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Mathematics in Engineering

2021, Volume 3, Issue 3: 1-37. doi: 10.3934/mine.2021021
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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
  • Received: 16 December 2019 Accepted: 16 March 2020 Published: 15 July 2020

  • 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.

    Citation: Isabeau Birindelli, Kevin R. Payne. Principal eigenvalues for k-Hessian operators by maximum principle methods[J]. Mathematics in Engineering, 2021, 3(3): 1-37. doi: 10.3934/mine.2021021

    Related Papers:

  • 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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