论文标题
Dirichlet问题的特征值的范围是对数Laplacian的
Bounds for eigenvalues of the Dirichlet problem for the logarithmic Laplacian
论文作者
论文摘要
我们为dirichlet问题的eigenvalues $ \ {λ_i（ω）\} _ i $提供界限\ mathbb {r}^n \setMinusΩ，$ $l_δ$是具有傅立叶变换符号$ 2 \ ln |ζ| $的对数laplacian运算符。如果域的体积足够大，则对数拉普拉斯运算符对确定性并不确定。在本文中，我们分别扩展了Li-Yau方法和Kröger的方法，为第一个$ k $ eigenvalues的总和获得了上限和下限。此外，我们显示了第一个$ k $ eigenvalues的限制，该总和与域的体积无关。最后，我们讨论了$ k $ th原理特征值的下限和上限，即特征值极限的渐近行为。
We provide bounds for the sequence of eigenvalues $\{λ_i(Ω)\}_i$ of the Dirichlet problem $$ L_Δu=λu\ \ {\rm in}\ \, Ω,\quad\quad u=0\ \ {\rm in}\ \ \mathbb{R}^N\setminus Ω,$$ where $L_Δ$ is the logarithmic Laplacian operator with Fourier transform symbol $2\ln |ζ|$. The logarithmic Laplacian operator is not positively definitive if the volume of the domain is large enough. In this article, we obtain the upper and lower bounds for the sum of the first $k$ eigenvalues by extending the Li-Yau method and Kröger's method respectively. Moreover, we show the limit of the sum of the first $k$ eigenvalues, which is independent of the volume of the domain. Finally, we discuss the lower and upper bounds of the $k$-th principle eigenvalue, the asymptotic behavior of the limit of eigenvalues.
