  In this article we discuss quantitative properties of convex integration
solutions arising in problems modeling shape-memory materials. For a
two-dimensional, geometrically linearized model case, the hexagonal-to-rhombic
phase transformation, we prove the existence of convex integration solutions
$u$ with higher Sobolev regularity, i.e. there exists $\theta_0>0$ such that
$\nabla u \in W^{s,p}_{loc}(\mathbb{R}^2)\cap L^{\infty}(\mathbb{R}^2)$ for
$s\in(0,1)$, $p\in(1,\infty)$ with $0<sp < \theta_0$. We also recall a
construction, which shows that in situations with additional symmetry much
better regularity properties hold.
