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.
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.
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Publié le :
Antoine Lemenant 1 ; Filippo Santambrogio 2
@article{CRMATH_2014__352_5_451_0, author = {Antoine Lemenant and Filippo Santambrogio}, title = {A {Modica{\textendash}Mortola} approximation for the {Steiner} {Problem}}, journal = {Comptes Rendus. Math\'ematique}, pages = {451--454}, publisher = {Elsevier}, volume = {352}, number = {5}, year = {2014}, doi = {10.1016/j.crma.2014.03.008}, language = {en}, }
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/
[1] A variational model for plastic slip and its regularization via gamma-convergence, J. Elasticity, Volume 110 (2013) no. 2, pp. 201-235
[2] On the approximation of free discontinuity problems, Boll. Un. Mat. Ital. B (7), Volume 6 (1992) no. 1, pp. 105-123
[3] 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] Optimal transportation with traffic congestion and wardrop equilibria, SIAM J. Control Optim., Volume 47 (2008), pp. 1330-1350
[6] Steiner minimal trees, SIAM J. Appl. Math., Volume 16 (1968), pp. 1-29
[7] Reducibility among combinatorial problems, Complexity of Computer Computations, Plenum Press, 1972, pp. 85-103
[8] Fracture models as Γ-limits of damage models, Comm. Pure Appl. Anal., Volume 12 (2013) no. 4, pp. 1657-1686
[9] 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] Approximation of partitions of least perimeter by Γ-convergence: around Kelvin's conjecture, Exp. Math., Volume 20 (2011) no. 3, pp. 260-270
[11] A Modica–Mortola approximation for branched transport and applications, Arch. Ration. Mech. Anal., Volume 201 (2011) no. 1, pp. 115-142
[12] Existence and regularity results for the Steiner problem, Calc. Var. Partial Differential Equations, Volume 46 (2013) no. 3, pp. 837-860
[13] A Modica–Mortola approximation for branched transport, C. R. Acad. Sci. Paris, Ser. I, Volume 348 (2010) no. 15–16, pp. 941-945
[14] Level Set Methods and Fast Marching Methods, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, 1999
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☆ 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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