Existence theorem for the Griffith static fracture model in dimension two
Sergio Conti, Matteo Focardi, Flaviana Iurlano · arXiv (Cornell University) · 2016
We consider the Griffith fracture model in two spatial dimensions, and prove existence of strong minimizers, with closed jump set and continuously differentiable deformation fields. One key ingredient is a generalization of the decay estimate by De Giorgi, Carriero, and Leaci to the vectorial situation, based on replacing the coarea formula by a method to approximate $SBD^p$ functions with small jump set by Sobolev functions, which is restricted to two dimensions. The second ingredient is given by regularity results for vectorial elliptic problems of the elasticity type. The third is a method to approximate in energy $GSBD^p$ functions by $SBD^p$ ones.