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

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 \(f :\Omega \rightarrow \Omega \) that is Holder 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 \(Df^{-1}\in W^{1,q}\) 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.