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 2 log | 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 -normalized Dirichlet eigenfunctions of ( - Δ ) 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.