The origin of the rigidity theory as presented in this chapter lies in the Murat–Tartar Div-Curl Lemma

8.32, first published in [209] (but established four years before in 1974) and the Ball–James Rigidity Theorem

8.1 from [30]. The latter can further be traced back to

**Hadamard’s jump condition** : For a matrix-valued function

\(V :\mathbb {R}^d \rightarrow \mathbb {R}^{m \times d}\) of the form

$$ V(x) = {\left\{ \begin{array}{ll} A &{} \text {if}~x \cdot n \le 0, \\ B &{} \text {if}~x \cdot n > 0, \end{array}\right. } $$

where

\(A, B \in \mathbb {R}^{m \times d}\) and

\(n \in \mathbb {S}^{d-1}\), to be the gradient of a function

\(u :\mathbb {R}^d \rightarrow \mathbb {R}^m\), it is necessary and sufficient that

$$ A - B = a \otimes n \qquad \text {for some}~a \in \mathbb {R}^m. $$

Many related rigidity and compensated compactness theorems have been proved since, of which we could only present a selection.

The term “rigidity” itself is unfortunately used in different ways by different authors. In fact, *any* kind of restriction on the shape of a map is sometimes called “rigidity”. Here, however, we reserve this term for the conclusion that a map is affine. Our definitions of rigidity for exact and for approximate solutions follow most closely those in Kirchheim’s influential lecture notes [160]. In particular, we require linear boundary values along approximate solutions, but no boundary condition for exact solutions. This is explained as follows: Many differential inclusions with a discrete set *K* are trivially rigid when we impose affine boundary conditions, even if there are rank-one connections in *K*. On the other hand, rigidity for approximate solutions is most interesting when imposing linear (or affine) boundary conditions, see the discussion in Section 8.3.

Theorem 8.3 is originally contained in a more general result of Tartar [268]. It is called the “span restriction” in [43]. A special case of Theorem 8.5 (i) for \(d=m=2\) was shown in Lemma 1.4 of [90], also see the proof of Theorem 3.95 in [15]. The “blow-up technique” mentioned in connection with the polar inclusion (8.10) is systematically explained in great detail in [123, 124]. Theorem 8.11 was first established in full in [249], which was never published. The rigidity for exact solutions was known before, namely through more general results in [154] and an unpublished manuscript by Zhang. The presented proof of rigidity for exact solutions, however, is due to Kirchheim and reproduced in [203]. Our proof of approximate rigidity follows an idea from [251] and also uses Lemma 8.13, which is from [253]. Another proof is in [17] based on the theory of quasiregular mappings (which also mentions an unpublished similar argument by Ball and James).

Constructions similar to the fundamental \(T_4\)-configuration were probably first employed by Scheffer [239], but its importance in the present context was only realized after Tartar’s work, see [271], which refers to his work of 1983. Similar examples to Tartar’s were found in [21, 43, 215] (in particular, [43] adapted Tartar’s original example to the present version with diagonal matrices).

The Young measure approach of Section 8.3 and the general compensated compactness philosophy discussed at the beginning of Section 8.8 is again mostly due to Tartar [267, 268, 270, 271]. This theory has also proved to be very fruitful in the study of hyperbolic conservation laws, see Chapter XVI in [81] for an overview and many references to the vast literature. Some recent investigations into various notions of “incompatibility” between several sets \(K_1,\ldots , K_n\), which generalizes our notions of rigidity, can be found in [32] and the references cited therein.

The first part of Theorem 8.20 is (a version of) the classical Liouville theorem (also see Problem 8.2 (ii)); the extension to Sobolev functions as well as part (ii) is the work of Reshetnyak [225]. Our proof is due to Kinderlehrer [156].

The rigidity for the two-well problem in two dimensions and its extension to the *N*-well problem are from [253, 254]. More general *N*-well problems for \(N \ge 3\) in three dimensions were investigated by Kirchheim [159, 160].

A direct proof of the div-curl lemma using elliptic regularity theory can be found in Theorem 16.2.1 of [81]. In the context of such compensated compactness problems, extensions of Young measure theory that allow one to pass to the limit in quadratic expressions have been developed by Tartar [269] under the name “H-measures” and, independently, by Gérard [130], who called them “micro-local defect measures”, cf. the survey articles [126, 272].