Geodesic
Explicit Construction of Piecewise Affine Mappings with Constraints
Waldemar Pompe1
1 Institute of Mathematics Department of Mathematics, Informatics and Mechanics University of Warsaw Banacha 2 02-097 Warszawa, Poland
Bulletin of the Polish Academy of Sciences. Mathematics, Tome 58 (2010) no. 3, pp. 209-220

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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 $\det Du=1$.
DOI : 10.4064/ba58-3-4
Keywords: construct explicitly piecewise affine mappings colon mathbb mathbb affine boundary satisfying constraint mathop div application construction short direct proofs main approximation lemmas constraints convex integration theory approach provides direct proofs avoiding approximation smooth mappings works dimensions geq after slight modification construction constraint mathop div turned det giving examples piecewise affine mappings det

Waldemar Pompe 1

1 Institute of Mathematics Department of Mathematics, Informatics and Mechanics University of Warsaw Banacha 2 02-097 Warszawa, Poland 
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Waldemar Pompe. Explicit Construction of Piecewise Affine Mappings with Constraints. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 58 (2010) no. 3, pp. 209-220. doi: 10.4064/ba58-3-4

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