Oscillatory Phenomena for Higher-Order Fractional Laplacians

TL;DR

This paper explores the oscillatory behaviors of higher-order fractional Laplacians, revealing their unique properties and demonstrating that classical inequalities like Faber-Krahn still hold in one dimension despite these peculiarities.

Contribution

It provides new insights into the oscillatory nature of fractional Laplacians for s>1 and proves the Faber-Krahn inequality in one dimension for these operators.

Findings

Fractional Laplacians s>1 exhibit oscillatory behavior.

Polarization and Pólya-Szegő inequalities fail for these operators.

Faber-Krahn inequality holds in one dimension despite oscillations.

Abstract

We collect some peculiarities of higher-order fractional Laplacians $(-\Delta)^s$, $s>1$, with special attention to the range $s\in(1,2)$, which show their oscillatory nature. These include the failure of the polarization and P\'olya-Szeg\"o 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 $s>1$ in dimension one.