Abstract
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 <i>admissible viscosity supersolutions</i> 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 <i>elliptic set</i> in the sense of Krylov <sup>[<span class="xref"><a href="#b23" ref-type="bibr">23</a></span>]</sup> 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.
Highlights
For each 1 ≤ k ≤ N, the k-Hessian operator acting on u ∈ C2(Ω) with Ω ⊆ RN open is defined byS k(D2u) := σk(λ(D2u)) :=λi1(D2u) · · · λik (D2u)1≤i1
For fully nonlinear k-Hessian operators on bounded strictly (k − 1)-convex domains Ω of RN, 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 Σk ⊂ 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
Summary
For each 1 ≤ k ≤ N, the k-Hessian operator acting on u ∈ C2(Ω) with Ω ⊆ RN open is defined by. Λk := {λ ∈ R : ∃ψ ∈ Φ−k (Ω) with S k(D2ψ) + λψ|ψ|k−1 ≥ 0 in Ω}, where the inequality above is again in the admissible viscosity sense, and define our candidate for a (generalized) principal eigenvalue by λ−1 := sup Λk. We should mention that the k-Hessians are variational and it is possible to give a variational characterization of the principal eigenvalue through a generalized Rayleigh quotient. It allows one to prove the minimum principle for λ below λ−1 while on the other hand, it strongly suggests that it may be possible to extend the results to a class of fully non linear operators that are not variational, but which may include the k-Hessians. See Remark 6.4 for a discussion of this point, including the use of some measure theoretic techniques (to augment the maximum principle techniques developed ) in order establish existence in the nonlinear range
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