Upper semicontinuity of the lamination hull

Harris, Terence L.J.¹ 

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 $K^{rc}$ be its rank-one convex hull, and let $L\left(K\right)$ be its lamination convex hull. It is shown that the mapping $K\mapsto \overline{L\left(K\right)}$ 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 $2\hspace{0.17em}\times 2$ symmetric matrices with non-compact lamination hull. Finally, another $5$-point set $K$ is constructed, which has $L\hspace{0.17em}\left(K\right)$ connected, compact and strictly smaller than $K^{rc}$.

Détail

DOI : 10.1051/cocv/2017033

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