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

In [Dal Maso and Toader, NoDEA 2022], the authors introduced the space $GB{V}_{\star }(A)$ to minimise a class of functionals whose study is motivated by fracture mechanics. In this paper, we extend the definition of $GB{V}_{\star }(A)$ to the vectorial case, introducing the space $GB{V}_{\star }(A;{\mathbb{R}}^k)$. We study the main properties of $GB{V}_{\star }(A;{\mathbb{R}}^k)$ and prove a lower semicontinuity result useful for minimisation purposes. With the Direct Method in mind, we adapt the arguments of [Dal Maso and Toader, NoDEA 2022] to show that minimising sequences in $GB{V}_{\star }(A;{\mathbb{R}}^k)$ can be modified to obtain a minimising sequence converging ${\mathcal{L}}^d$-a.e in $A$.