Reverse Faber-Krahn inequality for a truncated Laplacian operator

Abstract

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.