Principal eigenvalues for k-Hessian operators by maximum principle methods

For fully nonlinear k-Hessian operators on bounded strictly (k - 1)-convex domains Ω of RN, a characterization of the principal eigenvalue associated to a k-convex and negative principal eigenfunction will be given as the supremum over values of a spectral parameter for which admissible viscosity supersolutions obey a minimum principle. The admissibility condition is phrased in terms of the natural closed convex cone Σk ⊂ S(N) which is an elliptic set in the sense of Krylov [23] which corresponds to using k-convex functions as admissibility constraints in the formulation of viscosity subsolutions and supersolutions. Moreover, the associated principal eigenfunction is constructed by an iterative viscosity solution technique, which exploits a compactness property which results from the establishment of a global Hölder estimate for the unique k-convex solutions of the approximating equations.