REVERSE FABER-KRAHN INEQUALITY FOR A TRUNCATED LAPLACIAN OPERATOR

In this paper we prove a reverse Fab er-Krahn inequality for the principal eigenvalue mu 1(S2) of the fully nonlinear eigenvalue problem (-lambda(N) (D(2)u) = mu u in Omega, u = 0 on partial derivative Omega. 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(2) subset of R-N, the inequality mu(1)(S2) <= pi 2/[diam(S Omega)](2) = mu(1)(B-diam(Omega)/2), where 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 con-straints.