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Connectivity
conditions and boundary Poincaré inequalities
Olli Tapiola and Xavier Tolsa |
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Vol. 17 (2024), No. 5, 1831–1870
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Abstract |
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Inspired by recent work of Mourgoglou and the second author, and earlier work of Hofmann, Mitrea and Taylor, we consider connections between the local John condition, the Harnack chain condition and weak boundary Poincaré inequalities in open sets , with codimension- Ahlfors–David regular boundaries. First, we prove that if satisfies both the local John condition and the exterior corkscrew condition, then also satisfies the Harnack chain condition (and hence is a chord-arc domain). Second, we show that if is a -sided chord-arc domain, then the boundary supports a Heinonen–Koskela-type weak -Poincaré inequality. We also construct an example of a set such that the boundary is Ahlfors–David regular and supports a weak boundary -Poincaré inequality but is not a chord-arc domain. Our proofs utilize significant advances in particularly harmonic measure, uniform rectifiability and metric Poincaré theories. |
Keywords
John-type conditions, Harnack chains, boundary Poincaré
inequalities, chord-arc domains, Sobolev spaces
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Mathematical Subject Classification
Primary: 28A75, 46E35
Secondary: 35J25
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Milestones
Received: 19 June 2022
Accepted: 21 February 2023
Published: 20 June 2024
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Authors |
| Olli Tapiola | |
| Departament de Matemàtiques Universitat Autònoma de Barcelona Barcelona Catalonia |
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| Xavier Tolsa | |
| ICREA, Departament de
Matemàtiques Universitat Autònoma de Barcelona and Centre de Recerca Matemàtica Barcelona Catalonia |
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| © 2024 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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