Connectivity conditions and boundary Poincaré inequalities
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916299398971392 |
|---|---|
| author | Tapiola, Olli Tolsa, Xavier |
| author_facet | Tapiola, Olli Tolsa, Xavier |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_11667 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Connectivity conditions and boundary Poincaré inequalities Tapiola, Olli Tolsa, Xavier Analysis of PDEs Classical Analysis and ODEs 28A75, 46E35, 35J25 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. |
| title | Connectivity conditions and boundary Poincaré inequalities |
| topic | Analysis of PDEs Classical Analysis and ODEs 28A75, 46E35, 35J25 |
| url | https://arxiv.org/abs/2205.11667 |