Calculus of Variations and Geometric Measure Theory

D. Campbell - A. Kauranen - E. Radici

Minimal Extension for the $α$-Manhattan norm

created by radici on 09 Jan 2023
modified on 05 Jun 2024


Published Paper

Inserted: 9 jan 2023
Last Updated: 5 jun 2024

Journal: Rend. Lincei Mat. Apple.
Year: 2024

ArXiv: 2212.08367 PDF


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.