Abstract
We revisit the k -Hessian eigenvalue problem on a smooth and bounded {(k-1)} -convex domain in \mathbb{R}^n . First, we obtain a spectral characterization of the k -Hessian eigenvalue as the infimum of the first eigenvalues of linear second-order elliptic operators whose coefficients belong to the dual of the corresponding Gårding cone. Second, we introduce a non-degenerate inverse iterative scheme to solve the eigenvalue problem for the k -Hessian operator. We show that the scheme converges, with a rate, to the k -Hessian eigenvalue for all k . When 2\leq k\leq n , we also prove a local L^1 convergence of the Hessian of solutions of the scheme. Hyperbolic polynomials play an important role in our analysis.
Highlights
Introduction and statements of the main resultsWe consider the k-Hessian counterparts of some results on the Monge-Ampere eigenvalue problem
We revisit the k-Hessian eigenvalue problem on a smooth, bounded, (k − 1)-convex domain in Rn
We consider the k-Hessian counterparts of some results on the Monge-Ampere eigenvalue problem
Summary
We consider the k-Hessian counterparts of some results on the Monge-Ampere eigenvalue problem. Eigenvalue problem, k-Hessian operator, spectral characterization, non-degenerate iterative scheme, hyperbolic polynomial. The author [15] studied the Monge-Ampere eigenvalue problem for general open bounded convex domains and established the singular counterparts of previous results by Lions and Tso. Let Ω be a bounded open convex domain in Rn. Define the constant λ = λ[n; Ω] via infimum of the Rayleigh quotient by (1.4) (λ[n; Ω])n = inf Rn(u) : u ∈ C(Ω), u is convex, nonzero in Ω, u = 0 on ∂Ω. (Theorem 1.1), and study a non-degenerate inverse iterative scheme (1.15), similar to (1.5), to solve the k-Hessian eigenvalue problem. Theorem 1.2 (Convergence to the Hessian eigenvalue of the non-degenerate inverse iterative scheme).
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