Calculus of Variations and Geometric Measure Theory

A. Chambolle - V. Crismale

Compactness and lower semicontinuity in $GSBD$

created by crismale on 08 Feb 2018
modified on 30 Jan 2021


Published Paper

Inserted: 8 feb 2018
Last Updated: 30 jan 2021

Journal: J. Eur. Math. Soc. (JEMS)
Year: 2021
Doi: 10.4171/JEMS/1021

ArXiv: 1802.03302 PDF

Published online


In this paper, we prove a compactness and semicontinuity result in $GSBD$ for sequences with bounded Griffith energy. This generalises classical results in $(G)SBV$ by Ambrosio and $SBD$ by Bellettini-Coscia-Dal Maso. As a result, the static problem in Francfort-Marigo's variational approach to crack growth admits (weak) solutions. Moreover, we obtain a compactness property for minimisers of suitable Ambrosio-Tortorelli's type energies, for which we have recently shown the $\Gamma$-convergence to Griffith energy.