Rigidity of Euclidean product structure: breakdown for low Sobolev exponents

We develop a general toolbox to study $W^{1,p}$ solutions of differential inclusions $\mathrm{\nabla }u\in K$ for unbounded sets $K$. A key notion is the concept that a subset $K$ of the space ${\mathbb{R}}^{d\times m}$ of $d\times m$ matrices can be reduced to another set $K^{\mathrm{\prime }}$. We then use this framework to show that the product rigidity for Sobolev maps fails for $p<2$, and also apply our toolbox to simplify several examples from the literature.