论文标题
连通性条件和边界庞加莱不平等现象
Connectivity conditions and boundary Poincaré inequalities
论文作者
论文摘要
受穆尔戈格洛（Mourgoglou）和第二名作者的最新作品的启发，以及霍夫曼（Hofmann），米特雷亚（Mitrea）和泰勒（Taylor）的早期作品，我们考虑了当地的约翰条件，harnack链条条件和弱边界庞加莱在开放式$ω\ subset \ subset \ subset \ subset \ mathbb {r}^r}^r}^{n+1} $ - $ 1 $ 1 $ 1 $ 1 $ 1 $ 1 $ 1之间的联系。首先，我们证明，如果$ω$同时满足当地的约翰条件和外部开瓶器条件，那么$ω$也可以满足Harnack链条条件（因此，是Chord-Arc域）。其次，我们表明，如果$ω$是一个$ 2 $侧的和弦 - arc域，则边界$ \ partialω$支持Heinonen-Koskela型弱$ 1 $-POINCARé不平等。 We also construct an example of a set $Ω\subset \mathbb{R}^{n+1}$ such that the boundary $\partial Ω$ is Ahlfors--David regular and supports a weak boundary $1$-Poincaré inequality but $Ω$ is not a chord-arc domain.我们的证据在特别谐波措施，统一的重新讨论和公制的庞加莱理论方面采取了重大进展。
Inspired by recent work of Mourgoglou and the second named author, and earlier work of Hofmann, Mitrea and Taylor, we consider connections between the local John condition, the Harnack chain condition and weak boundary Poincaré inequalities in open sets $Ω\subset \mathbb{R}^{n+1}$, with codimension $1$ Ahlfors--David regular boundaries. First, we prove that if $Ω$ satisfies both the local John condition and the exterior corkscrew condition, then $Ω$ also satisfies the Harnack chain condition (and hence, is a chord-arc domain). Second, we show that if $Ω$ is a $2$-sided chord-arc domain, then the boundary $\partial Ω$ supports a Heinonen--Koskela type weak $1$-Poincaré inequality. We also construct an example of a set $Ω\subset \mathbb{R}^{n+1}$ such that the boundary $\partial Ω$ is Ahlfors--David regular and supports a weak boundary $1$-Poincaré inequality but $Ω$ is not a chord-arc domain. Our proofs utilize significant advances in particularly harmonic measure, uniform rectifiability and metric Poincaré theories.
