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Abstract

We construct explicitly piecewise affine mappings $u\colon\mathbb R^n\to\mathbb R^n$ with affine boundary data satisfying the constraint $\mathop{\rm div} u=0$. As an application of the construction we give short and direct proofs of the main approximation lemmas with constraints in convex integration theory. Our approach provides direct proofs avoiding approximation by smooth mappings and works in all dimensions $n\geq2$. After a slight modification of our construction, the constraint $\mathop{\rm div} u=0$ can be turned into $\det Du=1$, giving new examples of piecewise affine mappings $u$ with $\det Du=1$.