Minimal Extension for the $α$-Manhattan norm

Abstract

Let $\partial \mathcal{Q}$ be the boundary of a convex polygon in $\mathbb{R}^2$, $e_α= (\cosα, \sin α)$ and $e_α^{\bot} = (-\sinα, \cos α)$ be a basis of $\mathbb{R}^2$ for some $α\in[0,2π)$ and $ϕ:\partial\mathcal{Q} \to\mathbb{R}^2$ be a continuous, finitely piecewise linear injective map. We construct a finitely piecewise affine homeomorphism $v: \mathcal{Q} \to \mathbb{R}^2$ coinciding with $ϕ$ on $\partial \mathcal{Q}$ such that the following property holds: $|\langle Dv, e_α\rangle|(\mathcal{Q})$ (resp. $\langle Dv, e_α^{\bot}\rangle|(\mathcal{Q})$) is as close as we want to $\inf |\langle Du, e_α\rangle|(\mathcal{Q})$ (resp. $\inf |\langle Du, e_α^{\bot}\rangle|(\mathcal{Q})$) where the infimum is meant over the class of all $BV$ homeomorphisms $u$ extending $ϕ$ inside $\mathcal{Q}$. This result extends that already proven in [14] in the shape of the domain.