On Faber-Krahn inequality for Dirichlet Fractional-Logarithmic Laplacian

Abstract

We study the Faber--Krahn problem for the fractional--logarithmic Laplacian $(-\Delta {)}^{s+\ln }$, $0<s<1$, with Fourier symbol $|\xi {|}^{2s}\ln |\xi {|}^2$. We prove a small-volume and large-volume dichotomy. For sufficiently small volume, balls minimize the first Dirichlet eigenvalue, with rigidity up to translation.

arXiv comment21