Abstract
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 rightarrow 0^+. While it is easy to see that all eigenvalues converge to 1 as s rightarrow 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 2log |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 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.
Highlights
The fractional Laplacian has received by far the most attention, see e.g. [2–8,25,29] and the references therein
The present paper is concerned with the small order asymptotics s → 0+ of the Dirichlet eigenvalue problem
In the following Proposition we collect the known properties on the eigenvalues and eigenfunctions of the fractional Laplacian and the logarithmic Laplacian, see e.g. [4, Proposition 3.1] and the references in there for the fractional Laplacian and [9, Theorem 3.4]) for the logarithmic Laplacian
Summary
Fueled by various applications and important links to stochastic processes and partial differential equations, the interest in nonlocal operators and associated Dirichlet problems has been growing rapidly in recent years. For general bounded open sets with Lipschitz boundary, the only available result regarding these asymptotics is contained in [9], where Chen and the third author introduced the Dirichlet problem for the logarithmic Laplacian operator L to give a more detailed description of the first eigenvalue λ1,s and the corresponding eigenfunction φ1,s as s → 0+. As a consequence of (1.9), L is an operator of logarithmic order, and it belongs to a class of weakly singular integral operators having an intrinsic scaling property Operators of this type have been studied e.g. in [11,16–19,21–23,28], while the most attention has been given to Lévy generators of geometric stable processes.
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