Bounds for eigenvalues of the Dirichlet problem for the logarithmic Laplacian - Archive ouverte HAL Access content directly
Journal Articles Advances in Calculus of Variation Year : 2021

Bounds for eigenvalues of the Dirichlet problem for the logarithmic Laplacian

Abstract

We provide bounds for the sequence of eigenvalues {λ i (Ω)} i of the Dirichlet problem L ∆ u = λu in Ω, u = 0 in R N \ Ω, where L ∆ 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, hence the principle eigenvalue is no longer always positive. We also give asymptotic estimates of the sum of the first k eigenvalues. To study the principle eigenvalue, we construct lower and upper bounds by a Li-Yau type method and calculate the Rayleigh quotient for some particular functions respectively. Our results point out the role of the volume of the domain in the bound of the principle eigenvalue. For the asymptotic of sum of eigenvalues, lower and upper bounds are built by a duality argument and by Kröger's method respectively. Finally, we obtain the limit of eigenvalues and prove that the limit is independent of the volume of the domain.
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Dates and versions

hal-02977991 , version 1 (26-10-2020)
hal-02977991 , version 2 (09-11-2020)
hal-02977991 , version 3 (04-11-2021)

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Huyuan Chen, Laurent Véron. Bounds for eigenvalues of the Dirichlet problem for the logarithmic Laplacian. Advances in Calculus of Variation, 2021, 15, ⟨10.1515/acv-2021-0025⟩. ⟨hal-02977991v3⟩
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