Quantitative aspects of the rigidity of branching microstructures in shape memory alloys via H-measures

We quantify the rigidity of branching microstructures in shape memory alloys undergoing cubic-to-tetragonal transformations in the geometrically linearized theory by making use of Tartar’s H-measures. The main result is a $B_{1,\mathrm{\infty }}^{2\mathrm{/}3}$-estimate for the characteristic functions of twins, which heuristically suggests that the larger-scale interfaces can cluster on a set of Hausdorff-dimension $3-\frac{2}{3}$. We provide evidence indicating that the dimension is optimal. Furthermore, we get an essentially local lower bound for the blow-up behavior of the limiting energy density close to a habit plane.