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

• 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

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.