -convergence for beginners. Vol. 22 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford (2002). | MR | Zbl
A Benamou-Brenier approach to branched transport. SIAM J. Math. Anal. 43 (2011) 1023–1040. | DOI | MR | Zbl
, and ,Free energy of a nonuniform system. i. interfacial free energy. J. Chem. Phys. 28 (1958) 258–267. | DOI | Zbl
and ,G. Dal Maso, An introduction to -convergence. Vol. 8 of Progress in Nonlinear Differential Equations and their Applications. Birkhäuser Boston, Inc., Boston, MA (1993). | MR | Zbl
Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 58 (1975) 842–850. | MR | Zbl
and ,C. Dubs, Problèmes de perturbations singulières avec un potentiel dégénéré à l’infini. Ph.D. thesis, Université de Toulon et du Var (1998).
Minimum cost communication networks. Bell Syst. Tech. J. 46 (1967) 2209–2227. | DOI
,A variational model of irrigation patterns. Interfaces Free Bound. 5 (2003) 391–415. | DOI | MR | Zbl
, and ,Un esempio di -convergenza. Boll. Un. Mat. Ital. B 14 (1977) 285–299. | MR | Zbl
and ,Comparison of distances between measures. Appl. Math. Lett. 20 (2007) 427–432. | DOI | MR | Zbl
and ,A Modica–Mortola approximation for branched transport and applications. Arch. Ration. Mech. Anal. 201 (2011) 115–142. | DOI | MR | Zbl
and ,P. Pegon, Equivalence between branched transport models by Smirnov decomposition. To appear in RICAM (2017).
A Modica–Mortola approximation for branched transport. C. R. Math. Acad. Sci. Paris 348 (2010) 941–945. | DOI | MR | Zbl
,F. Santambrogio, Optimal transport for applied mathematicians. Calculus of variations, PDEs and modeling. Vol. 87 (2015). | MR
C. Villani, Topics in optimal transportation. Number 58 in Graduate Studies in Mathematics. American Mathematical Society, cop. (2003). | 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. Eq. 20 (2004) 283–299. | DOI | MR | Zbl
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