Calculus of Variations pp 135-151 | Cite as

# Polyconvexity

Chapter

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## Abstract

At the beginning of the previous chapter we saw that convexity cannot hold concurrently with frame-indifference (and a mild non-degeneracy condition). Thus, we were led to consider quasiconvex integrands. However, while quasiconvexity is of tremendous importance in the theory of the calculus of variations, our Lower Semicontinuity Theorem 5.16 has one major drawback: we needed to require the

*p*-growth bound$$ |f(x, A)| \le M(1+|A|^p), \qquad (x, A) \in \varOmega \times \mathbb {R}^{m \times d}, $$

## Keywords

Quasiconvex Integrands Lower Semicontinuity Theorem Nonlinear Elasticity Theory Bounded Lipschitz Domain Polyconvex Integrands
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