  Let $\Omega\subset\mathbb R^n$ be a Lipschitz domain. Given $1\leq p<k\leq n$
and any $u\in W^{2,p}(\Omega)$ belonging to the little H\"older class
$c^{1,\alpha}$, we construct a sequence $u_j$ in the same space with
$\operatorname{rank}D^2u_j<k$ almost everywhere such that $u_j\to u$ in
$C^{1,\alpha}$ and weakly in $W^{2,p}$. This result is in strong contrast with
known regularity behavior of functions in $W^{2,p}$, $p\geq k$, satisfying the
same rank inequality.
