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Reverse Faber-Krahn inequality for a truncated laplacian operator

In this paper we prove a reverse Faber-Krahn inequality for the principal eigenvalue $μ_1(Ω)$ of the fully nonlinear eigenvalue problem \[ \label{eq} \left\{\begin{array}{r c l l} -λ_N(D^2 u) & = & μu & \text{in }Ω, \\ u & = & 0 & \text{on }\partial Ω. \end{array}\right. \] Here $ λ_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 $Ω\subset \mathbb{R}^N$, the inequality \[ μ_1(Ω) \leq \frac{π^2}{[\text{diam}(Ω)]^2} = μ_1(B_{\text{diam}(Ω)/2}),\] where $\text{diam}(Ω)$ is the diameter of $Ω$, 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 $μ_1(Ω)$ under different kinds of constraints.