Bi-Sobolev homeomorphisms $f$ with $Df$ and $D{f}^{-1}$ of low rank using laminates

Abstract

Let $\Omega \subset {\mathbb{R}}^n$ be a bounded open set. Given $1\le m_1,m_2\le n-2$, we construct a homeomorphism $f:\Omega \to \Omega$ that is Hölder continuous, $f$ is the identity on $\partial \Omega$, the derivative $Df$ has rank $m_1$ a.e.\ in $\Omega$, the derivative $D{f}^{-1}$ of the inverse has rank $m_2$ a.e.\ in $\Omega$, $Df\in W^{1,p}$ and $D{f}^{-1}\in W^{1,q}$ for $p<\min \{m_1+1,n-m_2\}$, $q<\min \{m_2+1,n-m_1\}$. The proof is based on convex integration and laminates. We also show that the integrability of the function and the inverse is sharp.