Small order asymptotics of the Dirichlet eigenvalue problem for the fractional Laplacian

TL;DR

This paper investigates the asymptotic behavior of Dirichlet eigenvalues and eigenfunctions of the fractional Laplacian as the order parameter approaches zero, revealing the first-order correction involves the logarithmic Laplacian.

Contribution

It extends previous results by deriving first-order asymptotics for all eigenvalues and eigenfunctions of the fractional Laplacian using the logarithmic Laplacian operator.

Findings

Eigenvalues converge to 1 as s→0+

First-order correction involves the logarithmic Laplacian eigenvalues

Eigenfunctions are uniformly bounded and converge to those of the logarithmic Laplacian

Abstract

In this article, we study the asymptotics of Dirichlet eigenvalues and eigenfunctions of the fractional Laplacian $(-\Delta)^s$ in bounded open Lipschitz sets in the small order limit $s \to 0^+$. While it is easy to see that all eigenvalues converge to $1$ as $s \to 0^+$, 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 symbol $2\log|\xi|$. By this we generalize a result of Chen and the third author which was restricted to the principal eigenvalue. Moreover, we show that $L^2$-normalized Dirichlet eigenfunctions of $(-\Delta)^s$ corresponding to the $k$-th eigenvalue are uniformly bounded and converge to the set of $L^2$-normalized eigenvalues of the logarithmic Laplacian. In order to derive these spectral asymptotics, we need to 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.