Small order asymptotics of the Dirichlet eigenvalue problem for the fractional Laplacian
- We study the asymptotics of Dirichlet eigenvalues and eigenfunctions of the fractional Laplacian (−Δ)s in bounded open Lipschitz sets in the small order limit s→0+. While it is easy to see that all eigenvalues converge to 1 as 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 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 L2-normalized Dirichlet eigenfunctions of (−Δ)s corresponding to the k-th eigenvalue are uniformly bounded and converge to the set of 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.
Author: | Pierre Aimé FeulefackORCiDGND, Sven JarohsORCiDGND, Tobias WethORCiDGND |
---|---|
URN: | urn:nbn:de:hebis:30:3-680967 |
DOI: | https://doi.org/10.1007/s00041-022-09908-8 |
ISSN: | 1531-5851 |
Parent Title (German): | The journal of Fourier analysis and applications |
Publisher: | Birkhäuser Boston |
Place of publication: | Cambridge, Mass. |
Document Type: | Article |
Language: | English |
Date of Publication (online): | 2022/03/01 |
Date of first Publication: | 2022/03/01 |
Publishing Institution: | Universitätsbibliothek Johann Christian Senckenberg |
Release Date: | 2022/05/18 |
Tag: | Fractional Laplacian; Logarithmic Laplacian; Small order expansion; Uniform regularity |
Volume: | 28 |
Issue: | 18 |
Page Number: | 44 |
HeBIS-PPN: | 496067583 |
Institutes: | Informatik und Mathematik / Mathematik |
Dewey Decimal Classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |
Sammlungen: | Universitätspublikationen |
Licence (German): | Creative Commons - Namensnennung 4.0 |