In [18], the authors introduced the space of scalar-valued functions \(GBV_\star (A)\) to minimise a class of functionals whose study is motivated by fracture mechanics. In this paper, we extend the definition of \(GBV_\star (A)\) to the vectorial case, introducing the space \(GBV_\star (A;{{\mathbb {R}}}^k)\) . We study the main properties of \(GBV_\star (A;{{\mathbb {R}}}^k)\) and prove a lower semicontinuity result useful for minimisation purposes. With the Direct Method in mind, we adapt the arguments of [18] to show that minimising sequences in \(GBV_\star (A;{{\mathbb {R}}}^k)\) can be modified to obtain a minimising sequence converging \({\mathcal {L}}^d\) -a.e in A.
