Equivalence between validity of the \(p\)-Poincaré inequality and finiteness of the strict \(p\)-capacitary inradius

It is shown that the \(p\)-Poincaré inequality holds on an open set \(\Omega\) in \(\mathbb{R}^n\) if and only if the strict \(p\)-capacitary inradius of \(\Omega\) is finite. To that end, new upper and lower bounds for the infimum for the associated nonlinear Rayleigh quotients are derived.

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Bibliographic Details
Published in	arXiv.org
Main Author	A -K Gallagher
Format	Paper
Language	English
Published	Ithaca Cornell University Library, arXiv.org 15.08.2024
Subjects	Infimum Lower bounds
Online Access	Get full text

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Summary:	It is shown that the \(p\)-Poincaré inequality holds on an open set \(\Omega\) in \(\mathbb{R}^n\) if and only if the strict \(p\)-capacitary inradius of \(\Omega\) is finite. To that end, new upper and lower bounds for the infimum for the associated nonlinear Rayleigh quotients are derived.
ISSN:	2331-8422