On the lower bounds of $p$-modulus of families of paths and a finite connectedness

Evgeny Sevost'yanov; Zarina Kovba; Heorhii Nosal; Nataliya Ilkevych

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arXiv · 2408.01771

On the lower bounds of $p$-modulus of families of paths and a finite connectedness

Evgeny Sevost'yanov·Zarina Kovba·Heorhii Nosal·Nataliya Ilkevych

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Abstract

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 N\"akki'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 N\"akki's result, which was proved for uniform domains in the case of a conformal modulus.

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BibTeX RIS

Evgeny Sevost'yanov, Zarina Kovba, Heorhii Nosal, Nataliya Ilkevych. 2024-08-03. On the lower bounds of $p$-modulus of families of paths and a finite connectedness. https://arxiv.org/abs/2408.01771

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