On the lower bounds of $p$-modulus of families of paths and a finite connectedness
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913572459642880 |
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| author | Sevost'yanov, Evgeny Kovba, Zarina Nosal, Heorhii Ilkevych, Nataliya |
| author_facet | Sevost'yanov, Evgeny Kovba, Zarina Nosal, Heorhii Ilkevych, Nataliya |
| contents | We study the problem of the lower bounds of the modulus of families of paths of order $p,$ $p>n-1,$ and their connection with the geometry of domains containing the specified families. Among other things, we have proved an analogue of Näkki's theorem on the positivity of the $p$-module of families of paths joining a pair of continua in the given domain. The geometry of domains with a strongly accessible boundary in the sense of the $p$-modulus of families of paths was also studied. We show that domains with a $p$-strongly accessible boundary with respect to a $p$-modulus, $p>n-1,$ are are finitely connected at their boundary. The mentioned result generalizes Näkki's result, which was proved for uniform domains in the case of a conformal modulus. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_01771 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the lower bounds of $p$-modulus of families of paths and a finite connectedness Sevost'yanov, Evgeny Kovba, Zarina Nosal, Heorhii Ilkevych, Nataliya Complex Variables 31A15, 30C65 We study the problem of the lower bounds of the modulus of families of paths of order $p,$ $p>n-1,$ and their connection with the geometry of domains containing the specified families. Among other things, we have proved an analogue of Näkki's theorem on the positivity of the $p$-module of families of paths joining a pair of continua in the given domain. The geometry of domains with a strongly accessible boundary in the sense of the $p$-modulus of families of paths was also studied. We show that domains with a $p$-strongly accessible boundary with respect to a $p$-modulus, $p>n-1,$ are are finitely connected at their boundary. The mentioned result generalizes Näkki's result, which was proved for uniform domains in the case of a conformal modulus. |
| title | On the lower bounds of $p$-modulus of families of paths and a finite connectedness |
| topic | Complex Variables 31A15, 30C65 |
| url | https://arxiv.org/abs/2408.01771 |