Principal eigenvalues for k-Hessian operators by maximum principle methods

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 adm...

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Bibliographic Details
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
Online Access:	Get full text
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Summary:	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.
ISSN:	2640-3501 2640-3501
DOI:	10.3934/mine.2021021