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Comptes Rendus Mathématique
Volume 354, n° 11
pages 1055-1059 (novembre 2016)
Doi : 10.1016/j.crma.2016.09.003
Received : 4 Mars 2016 ;  accepted : 12 September 2016
Existence of minimizers for the 2d stationary Griffith fracture model
Existence de déformations minimisant le modèle de Griffith des fractures 2d -stationnaires
 

Sergio Conti a , Matteo Focardi b , Flaviana Iurlano a
a Institut für Angewandte Mathematik, Universität Bonn, 53115 Bonn, Germany 
b DiMaI, Università di Firenze, 50134 Firenze, Italy 

Abstract

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   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.

The full text of this article is available in PDF format.
Résumé

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  , 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.

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