Bounds for eigenvalues of the Dirichlet problem for the logarithmic Laplacian

Huyuan Chen; Laurent Véron

doi:10.1515/acv-2021-0025

Published by De Gruyter January 6, 2022

Bounds for eigenvalues of the Dirichlet problem for the logarithmic Laplacian

Huyuan Chen and Laurent Véron

From the journal Advances in Calculus of Variations

https://doi.org/10.1515/acv-2021-0025

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Abstract

We provide bounds for the sequence of eigenvalues { λ i ⁢ ( Ω ) } i of the Dirichlet problem

L Δ ⁢ u = λ ⁢ u ⁢ in ⁢ Ω , u = 0 ⁢ in ⁢ ℝ N ∖ Ω ,

where L Δ is the logarithmic Laplacian operator with Fourier transform symbol 2 ⁢ ln ⁡ | ζ | . 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 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.

Keywords: Dirichlet eigenvalues; logarithmic Laplacian

MSC 2010: 35P15; 35R09

Communicated by Hitoshi Ishii

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 12071189

Award Identifier / Grant number: 12001252

Funding source: Natural Science Foundation of Jiangxi Province

Award Identifier / Grant number: 20202BAB201005

Award Identifier / Grant number: 20202ACBL201001

Funding statement: This work is supported by NNSF of China, Nos. 12071189, 12001252, by the Jiangxi Provincial Natural Science Foundation, Nos. 20202BAB201005, 20202ACBL201001.

A Appendix: definedness of L Δ ⁢ μ z ⁢ ( x )

Lemma A.1.

Let z ∈ R N ∖ { 0 } and μ z ⁢ ( x ) = e - i ⁢ x ⋅ z for all x ∈ R N . Then L Δ ⁢ μ z ⁢ ( x ) is well defined for any x ∈ R N , and x ↦ L Δ ⁢ μ z ⁢ ( x ) is continuous and bounded in R N .

Proof.

Since the function μ z is bounded and smooth in ℝ N , L Δ ⁢ μ z ⁢ ( x ) is well defined if the following limit exists:

lim r → + ∞ ⁡ ∫ B r ⁢ ( x ) ∖ B 1 ⁢ ( x ) μ z ⁢ ( y ) | x - y | N ⁢ 𝑑 y .

If it holds, we have

L Δ ⁢ μ z ⁢ ( x ) = c N ⁢ ∫ B 1 ⁢ ( x ) μ z ⁢ ( x ) - μ z ⁢ ( y ) | x - y | N ⁢ 𝑑 y - c N ⁢ lim r → + ∞ ⁡ ∫ B r ⁢ ( x ) ∖ B 1 ⁢ ( x ) μ z ⁢ ( y ) | x - y | N ⁢ 𝑑 y + ρ N ⁢ μ z ⁢ ( x ) .

We introduce the Laplacian of μ z :

Δ ⁢ μ z ⁢ ( x ) = Δ ⁢ e - i ⁢ ∑ j x j ⁢ z j = Δ ⁢ ( ∏ j e - i ⁢ x j ⁢ z j ) = - ( ∑ j z j 2 ) ⁢ ∏ j e - i ⁢ x j ⁢ z j = - | z | 2 ⁢ μ z ⁢ ( x ) .

There holds

∫ B r ⁢ ( x ) ∖ B 1 ⁢ ( x ) μ z ⁢ ( y ) | x - y | N ⁢ 𝑑 y = μ z ⁢ ( x ) ⁢ ∫ B r ∖ B 1 μ z ⁢ ( ζ ) | ζ | N ⁢ 𝑑 ζ = - μ z ⁢ ( x ) | z | 2 ⁢ ∫ B r ∖ B 1 Δ ⁢ μ z ⁢ ( ζ ) | ζ | N ⁢ 𝑑 ζ .

By Green’s formula,

∫ B r ∖ B 1 | ζ | - N ⁢ Δ ⁢ μ z ⁢ ( ζ ) ⁢ 𝑑 ζ = ∫ B r ∖ B 1 μ z ⁢ ( ζ ) ⁢ Δ ⁢ | ζ | - N ⁢ 𝑑 ζ + ∫ ∂ ⁡ ( B r ∖ B 1 ) ( | ζ | - N ⁢ ∂ ⁡ μ z ⁢ ( ζ ) ∂ ⁡ n - μ z ⁢ ( ζ ) ⁢ ∂ | ζ | - N ∂ ⁡ n ) ⁢ 𝑑 S ⁢ ( ζ ) .

We have

Δ ⁢ | ζ | - N = 2 ⁢ N ⁢ | ζ | - N - 2 ,

∂ | ζ | - N ∂ ⁡ n = { - N ⁢ r - N - 1 on ⁢ ∂ ⁡ B r , N on ⁢ ∂ ⁡ B 1 ,

and

∂ ⁡ μ z ⁢ ( ζ ) ∂ ⁡ n = { - i ⁢ ζ ⋅ z r ⁢ μ z ⁢ ( ζ ) on ⁢ ∂ ⁡ B r , i ⁢ ζ ⋅ z ⁢ μ z ⁢ ( ζ ) on ⁢ ∂ ⁡ B 1 ,

since n is the unit outward normal vector either on ∂ ⁡ B r or on ∂ ⁡ B 1 . Letting r → ∞ , we obtain

lim r → + ∞ ⁡ ∫ B r ⁢ ( x ) ∖ B 1 ⁢ ( x ) μ z ⁢ ( y ) | x - y | N ⁢ 𝑑 y = - μ z ⁢ ( x ) | z | 2 ⁢ ( 2 ⁢ N ⁢ ∫ ℝ N ∖ B 1 μ z ⁢ ( ζ ) | ζ | N + 2 ⁢ 𝑑 ζ - ∫ ∂ ⁡ B 1 ( N - i ⁢ ζ ⋅ z ) ⁢ μ z ⁢ ( ζ ) ⁢ 𝑑 S ⁢ ( ζ ) ) .

Finally, we obtain

L Δ ⁢ μ z ⁢ ( x ) = c N ⁢ ∫ B 1 ⁢ ( x ) μ z ⁢ ( x ) - μ z ⁢ ( y ) | x - y | N ⁢ 𝑑 y + ρ N ⁢ μ z ⁢ ( x )

+ c N ⁢ μ z ⁢ ( x ) | z 2 | ⁢ ( 2 ⁢ N ⁢ ∫ ℝ N ∖ B 1 μ z ⁢ ( ζ ) | ζ | N + 2 ⁢ 𝑑 ζ - ∫ ∂ ⁡ B 1 ( N - i ⁢ ζ ⋅ z ) ⁢ μ z ⁢ ( ζ ) ⁢ 𝑑 S ⁢ ( ζ ) ) .

The boundedness and the continuity of x ↦ L Δ ⁢ μ z ⁢ ( x ) follow from this formula. ∎

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Received: 2021-03-16

Revised: 2021-09-01

Accepted: 2021-11-02

Published Online: 2022-01-06

Published in Print: 2023-07-01

Bounds for eigenvalues of the Dirichlet problem for the logarithmic Laplacian

Abstract

A Appendix: definedness of L Δ ⁢ μ z ⁢ ( x )

Lemma A.1.

Proof.

References

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