Principal eigenvalues for k-Hessian operators by maximum principle methods

Abstract

For fully nonlinear $k$-Hessian operators on bounded strictly $(k-1)$-convex domains $\Omega$ of $\R^N$, 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 {\em admissible viscosity supersolutions} obey a minimum principle. The admissibility condition is phrased in terms of the natural closed convex cone $\Sigma_k \subset \Ss(N)$ which is an {\em elliptic set} in the sense of Krylov \cite{Kv95} 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 H\"older 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