A Modica–Mortola approximation for the Steiner Problem

Antoine Lemenant; Filippo Santambrogio

doi:10.1016/j.crma.2014.03.008

Calculus of variations

A Modica–Mortola approximation for the Steiner Problem

Presented by: Yves Meyer

Antoine Lemenant ¹; Filippo Santambrogio ²

¹ Université Paris-Diderot, Laboratoire Jacques-Louis-Lions, France
² Université Paris-Sud, Laboratoire de mathématiques d'Orsay, France

Comptes Rendus. Mathématique, Volume 352 (2014) no. 5, pp. 451-454.

Abstract
Résumé

In this note we present a way to approximate the Steiner Problem by a family of elliptic energies of Modica–Mortola type, with an additional term relying on a weighted geodesic distance which takes care of the connectedness constraint.

Dans cette note, nous présentons une méthode d'approximation du problème de Steiner par une famille de fonctionnelles de type Modica–Mortola, avec un terme additionnel basé sur une distance géodésique à poids, pour prendre en compte la contrainte de connexité.

Received: 2013-11-13
Accepted: 2014-03-11
Published online: 2014-04-02

DOI: 10.1016/j.crma.2014.03.008

Author's affiliations:

Antoine Lemenant ¹; Filippo Santambrogio ²

¹ Université Paris-Diderot, Laboratoire Jacques-Louis-Lions, France
² Université Paris-Sud, Laboratoire de mathématiques d'Orsay, France

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     author = {Antoine Lemenant and Filippo Santambrogio},
     title = {A {Modica{\textendash}Mortola} approximation for the {Steiner} {Problem}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {451--454},
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     language = {en},
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Antoine Lemenant; Filippo Santambrogio. A Modica–Mortola approximation for the Steiner Problem. Comptes Rendus. Mathématique, Volume 352 (2014) no. 5, pp. 451-454. doi : 10.1016/j.crma.2014.03.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.03.008/

References
Cited by

[1] L. Ambrosio; A. Lemenant; G. Royer-Carfagni A variational model for plastic slip and its regularization via gamma-convergence, J. Elasticity, Volume 110 (2013) no. 2, pp. 201-235

[2] L. Ambrosio; V.M. Tortorelli On the approximation of free discontinuity problems, Boll. Un. Mat. Ital. B (7), Volume 6 (1992) no. 1, pp. 105-123

[3] F. Benmansour; G. Carlier; G. Peyré; F. Santambrogio Derivatives with respect to metrics and applications: subgradient marching algorithm, Numer. Math., Volume 116 (2010) no. 3, pp. 357-381

[4] M. Bonnivard, A. Lemenant, F. Santambrogio, Approximation of length minimization problems among compact connected sets, preprint available on CVGMT.

[5] G. Carlier; C. Jimenez; F. Santambrogio Optimal transportation with traffic congestion and wardrop equilibria, SIAM J. Control Optim., Volume 47 (2008), pp. 1330-1350

[6] E.N. Gilbert; H.O. Pollak Steiner minimal trees, SIAM J. Appl. Math., Volume 16 (1968), pp. 1-29

[7] R. Karp Reducibility among combinatorial problems, Complexity of Computer Computations, Plenum Press, 1972, pp. 85-103

[8] G. Dal Maso; F. Iurlano Fracture models as Γ-limits of damage models, Comm. Pure Appl. Anal., Volume 12 (2013) no. 4, pp. 1657-1686

[9] L. Modica; S. Mortola Il limite nella Γ-convergenza di una famiglia di funzionali ellittici, Boll. Un. Mat. Ital. A (5), Volume 14 (1977) no. 3, pp. 526-529

[10] É. Oudet Approximation of partitions of least perimeter by Γ-convergence: around Kelvin's conjecture, Exp. Math., Volume 20 (2011) no. 3, pp. 260-270

[11] É. Oudet; F. Santambrogio A Modica–Mortola approximation for branched transport and applications, Arch. Ration. Mech. Anal., Volume 201 (2011) no. 1, pp. 115-142

[12] E. Paolini; E. Stepanov Existence and regularity results for the Steiner problem, Calc. Var. Partial Differential Equations, Volume 46 (2013) no. 3, pp. 837-860

[13] F. Santambrogio A Modica–Mortola approximation for branched transport, C. R. Acad. Sci. Paris, Ser. I, Volume 348 (2010) no. 15–16, pp. 941-945

[14] J.A. Sethian Level Set Methods and Fast Marching Methods, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, 1999

Cited by Sources:

^☆ This work has been partially supported by the Agence Nationale de la Recherche, through the project ANR-12-BS01-0014-01 GEOMETRYA, and by The Gaspard Monge Program for Optimization and operations research (PGMO) via the project MACRO.

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