The Dirichlet problem for the logarithmic Laplacian

Abstract

In this article, we study the logarithmic Laplacian operator which is a singular integral operator with symbol We show that this operator has the integral representationwith and where Γ is the Gamma function, is the Digamma function and is the Euler Mascheroni constant. This operator arises as formal derivative of fractional Laplacians at 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 as As a byproduct, we then derive a Faber-Krahn type inequality for the principal Dirichlet eigenvalue of 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.