论文标题
dirichlet特征值问题的小订单渐近学针对分数laplacian
Small order asymptotics of the Dirichlet eigenvalue problem for the fractional Laplacian
论文作者
论文摘要
在本文中，我们研究了有限的开放lipschitz集合中的分数laplacian $（ - δ）^s $的dirichlet特征值和特征函数的渐近学。虽然很容易看到所有特征值将$ 1 $作为$ s \ to 0^+$收敛，但我们表明，这些渐近学中的第一阶校正是由对数laplacian操作员的特征值给出的，即，符号$ 2 \ log log $ log 2 \ log | endull operator，即单独的整体操作员。通过这种情况，我们概括了陈的结果，而第三作者则仅限于主要特征值。此外，我们表明，$ l^2 $ normalized dirichlet eigenfunctions的$（ - δ）^s $对应于$ k $ - th eigenvalue均匀界限，并汇聚到$ l^2 $ normalitormatization the Goolegarithmic laplacian的$ l^2 $ normalitormatization。为了得出这些光谱渐进性，我们需要为分数拉普拉斯式的dirichlet本征函数建立新的均匀规律性和边界衰减估计。作为副产品，我们还获得了对数laplacian的征征性特征的相应规律性。
In this article, 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 $s \to 0^+$. While it is easy to see that all eigenvalues converge to $1$ as $s \to 0^+$, 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 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 eigenvalues of the logarithmic Laplacian. In order to derive these spectral asymptotics, we need to 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.
