Principal eigenvalues for k-Hessian operators by maximum principle methods

• Published in: Mathematics in engineering Vol. 3; no. 3; pp. 1 - 37
• Main Authors: Birindelli, Isabeau, R. Payne, Kevin
• Format: Journal Article
• Language: English
• Published: AIMS Press 01.01.2021
• Subjects: admissible viscosity solutions; comparison principles; elliptic sets; k-convex functions; k-hessian operators; maximum principles; principal eigenvalues
• ISSN: 2640-3501; 2640-3501
• DOI: 10.3934/mine.2021021

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.