Small Order Asymptotics of the Dirichlet Eigenvalue Problem for the Fractional Laplacian
Pierre Aime Feulefack, Sven Jarohs, Tobias Weth
Abstract
Abstract We study the asymptotics of Dirichlet eigenvalues and eigenfunctions of the fractional Laplacian $$(-\Delta )^s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mo>-</mml:mo><mml:mi>Δ</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>s</mml:mi></mml:msup></mml:math> in bounded open Lipschitz sets in the small order limit $$s \rightarrow 0^+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>s</mml:mi><mml:mo>→</mml:mo><mml:msup><mml:mn>0</mml:mn><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math> . While it is easy to see that all eigenvalues converge to 1 as $$s \rightarrow 0^+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>s</mml:mi><mml:mo>→</mml:mo><mml:msup><mml:mn>0</mml:mn><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math> , 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 $$2\log |\xi |$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>2</mml:mn><mml:mo>log</mml:mo><mml:mo>|</mml:mo><mml:mi>ξ</mml:mi><mml:mo>|</mml:mo></mml:mrow></mml:math> . By this we generalize a result of Chen and the third author which was restricted to the principal eigenvalue. Moreover, we show that $$L^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>L</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math> -normalized Dirichlet eigenfunctions of $$(-\Delta )^s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mo>-</mml:mo><mml:mi>Δ</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>s</mml:mi></mml:msup></mml:math> corresponding to the k -th eigenvalue are uniformly bounded and converge to the set of $$L^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>L</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math> -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.