On the Poincar\'e inequality on open sets in $\mathbb{R}^n$

TL;DR

This paper characterizes when the Poincaré inequality holds on open sets in R^n by linking it to the existence of smooth, bounded, strictly subharmonic functions and introduces a capacity-based inradius concept to describe geometric conditions.

Contribution

It establishes a new equivalence between the Poincare9 inequality, subharmonic functions, and a capacity-based inradius, providing sharp bounds for eigenvalues.

Findings

Poincare9 inequality holds iff a bounded, smooth, subharmonic function exists on the set.

Existence of such a function is equivalent to finiteness of a capacity-based inradius.

Provides a sharp upper bound for the smallest Dirichlet-Laplacian eigenvalue.

Abstract

We show that the Poincar\'{e} inequality holds on an open set $D\subset\mathbb{R}^n$ if and only if $D$ admits a smooth, bounded function whose Laplacian has a positive lower bound on $D$. Moreover, we prove that the existence of such a bounded, strictly subharmonic function on $D$ is equivalent to the finiteness of the strict inradius of $D$ measured with respect to the Newtonian capacity. We also obtain a sharp upper bound, in terms of this notion of inradius, for the smallest eigenvalue of the Dirichlet--Laplacian.