| Abstract: |
| We study the logarithmic $p$-Laplacian $L_{\Delta_p}$, which arises as formal derivative of the fractional $p$-Laplacian $(-\Delta_p)^s$ at $s=0$. We present a variational framework to study the Dirichlet problems involving the $L_{\Delta_p}$ in bounded domains and use it to characterize the asymptotics of principal Dirichlet eigenvalues
and eigenfunctions of $(-\Delta_p)^s$ as $s\to 0$. As a byproduct, we then derive a Faber-Krahn type
inequality for the principal Dirichlet eigenvalue of $L_{\Delta_p}$. In addition, we discuss a boundary Hardy-type inequality for the spaces associated with the weak formulation of the logarithmic $p$-Laplacian. This talk is based on joint work with B. Dyda(Wroclaw) and S. Jarohs(Frankfurt). |
|