On fractional Poincaré inequality for unbounded domains with finite ball conditions: Counter example. (March 2023) - Chowdhury, Indranil; Roy, Prosenjit
Abstract: In this paper we investigate the fractional Poincaré inequality on unbounded domains. In the local case, Mancini and Sandeep (2010) showed that in the class of simply connected domains, Poincaré inequality holds if and only if the domain satisfies finite ball condition. We prove that such a result cannot be true in the 'nonlocal/fractional' setting even if finite ball condition is replaced by a related stronger condition. We further provide some sufficient criterions on domains for fractional Poincaré inequality to hold. In the end, asymptotic behaviour of all eigenvalues of fractional Dirichlet problems on long cylindrical domains is addressed.