The Dirichlet Problem for the Logarithmic Laplacian

2017/10/10 by Huyuan Chen, Chen, Huyuan, Tobias Weth +1 · 7 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1710.03416

openalex publication_date 2017/10/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the logarithmic Laplacian operator L_Δ, which is a singular integral operator with symbol 2log |ζ|. We show that this operator has the integral representation L_Δu(x) = c_N ∫_ℝ^N \frac u(x)1_B1(x)(y)-u(y)|x-y|^N dy + ρ_N u(x) with c_N = π^{- (N)/(2)} Γ((N)/(2)) and ρ_N=2 log 2 + ψ((N)/(2)) -γ, where Γ is the Gamma function, ψ= \fracΓ'Γ is the Digamma function and γ= -Γ'(1) is the Euler Mascheroni constant. This operator arises as formal derivative ∂_s |_s=0 (-Δ)^s of fractional Laplacians at s= 0. We develop the functional analytic framework for Dirichlet problems involving the logarithmic Laplacian on bounded domains and use it to characterize the asymptotics of principal Dirichlet eigenvalues and eigenfunctions of (-Δ)^s as s → 0. As a byproduct, we then derive a Faber-Krahn type inequality for the principal Dirichlet eigenvalue of L_Δ. Using this inequality, we also establish conditions on domains giving rise to the maximum principle in weak and strong forms. This allows us to also derive regularity up to the boundary of solutions to corresponding Poisson problems.

Citations

• The Pohozaev Identity for the Fractional Laplacian
• Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian

Cited by

• Nonlocal Dirichlet problems involving the Logarithmic p-Laplacian
• Flat flows of periodic Lipschitz subgraphs for generalized nonlocal perimeters
• Sharp embeddings and existence results for Logarithmic p-Laplacian equations with critical growth
• Numerical Approximation of the logarithmic Laplacian via sinc-basis
• The conformal logarithmic Laplacian on the sphere: Yamabe-type problems and Sobolev spaces
• The Logarithmic Laplacian on General Graphs
• Fundamental Solutions of the Logarithmic Laplacian: An Approach via the Division Problem

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