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.

翻译：我们在一个平滑的、捆绑的、$(k-1)$-convex域域上,用$mathbb Rún美元,重新审视了美元-赫斯-赫斯-赫斯-海格值问题。首先,我们获得了一个光谱特征,将美元-赫斯-赫斯-赫斯-海格值作为线性第二级椭圆操作者第一个伊格值的最小值,其系数属于相应的G\r{a}rding conpee的双重值。第二,我们引入了一种非半衰化的反迭接机制,以解决赫斯-赫斯操作者的乙基值问题。我们显示,以一个速率将该计划与所有美元-赫斯-赫斯-赫斯-伊格值相趋同。当2美元\leq kleqn美元时,我们也证明Hesian的解决方案的当地合产值为1美元。超双曲曲多球在我们的分析中发挥了重要作用。