Reverse Faber-Krahn inequality for a truncated Laplacian operator

In this paper we prove a reverse Faber–Krahn inequality for the principal eigenvalue µ1(Ω) of the fully nonlinear eigenvalue problem (−λN (D2u) = µu in Ω,u = 0 on ∂Ω.Here λN (D2u) stands for the largest eigenvalue of the Hessian matrix of u. More precisely, we prove that, for an open, bounded, convex domain Ω ⊂ RN , the inequality µ1(Ω) ≤π2[diam(Ω)]2= µ1(Bdiam(Ω)/2), where 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 µ1(Ω) under different kinds of constraints.