Rigidity of Euclidean product structure: breakdown for low Sobolev exponents

We develop a general toolbox to study W1,p solutions of differential inclusions ∇ u ∈ K for unbounded sets K. A key notion is the concept that a subset K of the space Rd × m of d × m matrices can be reduced to another set $K'$. 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.