Towards a reversed Faber–Krahn inequality for the truncated Laplacian

We consider the nonlinear eigenvalue problem, with Dirichlet boundary condition, for the very degenerate elliptic operator ${\mathcal{P}}_1^{+}$ mapping a function $u$ to the maximum eigenvalue of its Hessian matrix. The aim is to show that, at least for square type domains having fixed volume, the symmetry of the domain maximizes the principal eigenvalue, contrary to what happens for the Laplacian.