Sobolev homeomorphisms with gradients of low rank via laminates

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

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