OSCILLATORY PHENOMENA FOR HIGHER-ORDER FRACTIONAL LAPLACIANS

Abstract

We collect some peculiarities of higher-order fractional Laplacians $(-\mathrm{\Delta }{)}^s$, , with special attention to the range $s\in (1,2)$, which show their oscillatory nature. These include the failure of the polarization and Pólya–Szegő inequalities and the explicit example of a domain with sign-changing first eigenfunction. In spite of these fluctuating behaviours, we prove how the Faber–Krahn inequality still holds for any in dimension one.