Characterizations of generalized John domains in $\mathbb{R}^n$ via metric duality
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2017
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| _version_ | 1866929538780364800 |
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| author | Goldstein, Pawel Grochulska, Zofia Guo, Chang-Yu Koskela, Pekka Nandi, Debanjan |
| author_facet | Goldstein, Pawel Grochulska, Zofia Guo, Chang-Yu Koskela, Pekka Nandi, Debanjan |
| contents | In this paper, we extend the characterization of John disks obtained by Näkki and Väisälä [Exp. Math. 1991] to generalized John domains in higher dimensions under mild assumptions. The main ingredient in this characterization is to use the higher dimensional analogues of the local linear connectivity (LLC) and homological bounded turning properties introduced by Väisälä in his study of metric duality theory [Math. Scan. 1997].
Somewhat surprisingly, we constructed a uniform domain in $\R^3$, which is topologically simple, such that the complementary domain fails to be homotopically $1$-bounded turning. In particular, this shows that a similar characterization of generalized John domains in terms of higher dimensional homotopic bounded turning does not hold in dimension three. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1710_02050 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Characterizations of generalized John domains in $\mathbb{R}^n$ via metric duality Goldstein, Pawel Grochulska, Zofia Guo, Chang-Yu Koskela, Pekka Nandi, Debanjan General Topology Algebraic Topology Classical Analysis and ODEs 57N65, 55M05 In this paper, we extend the characterization of John disks obtained by Näkki and Väisälä [Exp. Math. 1991] to generalized John domains in higher dimensions under mild assumptions. The main ingredient in this characterization is to use the higher dimensional analogues of the local linear connectivity (LLC) and homological bounded turning properties introduced by Väisälä in his study of metric duality theory [Math. Scan. 1997]. Somewhat surprisingly, we constructed a uniform domain in $\R^3$, which is topologically simple, such that the complementary domain fails to be homotopically $1$-bounded turning. In particular, this shows that a similar characterization of generalized John domains in terms of higher dimensional homotopic bounded turning does not hold in dimension three. |
| title | Characterizations of generalized John domains in $\mathbb{R}^n$ via metric duality |
| topic | General Topology Algebraic Topology Classical Analysis and ODEs 57N65, 55M05 |
| url | https://arxiv.org/abs/1710.02050 |