Small Order Asymptotics of the Dirichlet Eigenvalue Problem for the Fractional Laplacian

Abstract

AbstractWe study the asymptotics of Dirichlet eigenvalues and eigenfunctions of the fractional Laplacian$$(-\Delta )^s$$(-Δ)sin bounded open Lipschitz sets in the small order limit$$s \rightarrow 0^+$$s→0+. While it is easy to see that all eigenvalues converge to 1 as$$s \rightarrow 0^+$$s→0+, we show that the first order correction in these asymptotics is given by the eigenvalues of the logarithmic Laplacian operator, i.e., the singular integral operator with Fourier symbol$$2\log |\xi |$$2log|ξ|. By this we generalize a result of Chen and the third author which was restricted to the principal eigenvalue. Moreover, we show that$$L^2$$L2-normalized Dirichlet eigenfunctions of$$(-\Delta )^s$$(-Δ)scorresponding to thek-th eigenvalue are uniformly bounded and converge to the set of$$L^2$$L2-normalized eigenfunctions of the logarithmic Laplacian. In order to derive these spectral asymptotics, we establish new uniform regularity and boundary decay estimates for Dirichlet eigenfunctions for the fractional Laplacian. As a byproduct, we also obtain corresponding regularity properties of eigenfunctions of the logarithmic Laplacian.