Ricerc@Sapienza

We prove that if Ω is an open bounded domain with smooth and connected boundary, for every p ∈(1,+∞) 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 (nonoptimal) lower bound for the eigenvalue in terms of the measure of Ω, and we address the open problem of proving a Faber-Krahn-type inequality with balls as optimal domains.

We prove that the principal eigenvalue of any fully nonlinear homogeneous elliptic operator which fulfills a very simple convexity assumption satisfies a Brunn-Minkowski type inequality on the class of open bounded sets in satisfying a uniform exterior sphere condition. In particular the result applies to the (possibly normalized) p-Laplacian, and to the minimal Pucci operator.