Mathematical analysis/Partial differential equations
Existence of minimizers for the 2d stationary Griffith fracture model
Comptes Rendus. Mathématique, Volume 354 (2016) no. 11, pp. 1055-1059.

We consider the variational formulation of the Griffith fracture model in two spatial dimensions and prove the existence of strong minimizers, that is deformation fields that are continuously differentiable outside a closed jump set and that minimize the relevant energy. To this aim, we show that minimizers of the weak formulation of the problem, set in the function space GSBD2 and whose existence is well known, are actually strong minimizers following the approach developed by De Giorgi, Carriero, and Leaci in the corresponding scalar setting of the Mumford–Shah problem.

Nous considérons la formulation variationnelle du modèle de fracture de Griffith en dimension spatiale 2. Nous montrons l'existence de champs de déformation continûment différentiables hors d'un ensemble fermé de sauts, minimisant l'énergie relevante. Pour ce faire, nous montrons que les déformations minimisant la formulation faible du problème, dont l'existence est bien connue, placés dans l'espace des fonctions GSBD2, minimisent de fait la formulation forte. Nous suivons l'approche développée par De Giorgi, Carriero et Leach dans le cadre scalaire correspondant du problème de Mumford–Shah.

Published online:
DOI: 10.1016/j.crma.2016.09.003
Conti, Sergio 1; Focardi, Matteo 2; Iurlano, Flaviana 1

1 Institut für Angewandte Mathematik, Universität Bonn, 53115 Bonn, Germany
2 DiMaI, Università di Firenze, 50134 Firenze, Italy
     author = {Conti, Sergio and Focardi, Matteo and Iurlano, Flaviana},
     title = {Existence of minimizers for the 2d stationary {Griffith} fracture model},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1055--1059},
     publisher = {Elsevier},
     volume = {354},
     number = {11},
     year = {2016},
     doi = {10.1016/j.crma.2016.09.003},
     language = {en},
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Conti, Sergio; Focardi, Matteo; Iurlano, Flaviana. Existence of minimizers for the 2d stationary Griffith fracture model. Comptes Rendus. Mathématique, Volume 354 (2016) no. 11, pp. 1055-1059. doi : 10.1016/j.crma.2016.09.003.

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