Study of fractional Poincar\'e inequalities on unbounded domains

TL;DR

This paper investigates fractional Poincaré inequalities on unbounded domains, establishing existence, non-existence, and optimal constants, thus advancing understanding of nonlocal inequalities in unbounded geometries.

Contribution

It provides new existence and non-existence results for fractional Poincaré inequalities on unbounded domains and characterizes the best constants for strip-like domains.

Findings

Established conditions for existence and non-existence of inequalities.

Characterized optimal constants for strip domains.

Extended understanding of fractional inequalities in unbounded settings.

Abstract

The central aim of this paper is to study (regional) fractional Poincar\'e type inequalities on unbounded domains satisfying the finite ball condition. Both existence and non existence type results are established depending on various conditions on domains and on the range of $s \in (0,1)$. The best constant in both regional fractional and fractional Poincar\'e inequality is characterized for strip like domains $(\omega \times \mathbb{R}^{n-1})$, and the results obtained in this direction are analogous to those of the local case. This settles one of the natural questions raised by K. Yeressian in [\textit{Asymptotic behavior of elliptic nonlocal equations set in cylinders, Asymptot. Anal. 89, (2014), no 1-2}].