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MiS Preprint
19/1998

Quasiconvexity is not invariant under transposition

Stefan Müller

Abstract

An example is given of a quasiconvex $f : M^{2 \times 3} \to \mathbb{R}$ such that the transposed function $\overline{\rm f} : M^{3 \times 2} \to \mathbb{R}$ given by $\overline{\rm f} (F) =f(F^T)$ is not quasiconvex. For $\overline{\rm f}$ one can take Sverák\'s quartic polynomial that is rank-one convex but not quasiconvex. The proof is closely related to the observation that the map $v \mapsto v^1 v^2 v^3$ is weakly continuous from $L^3(\mathbb{R}^3 ; \mathbb{R}^3)$ into distributions provided that $A(Pv) = (\partial_2 v^1,\partial_3 v^1,\partial_1 v^2, \partial_3 v^2, \partial_1 v^3, \partial_2 v^3 )$ is compact in $W^{-1,3} (\mathbb{R}^3;\mathbb{R}^6 )$.

Received:
18.06.98
Published:
18.06.98

Related publications

inJournal
2000 Repository Open Access
Stefan Müller

Quasiconvexity is not invariant under transposition

In: Proceedings of the Royal Society of Edinburgh / A, 130 (2000) 2, pp. 389-395