Principal eigenvalues for k-Hessian operators by maximum principle methods

For fully nonlinear k-Hessian operators on bounded strictly (k - 1)-convex domains [OMEGA] of [R.sup.N], 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 [[summation].sub.k] [subset] 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 Holder estimate for the unique k-convex solutions of the approximating equations. Keywords: maximum principles; comparison principles; principal eigenvalues; k-Hessian operators; k-convex functions; admissible viscosity solutions; elliptic sets