Connectivity conditions and boundary poincare inequalities
Tapiola, Olli Martti Nikolai (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Tolsa Domènech, Xavier (Universitat Autònoma de Barcelona. Departament de Matemàtiques)

Date:	2024
Abstract:	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 Poincare inequalities in open sets Ω ⊂ Rn+1, with codimension-1 Ahlfors-David regular boundaries. First, we prove that if Ω satisfies both the local John condition and the exterior corkscrew condition, then Ω also satisfies the Harnack chain condition (and hence is a chord-arc domain). Second, we show that if Ω is a 2-sided chord-arc domain, then the boundary ∂Ω supports a Heinonen-Koskela-type weak 1-Poincare inequality. We also construct an example of a set Ω ⊂Rn+1 such that the boundary ∂Ω is Ahlfors-David regular and supports a weak boundary 1-Poincare inequality but Ω is not a chord-arc domain. Our proofs utilize significant advances in particularly harmonic measure, uniform rectifiability and metric Poincare theories.
Grants:	European Commission 101018680 Generalitat de Catalunya 2021/SGR-00071 Agencia Estatal de Investigación PID2020-114167GB-I00 Agencia Estatal de Investigación CEX2020-001084-M
Rights:	Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, la comunicació pública de l'obra i la creació d'obres derivades, fins i tot amb finalitats comercials, sempre i quan es reconegui l'autoria de l'obra original.
Language:	Anglès
Document:	Article ; recerca ; Versió publicada
Subject:	John-type conditions ; Harnack chains ; Boundary Poincaré inequalities ; Chord-arc domains ; Sobolev spaces
Published in:	Analysis and PDE, Vol. 17, Num. 5 (2024) , p. 1831-1870, ISSN 1948-206X

DOI: 10.2140/apde.2024.17.1831

44 p, 1.3 MB

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