Upper semicontinuity of the lamination hull

TL;DR

This paper investigates the upper semicontinuity of the lamination hull mapping for sets of matrices, providing counterexamples and exploring properties of specific finite sets related to rank-one convexity.

Contribution

It demonstrates the failure of upper semicontinuity of the lamination hull mapping on certain matrix sets and constructs explicit examples illustrating these phenomena.

Findings

The mapping $K o ar{L(K)}$ is not upper semicontinuous on diagonal matrices.

Existence of a 5-point symmetric matrix set with a non-compact lamination hull.

Existence of a 5-point set with connected, compact lamination hull smaller than the rank-one convex hull.

Abstract

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