Classically, one often defines the quasiconvex envelope through (7.3) and not as we have done through (7.1). In this case, (7.1) is usually called *Dacorogna’s formula* , see Section 6.3 in [76] for further references.

The construction of Lemma 7.3 and Example 7.7 are from [250], but our proof also uses some ellipticity arguments similar to those in Lemma 2.7 of [203]. A different proof of Lemma 7.3 can be found in Section 5.3.9 of [76]. We refer to [33] for some regularity properties of quasiconvex envelopes.

Further relaxation formulas can be found in Chapter 11 of the textbook [19]; historically, Dacorogna’s lecture notes [75] were also influential.

The conditions (i)–(iii) at the beginning of Section 7.4 are modeled on the concept of \(\Gamma \)-convergence (introduced by De Giorgi), see Chapter 13 for more on this topic.

The Kinderlehrer–Pedregal theorem is conceptually very important. In particular, it entails that if we could understand the class of quasiconvex functions, then we also could understand gradient Young measures and thus the asymptotic “shape” of gradients. Unfortunately, our knowledge of quasiconvex functions, and hence of gradient Young measures, is limited at present. There is some further discussion on this point throughout [222].

The truncation argument used in Zhang’s Lemma 7.18 seems to be due originally to Acerbi–Fusco [1, 3]. The book [177] makes use of this technique in regularity theory and also contains several refinements.