Small Order Asymptotics of the Dirichlet Eigenvalue Problem for the Fractional Laplacian

Abstract We study the asymptotics of Dirichlet eigenvalues and eigenfunctions fractional Laplacian $$(-\Delta )^s$$ (-Δ)s in bounded open Lipschitz sets small order limit $$s \rightarrow 0^+$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">s→0+ . While it is easy to see that all converge 1 as , we show first correction these given by logarithmic operator, i.e., singular integral operator with Fourier symbol $$2\log |\xi |$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">2log|ξ| By this generalize a result Chen third author which was restricted principal eigenvalue. Moreover, $$L^2$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">L2 -normalized corresponding k -th eigenvalue are uniformly set Laplacian. In derive spectral asymptotics, establish new uniform regularity boundary decay estimates for As byproduct, also obtain properties