# Wydawnictwa / Czasopisma IMPAN / Bulletin Polish Acad. Sci. Math. / Wszystkie zeszyty

## Explicit Construction of Piecewise Affine Mappings with Constraints

### Tom 58 / 2010

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

#### Streszczenie

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 $\det Du=1$.

#### Autorzy

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

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