论文标题
Hessian特征值的光谱表征和近似方案
A spectral characterization and an approximation scheme for the Hessian eigenvalue
论文作者
论文摘要
我们在$ \ Mathbb r^n $中的平滑，有限的，$（k-1）$ - 凸面域中重新审视$ k $ -Hessian eigenvalue问题。首先，我们获得了$ k $ -Hessian特征值作为线性二阶椭圆算子的第一个特征值的光谱表征，其系数属于相应的gårding锥的双重。其次，我们引入了一个非分类反复迭代方案，以解决$ k $ -Hessian运营商的特征值问题。我们表明，该计划以$ k $ -Hessian eigenvalue汇总了所有$ k $。当$ 2 \ leq k \ leq n $时，我们还证明了该计划解决方案Hessian的本地$ l^1 $收敛。双曲线多项式在我们的分析中起重要作用。
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 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.
