Equivalence between validity of the $p$-Poincar\'e inequality and finiteness of the strict $p$-capacitary inradius

TL;DR

This paper establishes a precise equivalence between the validity of the $p$-Poincaré inequality on a domain and the finiteness of its strict $p$-capacitary inradius, using new bounds for nonlinear Rayleigh quotients.

Contribution

It introduces a novel equivalence linking the $p$-Poincaré inequality to the strict $p$-capacitary inradius and derives new bounds for related nonlinear Rayleigh quotients.

Findings

The $p$-Poincaré inequality holds iff the strict $p$-capacitary inradius is finite.

New bounds for the infimum of nonlinear Rayleigh quotients are established.

The results provide a geometric characterization of the Poincaré inequality.

Abstract

It is shown that the $p$-Poincar\'e 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.