Higher Sobolev Regularity of Convex Integration Solutions in Elasticity: The Planar Geometrically Linearized Hexagonal-to-Rhombic Phase Transformation
Angkana Rüland, Christian Zillinger, Barbara Zwicknagl · Journal of Elasticity · 2019
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$u$ with higher Sobolev regularity, i.e., there exists θ0>0$\theta _{0}>0$ such that ∇u∈Wlocs,p(R2)∩L∞(R2)$ abla u \in W^{s,p}_{loc}( \mathbb{R}^{2})\cap L^{\infty }(\mathbb{R}^{2})$ for s∈(0,1)$s\in (0,1)$, p∈(1,∞)$p\in (1,\infty )$ with 0<sp<θ0$0< sp < \theta _{0}$. We also recall a construction which shows that in very specific situations with additional symmetry much better regularity properties hold.