论文标题
$α$ -MANHATTAN NORM的最小扩展
Minimal Extension for the $α$-Manhattan norm
论文作者
论文摘要
令$ \ Mathcal {q} $为$ \ m athbb {r}^2 $，$e_α=（\cosα，\sinα）$和$e_α^{\ bot} =（ - $α\ in [0,2π）$和$ ϕ：\ partial \ mathcal {q} \ to \ mathbb {r}^2 $是连续的，有限的分段线性线性映射。我们构建了一个有限的分段仿射同构$ v：\ nathcal {q} \ to \ mathbb {r}^2 $与$ \ partial \ partial \ mathcal {q} $重合$ ϕ $，以下属性：$ | | | | | \ langle dv，e__α\ rangle | dv，e_α^{\ bot} \ rangle |（\ mathcal {q}）$）与我们想要的$ \ ing | \ langle du，e_α\ rangle |（\ mathcal {q}）一样接近e_α^{\ bot} \ rangle |（\ Mathcal {q}）$）其中，aftimum在所有$ bv $ assomormorphisms $ u $ u $ forting $ u $ forting $ necter $ \ nefencal $ \ mathcal {q} $上的含义。该结果扩展了已经在[14]中以域形状证明的。
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.
