Principal eigenvalues for k-Hessian operators by maximum principle methods

Abstract

For fully nonlinear k-Hessian operators on bounded strictly (k −1)-convex domains Ω of R N , a characterization of the principal eigenvalue associated to a kconvex 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 Σ k ⊂ S(N ) which is an elliptic set in the sense of Krylov [26] 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. Contents 1. Introduction 2. Preliminaries. 3. Comparison and maximum principles. 4. Boundary estimates 5. Characterization of the principal eigenvalue 6. Existence of the principal eigenfunction by maximum principle methods 7. Bounds on the principal eigenvalue. References