Upper semicontinuity of the lamination hull

Harris, Terence L.J.

doi:10.1051/cocv/2017033

Harris, Terence L.J.¹

ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 4, pp. 1503-1510.

Résumé

Let $K \subseteq ℝ^{2 \times 2}$ be a compact set, let $K^{r c}$ be its rank-one convex hull, and let $L (K)$ be its lamination convex hull. It is shown that the mapping $K \mapsto \bar{L (K)}$ is not upper semicontinuous on the diagonal matrices in $ℝ^{2 \times 2}$ , which was a problem left by Kolář. This is followed by an example of a $5$ -point set of $2 \times 2$ symmetric matrices with non-compact lamination hull. Finally, another $5$ -point set $K$ is constructed, which has $L (K)$ connected, compact and strictly smaller than $K^{r c}$ .

MR Zbl

DOI : 10.1051/cocv/2017033

Classification : 49J45, 52A30
Mots-clés : Lamination convexity, rank-one convexity

Affiliations des auteurs :

Harris, Terence L.J. ¹

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     title = {Upper semicontinuity of the lamination hull},
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     publisher = {EDP-Sciences},
     volume = {24},
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Harris, Terence L.J. Upper semicontinuity of the lamination hull. ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 4, pp. 1503-1510. doi : 10.1051/cocv/2017033. http://archive.numdam.org/articles/10.1051/cocv/2017033/

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