On the best constant in fractional $p$-Poincar\'e inequalities on cylindrical domains

TL;DR

This paper studies the optimal constants in fractional p-Poincaré inequalities within cylindrical domains, extending known results for p=2 to general p and analyzing eigenvalue asymptotics as domains grow unbounded.

Contribution

It provides new bounds and asymptotic analysis for fractional p-Poincaré inequalities in cylindrical domains, generalizing previous p=2 results.

Findings

Determined best constants for fractional p-Poincaré inequalities in cylindrical domains.

Analyzed asymptotic behavior of eigenvalues as domains become unbounded.

Extended known results from p=2 to general p cases.

Abstract

We investigate the best constants for the regional fractional $p$-Poincar\'e inequality and the fractional $p$-Poincar\'e inequality in cylindrical domains. For the special case $p=2$, the result was already known due to Chowdhury-Csat\'{o}-Roy-Sk [Study of fractional Poincar\'{e} inequalities on unbounded domains, Discrete Contin. Dyn. Syst., 41(6), 2021]. We addressed the asymptotic behaviour of the first eigenvalue of the nonlocal Dirichlet $p$-Laplacian eigenvalue problem when the domain is becoming unbounded in several directions.