A geometric characterization of planar Sobolev extension domains
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2015
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866914967178969088 |
|---|---|
| author | Koskela, Pekka Rajala, Tapio Zhang, Yi Ru-Ya |
| author_facet | Koskela, Pekka Rajala, Tapio Zhang, Yi Ru-Ya |
| contents | We characterize bounded simply-connected planar $W^{1,p}$-extension domains for $1 < p <2$ as those bounded domains $Ω\subset \mathbb R^2$ for which any two points $z_1,z_2 \in \mathbb R^2 \setminus Ω$ can be connected with a curve $γ\subset \mathbb R^2 \setminus Ω$ satisfying $$\int_γ dist(z,\partial Ω)^{1-p}\, dz \lesssim |z_1-z_2|^{2-p}.$$ Combined with known results, we obtain the following duality result: a Jordan domain $Ω\subset \mathbb R^2$ is a $W^{1,p}$-extension domain, $1 < p < \infty$, if and only if the complementary domain $\mathbb R^2 \setminus \barΩ$ is a $W^{1,p/(p-1)}$-extension domain. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1502_04139 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | A geometric characterization of planar Sobolev extension domains Koskela, Pekka Rajala, Tapio Zhang, Yi Ru-Ya Classical Analysis and ODEs Functional Analysis 46E35 We characterize bounded simply-connected planar $W^{1,p}$-extension domains for $1 < p <2$ as those bounded domains $Ω\subset \mathbb R^2$ for which any two points $z_1,z_2 \in \mathbb R^2 \setminus Ω$ can be connected with a curve $γ\subset \mathbb R^2 \setminus Ω$ satisfying $$\int_γ dist(z,\partial Ω)^{1-p}\, dz \lesssim |z_1-z_2|^{2-p}.$$ Combined with known results, we obtain the following duality result: a Jordan domain $Ω\subset \mathbb R^2$ is a $W^{1,p}$-extension domain, $1 < p < \infty$, if and only if the complementary domain $\mathbb R^2 \setminus \barΩ$ is a $W^{1,p/(p-1)}$-extension domain. |
| title | A geometric characterization of planar Sobolev extension domains |
| topic | Classical Analysis and ODEs Functional Analysis 46E35 |
| url | https://arxiv.org/abs/1502.04139 |