Bounds for eigenvalues of the Dirichlet problem for the logarithmic Laplacian

Huyuan Chen (1) , Laurent Véron (2)

1 Jiangxi Normal University
2 IDP - Institut Denis Poisson

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.

Mots clés: Logarithmic Laplacian; Dirichlet eigenvalues

Domaines: Equations aux dérivées partielles [math.AP]

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⟩