A spectral characterization and an approximation scheme for the Hessian eigenvalue

TL;DR

This paper provides a spectral characterization of the $k$-Hessian eigenvalue and introduces a convergent iterative scheme for approximating it, with convergence rates and local Hessian convergence results.

Contribution

It offers a novel spectral characterization of the $k$-Hessian eigenvalue and develops a new inverse iterative scheme with proven convergence properties.

Findings

Spectral characterization of the $k$-Hessian eigenvalue as an infimum over linear operators.

A non-degenerate inverse iterative scheme that converges to the eigenvalue.

Local $L^1$ convergence of the Hessian for solutions when $2 \,\leq\, k \leq n$.

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 G\r{a}rding 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.