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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
1/1999

On Conditions for Polyconvexity

Jan Kristensen

Abstract

We give an example of a smooth function $f:\mathbb{R}^{2 \times 2} \to \mathbb{R}$, which is not polyconvex and that has the property that its restriction to any ball $B \subset \mathbb{R}^{2 \times 2}$ of radius one can be extended to a smooth polyconvex function $f_B:\mathbb{R}^{2\times 2} \to \mathbb{R} $. In particular, it implies that there exists no 'local condition' which is necessary and sufficient for polyconvexity of functions $g:\mathbb{R}^{n\times m} \to \mathbb{R} $, where n, $m \geq 2$. We also briefly discuss connections with quasiconvexity.

Received:
24.01.99
Published:
24.01.99
MSC Codes:
49J10, 49J45
Keywords:
polyconvexity, quasiconvexity, rank-1 convexity

Related publications

inJournal
2000 Repository Open Access
Jan Kristensen

On conditions for polyconvexity

In: Proceedings of the American Mathematical Society, 128 (2000) 6, pp. 1793-1797