Sobolev homeomorphisms with gradients of low rank via laminates

Abstract

Let $\Omega \subset {\mathbb{R}}^n$ be a bounded open set. Given $2\le m\le n$, we construct a convex function $\varphi :\Omega \to \mathbb{R}$ whose gradient $f=\nabla \varphi$ is a Hölder continuous homeomorphism, $f$ is the identity on $\partial \Omega$, the derivative $Df$ has rank $m-1$ a.e.\ in $\Omega$ and $Df$ is in the weak $L^m$ space $L^{m,w}$. The proof is based on convex integration and staircase laminates.