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

Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ be a bounded open set. Given $1\leq m_1,m_2\leq n-2$,weconstructahomeomorphism$f :\Omega\to \Omega$ that is H"older continuous, $f$ is the identity on $\mathrm{\partial }\mathrm{\Omega }$, the derivative $Df$ has rank $m_1$ a.e.\ in $\mathrm{\Omega }$, the derivative $D{f}^{-1}$ of the inverse has rank $m_2$ a.e.\ in $\mathrm{\Omega }$, $Df\in W^{1,p}$ and $Df^{-1}\in W^{1,q}$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.