Higher Sobolev regularity of convex integration solutions in elasticity: the planar geometrically linearized hexagonal-to-rhombic phase transformation - Rüland, Angkana; Zillinger, Christian; Zwicknagl, Barbara (Journal of elasticity, 2020)
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 $θ _{0}>0$ such that $\nabla u \in W^{s,p}_{loc}( \mathbb{R}^{2})\cap L^{∞ }(\mathbb{R}^{2})$ for $s\in (0,1)$, $p\in (1,∞ )$ with $0< sp < θ _{0}$. We also recall a construction which shows that in very specific situations with additional symmetry much better regularity properties hold.