Principal eigenvalues for k-Hessian operators by maximum principle methods

TL;DR

This paper characterizes the principal eigenvalue of fully nonlinear k-Hessian operators on certain domains using maximum principle methods, and constructs the eigenfunction via iterative viscosity solutions.

Contribution

It provides a new characterization of the principal eigenvalue for k-Hessian operators through a maximum principle approach and develops an iterative viscosity solution method for eigenfunction construction.

Findings

Characterization of principal eigenvalue as a supremum over spectral parameters.

Development of an iterative viscosity solution technique for eigenfunction construction.

Establishment of a global Hölder estimate for k-convex solutions.

Abstract

For fully nonlinear $k$-Hessian operators on bounded strictly $(k-1)$-convex domains $\Omega$ 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 $\Sigma_k$ 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 H\"{o}lder estimate for the unique $k$-convex solutions of the approximating equations.