Affiliation:
1. Department of Mathematics , Jiangxi Normal University , Nanchang , Jiangxi 330022 , P. R. China
2. Laboratoire de Mathématiques et Physique Théorique , Université de Tours , 37200 Tours , France
Abstract
Abstract
We provide bounds for the sequence of eigenvalues
{
λ
i
(
Ω
)
}
i
{\{\lambda_{i}(\Omega)\}_{i}}
of the Dirichlet problem
L
Δ
u
=
λ
u
in
Ω
,
u
=
0
in
ℝ
N
∖
Ω
,
L_{\Delta}u=\lambda u\text{ in }\Omega,\quad u=0\text{ in }\mathbb{R}^{N}%
\setminus\Omega,
where
L
Δ
{L_{\Delta}}
is the logarithmic Laplacian operator with Fourier transform symbol
2
ln
|
ζ
|
{2\ln\lvert\zeta\rvert}
.
The logarithmic Laplacian operator is not positively defined 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 quotient of the sum of the first k eigenvalues by
k
ln
k
{k\ln k}
is independent of the volume of the domain. Finally, we discuss the lower and upper bounds of the k-th principle eigenvalue, and the asymptotic behavior of the limit of eigenvalues.
Funder
National Natural Science Foundation of China
Natural Science Foundation of Jiangxi Province
Subject
Applied Mathematics,Analysis
Cited by
2 articles.
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1. Small order limit of fractional Dirichlet sublinear-type problems;Fractional Calculus and Applied Analysis;2023-05-24
2. Small order asymptotics for nonlinear fractional problems;Calculus of Variations and Partial Differential Equations;2022-03-28