TY  - JOUR
A1  - Feulefack, Pierre Aimé
A1  - Jarohs, Sven
A1  - Weth, Tobias
T1  - Small order asymptotics of the Dirichlet eigenvalue problem for the fractional Laplacian
T2  - The journal of Fourier analysis and applications
N2  - 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.
KW  - Fractional Laplacian
KW  - Small order expansion
KW  - Logarithmic Laplacian
KW  - Uniform regularity
Y1  - 2022
UR  - http://publikationen.ub.uni-frankfurt.de/frontdoor/index/index/docId/68096
UR  - https://nbn-resolving.org/urn:nbn:de:hebis:30:3-680967
SN  - 1531-5851
VL  - 28
IS  - 18
PB  - Birkhäuser Boston
CY  - Cambridge, Mass.
ER  -