The proof of Lemma 11.1 is from the appendix of [168] (the case for Lipschitz domains can be found in Lemma B.1 of [44]). Theorem 11.2 in the extended form of Remark 11.3 is from [169]. Results in this direction already appear in [44].

The blow-up method in the proof of the Ambrosio–Dal Maso–Fonseca–Müller Theorem 11.7 was first introduced in [123], also see [48] for a systematic approach to the idea of proving lower semicontinuity and relaxation theorems for integral functionals via the auxiliary functional \(\mathscr {J}[u;U]\) from (11.9). In fact, the idea of Lemma 11.15 dates back to [86] where it was used in the context of Sobolev spaces. In the \({\mathrm {BV}}\)-context it seems to have been used for the first time in [47] and [48].

We note that the work by Fonseca & Müller [124] also considered *u*-dependent integrands. A more general approach to this problem using *liftings* can be found in [230].

The reader is pointed to Problem 11.3 and also to [284] for the construction of a non-convex quasiconvex function with linear growth and to [200] for an example of a non-convex quasiconvex function that is even positively 1-homogeneous. Theorem 8.1 of [164] shows, in a non-constructive fashion, that “many” quasiconvex functions with linear growth must exist.

The notation for recession functions is not consistent in the literature. In many works, the upper weak recession function \(f^\#\) is written as \(f^\infty \) and simply called the “recession function”. We refer to [22], in particular Section 2.5, for a more systematic approach to recession functions and their associated cones.