Higher Sobolev Regularity of Convex Integration Solutions in Elasticity

Abstract

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 $\nabla u\in W_{loc}^{s,p}({\mathbb{R}}^2)\cap L^{\infty }({\mathbb{R}}^2)$ 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.

69 pages, 27 figures

References in corpus (1)

• High dimensionality and h-principle in PDE

Cited by in corpus (1)

• Higher Sobolev Regularity of Convex Integration Solutions in Elasticity: The Dirichlet Problem with Affine Data in $\text{int}(K^{lc})$