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.
Categories: math.AP