Principal eigenvalues for k-Hessian operators by maximum principle methods

For fully nonlinear $k$-Hessian operators on bounded strictly $(k-1)$-convex domains $Ω$ in ${\mathbb R}^N$, 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 admissible viscosity supersolutions obey a minimum principle. The admissibility condition is phrased in terms of the natural closed convex cone $Σ_k$ in the space of symmetric N by N matrices, which is an elliptic set in the sense of Krylov [Trans. AMS, 1995] and 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.

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