Relaxation of variational problems in the space of functions with bounded $\mathcal{B}$-variation: interaction with measures and lack of concentration phenomena

We prove an integral representation result for variational functionals in the space $B{V}^{\mathcal{B}}$ of functions with bounded $\mathcal{B}$-variation where $\mathcal{B}$ denotes a $k$-th order, $\mathbb{C}$-elliptic, linear homogeneous differential operator. This result has been used as a key tool to get an explicit representation of relaxed energies with linear growth which lead to limiting generic measures. According to the space dimension and the order of the operator, concentration phenomena appear and an explicit interaction is featured. These results are complemented also with Sobolev-type counterparts. As a further application, a lower semicontinuity result in the space of fields with $p(\cdot )$-bounded $\mathcal{B}$-variation has also been obtained.