Principal eigenvalues for k-Hessian operators by maximum principle methods

2019/12/19 by Isabeau Birindelli, Birindelli, Isabeau, Kevin R. Payne +1 · 3 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1912.09226

openalex publication_date 2019/12/19 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

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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• Large dimension behavior of the Hessian eigenvalues of the unit balls

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