Capacitary inradius and Poincar\'e-Sobolev inequalities

TL;DR

This paper establishes two-sided estimates for the sharp $L^p$ Poincaré constant of open sets using a capacitary inradius, extending previous results to all $p$ including extremal cases, and discusses related Poincaré-Sobolev embeddings.

Contribution

It generalizes the estimate of Poincaré constants to all $p$, including extremal cases, using capacitary inradius, and extends the analysis to Poincaré-Sobolev inequalities.

Findings

Two-sided estimates for $L^p$ Poincaré constants in terms of capacitary inradius.

Extension of Maz'ya and Shubin's result to all $p$, including $p=1$ and $p=N$.

Discussion of superconformal case and examples.

Abstract

We prove a two-sided estimate on the sharp $L^p$ Poincar\'e constant of a general open set, in terms of a capacitary variant of its inradius. This extends a result by Maz'ya and Shubin, originally devised for the case $p=2$, in the subconformal regime. We cover the whole range of $p$, by allowing in particular the extremal cases $p=1$ (Cheeger's constant) and $p=N$ (conformal case), as well. We also discuss the more general case of the sharp Poincar\'e-Sobolev embedding constants and get an analogous result. Finally, we present a brief discussion on the superconformal case, as well as some examples and counter-examples.