Minimal extension for the $\alpha$-Manhattan norm

Abstract

Let $\partial \mathcal{Q}$ be the boundary of a convex polygon in ${\mathbb{R}}^2$, $e_{\alpha }=(\cos \alpha ,\sin \alpha )$ and $e_{\alpha }^{\perp }=(-\sin \alpha ,\cos \alpha )$ a basis of ${\mathbb{R}}^2$ for some $\alpha \in [0,2\pi )$ and $\phi :\partial \mathcal{Q}\to {\mathbb{R}}^2$ 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: $|⟨Dv,e_{\alpha }⟩|(\mathcal{Q})$ (resp., $⟨Dv,e_{\alpha }^{\perp }⟩|(\mathcal{Q})$) is as close as we want to $\inf |⟨Du,e_{\alpha }⟩|(\mathcal{Q})$ (resp., $\inf |⟨Du,e_{\alpha }^{\perp }⟩|(\mathcal{Q})$) where the infimum is meant over the class of all $BV$ homeomorphisms $u$ extending $\phi$ inside $\mathcal{Q}$. This result extends that already proven by Pratelli and the third author in [Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29 (2018), no. 3, 511–555] in the shape of the domain.