Submitted Paper
Inserted: 9 jan 2023
Last Updated: 9 jan 2023
Year: 2022
Abstract:
Let $\partial \mathcal{Q}$ be the boundary of a convex polygon in
$\mathbb{R}^2$, $e_\alpha = (\cos\alpha, \sin \alpha)$ and $e_{\alpha}^{\bot} =
(-\sin\alpha , \cos \alpha)$ be a basis of $\mathbb{R}^2$ for some
$\alpha\in[0,2\pi)$ and $\phi:\partial\mathcal{Q} \to\mathbb{R}^2$ be a
continuous, finitely piecewise linear injective map. We construct a finitely
piecewise affine homeomorphism $v: \mathcal{Q} \to \mathbb{R}^2$ coinciding
with $\phi$ on $\partial \mathcal{Q}$ such that the following property holds:
$
\langle Dv, e_{\alpha}\rangle
(\mathcal{Q})$ (resp. $\langle Dv,
e_{\alpha}^{\bot}\rangle
(\mathcal{Q})$) is as close as we want to $\inf
\langle Du, e_{\alpha}\rangle
(\mathcal{Q})$ (resp. $\inf
\langle Du,
e_{\alpha}^{\bot}\rangle
(\mathcal{Q})$) where the infimum is meant over the
class of all $BV$ homeomorphisms $u$ extending $\phi$ inside $\mathcal{Q}$.
This result extends that already proven in 14 in the shape of the domain.
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