On the best constant in fractional $p$-Poincaré inequalities on cylindrical domains

Abstract

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