Existence of strong solutions to the Dirichlet problem for the Griffith energy

In this paper we continue the study of the Griffith brittle fracture energy minimisation under Dirichlet boundary conditions, suggested by Francfort and Marigo (J Mech Phys Solids 46:1319-1342, 1998). In a recent paper (Chambolle and Crismale in J Eur Math Soc (JEMS), 2018) we proved the existence of weak minimisers of the problem. Now we show that these minimisers are indeed strong solutions, namely their jump set is closed and they are smooth away from the jump set and continuous up to the Dirichlet boundary. This is obtained by extending up to the boundary the recent regularity results of Conti et al. (Ann Inst H Poincare Anal Non Lineaire 36:455-474, 2019) and Chambolle et al. (J Math Pures Appl, 2019. 10.1016/j.matpur.2019.02.001).