Calculus of Variations and Geometric Measure Theory

S. Conti - M. Focardi - F. Iurlano

Existence of strong minimizers for the Griffith static fracture model in dimension two

created by iurlano on 10 Nov 2016
modified by focardi on 17 Dec 2018


Published Paper

Inserted: 10 nov 2016
Last Updated: 17 dec 2018

Journal: Ann. I. H. Poincaré
Year: 2018


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, which is the object of the present paper, is a generalization of the decay estimate by De Giorgi, Carriero, and Leaci to the vectorial situation. This is based on replacing the coarea formula by a method to approximate $SBD^p$ functions with small jump set by Sobolev functions and is restricted to two dimensions. The other two ingredients are contained in companion papers and consist respectively in regularity results for vectorial elliptic problems of the elasticity type and in a method to approximate in energy $GSBD^p$ functions by $SBV^p$ ones.