The Dirichlet problem for the logarithmic Laplacian
The Dirichlet problem for the logarithmic Laplacian
In this article, we study the logarithmic Laplacian operator LΔ, which is a singular integral operator with symbol 2 log |ζ|. We show that this operator has the integral representation LΔu(x)=cN∫RNu(x)1B1(x)(y)−u(y)|x−y|Ndy+ρNu(x) with cN=π−N2Γ(N2) and ρN=2 log 2+ψ(N2)−γ, where Γ is the Gamma function, ψ=Γ′Γ is the Digamma function and γ=−Γ′(1) …