Accessible parts of the boundary for domains in metric measure spaces

Ryan Gibara; Riikka Korte

doi:10.54330/afm.116365

Accessible parts of the boundary for domains in metric measure spaces

Ryan Gibara^*, Riikka Korte

^*Corresponding author for this work

University of Cincinnati

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Abstract

We prove in the setting of Q-Ahlfors regular PI-spaces the following result: if a domain has uniformly large boundary when measured with respect to the s-dimensional Hausdorff content, then its visible boundary has large t-dimensional Hausdorff content for every 0 < t < s ≤ Q − 1. The visible boundary is the set of points that can be reached by a John curve from a fixed point z0∈Ω. This generalizes recent results by Koskela–Nandi–Nicolau (from R²) and Azzam (Rⁿ). In particular, our approach shows that the phenomenon is independent of the linear structure of the space.

Translated title of the contribution	Alueen näkyvä reuna metrisissä avaruuksissa
Original language	English
Pages (from-to)	695-706
Number of pages	12
Journal	Annales Fennici Mathematici
Volume	47
Issue number	2
DOIs	https://doi.org/10.54330/afm.116365
Publication status	Published - 2022
MoE publication type	A1 Journal article-refereed

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title = "Accessible parts of the boundary for domains in metric measure spaces",

abstract = "We prove in the setting of Q-Ahlfors regular PI-spaces the following result: if a domain has uniformly large boundary when measured with respect to the s-dimensional Hausdorff content, then its visible boundary has large t-dimensional Hausdorff content for every 0 < t < s ≤ Q − 1. The visible boundary is the set of points that can be reached by a John curve from a fixed point z0∈Ω. This generalizes recent results by Koskela–Nandi–Nicolau (from R2) and Azzam (Rn). In particular, our approach shows that the phenomenon is independent of the linear structure of the space.",

keywords = "John domain, Metric measure space, Visible boundary",

author = "Ryan Gibara and Riikka Korte",

note = "Funding information: The first author was partially supported by the Fonds de recherche du Qu{\'e}bec – Nature et technologies (FRQNT). Part of the work for this project was done while the first author was visiting Aalto University; he would like to thank that institution for their kind hospitality. The second author was partially supported by Academy of Finland, project 308063. Publisher Copyright: {\textcopyright} 2022. The Finnish Mathematical Society",

year = "2022",

doi = "10.54330/afm.116365",

language = "English",

volume = "47",

pages = "695--706",

journal = "Annales Fennici Mathematici",

issn = "2737-0690",

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number = "2",

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TY - JOUR

T1 - Accessible parts of the boundary for domains in metric measure spaces

AU - Gibara, Ryan

AU - Korte, Riikka

N1 - Funding information: The first author was partially supported by the Fonds de recherche du Québec – Nature et technologies (FRQNT). Part of the work for this project was done while the first author was visiting Aalto University; he would like to thank that institution for their kind hospitality. The second author was partially supported by Academy of Finland, project 308063. Publisher Copyright: © 2022. The Finnish Mathematical Society

PY - 2022

Y1 - 2022

N2 - We prove in the setting of Q-Ahlfors regular PI-spaces the following result: if a domain has uniformly large boundary when measured with respect to the s-dimensional Hausdorff content, then its visible boundary has large t-dimensional Hausdorff content for every 0 < t < s ≤ Q − 1. The visible boundary is the set of points that can be reached by a John curve from a fixed point z0∈Ω. This generalizes recent results by Koskela–Nandi–Nicolau (from R2) and Azzam (Rn). In particular, our approach shows that the phenomenon is independent of the linear structure of the space.

AB - We prove in the setting of Q-Ahlfors regular PI-spaces the following result: if a domain has uniformly large boundary when measured with respect to the s-dimensional Hausdorff content, then its visible boundary has large t-dimensional Hausdorff content for every 0 < t < s ≤ Q − 1. The visible boundary is the set of points that can be reached by a John curve from a fixed point z0∈Ω. This generalizes recent results by Koskela–Nandi–Nicolau (from R2) and Azzam (Rn). In particular, our approach shows that the phenomenon is independent of the linear structure of the space.

KW - John domain

KW - Metric measure space

KW - Visible boundary

UR - http://www.scopus.com/inward/record.url?scp=85129700531&partnerID=8YFLogxK

UR - https://arxiv.org/abs/2103.14775

U2 - 10.54330/afm.116365

DO - 10.54330/afm.116365

M3 - Article

AN - SCOPUS:85129700531

SN - 2737-0690

VL - 47

SP - 695

EP - 706

JO - Annales Fennici Mathematici

JF - Annales Fennici Mathematici

IS - 2

ER -

Accessible parts of the boundary for domains in metric measure spaces

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Parabolic flows with variational methods

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Accessible parts of the boundary for domains in metric measure spaces

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Parabolic flows with variational methods

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