  In this paper we continue the study of the Griffith brittle fracture energy
minimisation under Dirichlet boundary conditions, suggested by Francfort and
Marigo in 1998. In a recent paper, 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, Focardi, Iurlano, and of
Chambolle, Conti, Iurlano.
