A spectral characterization and an approximation scheme for the Hessian eigenvalue

Nam Q Le

doi:10.4171/rmi/1306

Abstract

We revisit the $k$-Hessian eigenvalue problem on a smooth, 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 Garding 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

Introduction and statements of the main results

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).

Hyperbolic polynomials

Nonlinear integration by parts inequalities

A spectral characterization of the k-Hessian eigenvalue via dual Garding cone

Convergence to the k-Hessian eigenvalue