Compactness and Lower-Semicontinuity in GSBD

Antonin Chambolle (1) , Vito Crismale (1)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

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 [1, 2, 3] and SBD by Bellettini-Coscia-Dal Maso [9]. As a result, the static problem in Francfort-Marigo's variational approach to crack growth [27] admits (weak) solutions. Moreover, we obtain a compactness property for minimisers of suitable Ambrosio-Tortorelli's type energies [6], which have been shown to Γ-converge to Griffith energy in [16].

Domaines: Mathématiques [math]; Analyse fonctionnelle [math.FA]

Antonin Chambolle, Vito Crismale. Compactness and Lower-Semicontinuity in GSBD. Journal of the European Mathematical Society, 2020, 23 (3), pp.701--719. ⟨10.4171/jems/1021⟩. ⟨hal-01704866⟩