The notion of quasiconvexity was first introduced in Morrey’s seminal paper [195]. Lemma 5.6 is originally due to Morrey [196]; we follow the presentation in [33]. The results about null-Lagrangians, in particular Lemmas 5.8 and 5.10, go back to Morrey [196] and Ball [25]. The pivotal proof idea that certain combinations of derivatives might have good convergence properties even if the individual derivatives do not, is also the starting point for the theory of *compensated compactness* (see Section 8.8). A more general result on why convexity is inadmissible for realistic problems in nonlinear elasticity can be found in Section 4.8 of [64].

The convexity properties of quadratic forms have received considerable attention because they correspond to *linear* Euler–Lagrange equations. In this case, quasiconvexity and rank-one convexity are the same, see Problem 5.7. Moreover, for quadratic forms, even polyconvexity (see the next chapter) is equivalent to rank-one convexity if \(d = 2\) or \(m = 2\), but this does not hold for \(d, m \ge 3\). These results together with pointers to the literature can be found in Section 5.3.2 of [76].

The result that rank-one convex functions are locally Lipschitz continuous, Lemma 5.6, is well-known for convex functions, see, for example, Corollary 2.4 in [106] and an adapted version for rank-one convex (even separately convex) functions is in Theorem 2.31 of [76]. Our proof with a quantitative bound is from Lemma 2.2 in [33]. A more general version of this statement can be found in Lemma 2.3 of [162].

The Ball–James Rigidity Theorem 5.13 is from [30]. We will see much more general rigidity results in Chapter 8.

It is possible to prove Morrey’s Theorem 5.16 without the use of Young measures, see, for instance, Chapter 8 in [76] for such an approach. However, many of the ideas are essentially the same, they are just carried out directly without the Young measure intermediary (which obscures them somewhat). More on lower semicontinuity and Young measures can be found in the book [222].

All results in this chapter are formulated for Carathéodory integrands, but many continue to hold for \(f :\varOmega \times \mathbb {R}^N \rightarrow \mathbb {R}\) that are Borel-measurable and lower semicontinuous in the second argument, so called **normal integrands** , see [39] and [122].

An argument by Kružík [170], which was refined by Müller, shows the curious fact that for a quasiconvex \(h :\mathbb {R}^{m \times d} \rightarrow \mathbb {R}\) with \(m \ge 3\), \(d \ge 2\) the function \(A \mapsto h(A^T)\) may not be quasiconvex. The proof can be found in Section 4.7 of [203]; it is based on Šverák’s example of a rank-one convex function that is not quasiconvex (for the same dimensions as above), which we will present in Example 7.10 in Chapter 7.

For minimization problems where the integrand can take negative values one needs to look carefully at the negative part of the integrand, see Problem 5.6. If the integrand has critical negative growth, then lower semicontinuity only holds if the boundary values are fixed along a sequence or if one imposes *quasiconvexity at the boundary*, see [37] for a recent survey article discussing this topic.