We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.
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 $W^{-1,3} (\mathbb{R}^3;\mathbb{R}^6 )$.