Explicit Construction of Piecewise Affine Mappings with Constraints

Volume 58 / 2010

Waldemar Pompe Bulletin Polish Acad. Sci. Math. 58 (2010), 209-220 MSC: 26B25, 49K21. DOI: 10.4064/ba58-3-4


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 $\det Du=1$.


  • Waldemar PompeInstitute of Mathematics
    Department of Mathematics, Informatics and Mechanics
    University of Warsaw
    Banacha 2
    02-097 Warszawa, Poland

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