While solving a problem in calculus of variations, I came to the following question:
Let $A,B$ be two real $2 \times 2$ matrices, $\det A=0, \det B>0$, and suppose that $\operatorname{rank}(A-B)=1$. Let $D \subseteq \mathbb{R}^2$ be the closed unit disk.
Let $u:D \to \mathbb{R}^2$ be a Lipschitz map satisfying $\nabla u \in \{A,B\}$ a.e. in $D$, and that $ u$ is not affine (so $\nabla u$ equals $A$ on a set of positive measure).
Question: Let $1 \le p < 2$. Do there exist Lipschitz maps $u_k:D \to \mathbb{R}^2$ with $\det(\nabla u_k)>0$ everywhere, such that $u_k$ converges in $ \in W^{1,p}$ to $u$?