The Dirichlet Problem for the Logarithmic Laplacian

In this paper, we study the logarithmic Laplacian operator $L_{\mathrm{\Delta }}$, which is a singular integral operator with symbol $2\log \mathrm{\mid }\zeta \mathrm{\mid }$. We show that this operator has the integral representation $$L_{\mathrm{\Delta }}u(x)=c_N{\int }_{{\mathbb{R}}^N}\frac{u(x)1_{B_1(x)}(y)-u(y)}{\mathrm{\mid }x-y{\mathrm{\mid }}^N}dy+{\rho }_{N}u(x)$$ with $c_N={\pi }^{-\frac{N}{2}}\mathrm{\Gamma }(\frac{N}{2})$ and $\rho_N=2 \log 2 + \psi(\frac{N}{2}) -\gamma$,where$\Gamma$istheGammafunction,$\psi = \frac{\Gamma’}{\Gamma}$istheDigammafunctionand$\gamma= -\Gamma’(1)$ is the Euler Mascheroni constant. This operator arises as formal derivative ${\mathrm{\partial }}_s{\mid }_{s=0}(-\mathrm{\Delta }{)}^s$ 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 $(-\mathrm{\Delta }{)}^s$ as $s\to 0$. As a byproduct, we then derive a Faber-Krahn type inequality for the principal Dirichlet eigenvalue of $L_{\mathrm{\Delta }}$. 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.