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.
Categories: math.AP math.FA