A new space of generalised vector-valued functions of bounded variation

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 $GBV star(A;R^k)$ 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 $GBV_star(A;R^k)$ can be modified to obtain a minimising sequence converging $\mathcal L^d$-a.e in $A$.