Compactness and lower semicontinuity in $GSBD$

Antonin Chambolle; Vito Crismale

doi:10.4171/jems/1021

JournalsjemsVol. 23, No. 3pp. 701–719

Compactness and lower semicontinuity in $GSB D$

Antonin Chambolle
École Polytechnique, Palaiseau, France
Vito Crismale
École Polytechnique, Palaiseau, France

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Abstract

In this paper we prove a compactness and semicontinuity result in $GSB D$ for sequences with bounded Griffith energy. This generalises classical results in ( $G$ ) $SB V$ by Ambrosio [1, 2, 4] and SBD by Bellettini–Coscia–Dal Maso [9]. As a result, the static problem in Francfort–Marigo’s variational approach to crack growth [30] admits (weak) solutions.

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Antonin Chambolle, Vito Crismale, Compactness and lower semicontinuity in $GSB D$ . J. Eur. Math. Soc. 23 (2021), no. 3, pp. 701–719

DOI 10.4171/JEMS/1021