# Rigidity

Chapter
Part of the Universitext book series (UTX)

## Abstract

We noted in several places that oscillations may develop in minimizing sequences. Now we will embark on a more detailed study of these oscillations. Inspired by (but not limited to) the example on crystalline microstructure in Section , our overarching philosophy is the following: Assume that we are trying to minimize the functional
$$\mathscr {F}[u] := \int _\varOmega f(\nabla u(x)) \;\mathrm{d}x,$$
where $$f :\mathbb {R}^{m \times d} \rightarrow \mathbb {R}$$ ($$d, m \ge 2$$) is continuous, over a (Sobolev) class of functions $$u :\varOmega \rightarrow \mathbb {R}^m$$ with prescribed boundary values. Here, as usual, we assume that $$\varOmega \subset \mathbb {R}^d$$ is a bounded Lipschitz domain. We associate with $$\mathscr {F}$$ as above the pointwise differential inclusion
$$\nabla u(x) \in K := \bigl \{\, A \in \mathbb {R}^{m \times d} \ \ \mathbf : \ \ f(A) = \min f \,\bigr \}, \qquad x \in \varOmega ,$$
where $$\min f$$ denotes the pointwise minimum of f that we assume to exist in $$\mathbb {R}$$. Under a mild coercivity assumption on f we have that K is compact.