Reverse Faber-Krahn inequality for a truncated laplacian operator

In this paper we prove a reverse Faber-Krahn inequality for the principal eigenvalue $mu_1(Omega)$ of the fully nonlinear eigenvalue problem [ label{eq} left{begin{array}{r c l l} -lambda_N(D^2 u) & = & mu u & text{in }Omega, u & = & 0 & text{on }partial Omega. end{array}right. ] Here $ lambda_N(D^2 u)$ stands for the largest eigenvalue of the Hessian matrix of $u$. More precisely, we prove that, for an open, bounded, convex domain $Omega subset mathbb{R}^N$, the inequality [ mu_1(Omega) leq frac{pi^2}{[text{diam}(Omega)]^2} = mu_1(B_{text{diam}(Omega)/2}),] where $text{diam}(Omega)$ is the diameter of $Omega$, holds true. The inequality actually implies a stronger result, namely, the maximality of the ball under a diameter constraint. Furthermore, we discuss the minimization of $mu_1(Omega)$ under different kinds of constraints.