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Rigidity

  • Filip RindlerEmail author
Chapter
  • 3.2k Downloads
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  1.8, 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.

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK

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