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Principal eigenvalues for k-Hessian operators by maximum principle methods

1 Dipartimento di Matematica “G. Castelnuovo”, Sapienza Università di Roma, P.le Aldo Moro 2, 00185 Roma, Italy
2 Dipartimento di Matematica “F. Enriques”, Università di Milano, Via C. Saldini 50, 20133 Milano, Italy

This contribution is part of the Special Issue: Critical values in nonlinear pdes – Special Issue dedicated to Italo Capuzzo Dolcetta
Guest Editor: Fabiana Leoni
Link: www.aimspress.com/mine/article/5754/special-articles

Special Issues: Critical values in nonlinear pdes - Special Issue dedicated to Italo Capuzzo Dolcetta

For fully nonlinear $k$-Hessian operators on bounded strictly $(k-1)$-convex domains $\Omega$ of $\mathbb{R}^N$, 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 $\Sigma_k \subset {\cal S}(N)$ which is an elliptic set in the sense of Krylov [23] 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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