On the first eigenvalue of the normalized p-Laplacian

We prove that, if $\mathrm{\Omega }$ is an open bounded domain with smooth and connected boundary, for every $p\in (1,+\mathrm{\infty })$ the first Dirichlet eigenvalue of the normalized $p$-Laplacian is simple in the sense that two positive eigenfunctions are necessarily multiple of each other. We also give a (non-optimal) lower bound for the eigenvalue in terms of the measure of $\mathrm{\Omega }$, and we address the open problem of proving a Faber-Krahn type inequality with balls as optimal domains.