Sobolev homeomorphisms with gradients of low rank via laminates

Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ be a bounded open set. Given $2\leq m\leq n$,weconstructaconvexfunction$\phi :\Omega\to $\mathbb{R}$$whosegradient$f= \nabla \phi$isaH\ddot{o}ldercontinuoushomeomorphism,$f$istheidentityon$\partial \Omega$,thederivative$D f$hasrank$m-1$a\mathrm{.}e\mathrm{.}in$\Omega$and$D f$isintheweak$L^{m}$space$L^{m,w}$. The proof is based on convex integration and staircase laminates.