A spectral characterization and an approximation scheme for the Hessian eigenvalue

12/14/2020

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by   Nam Q. Le, et al.

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We revisit the k-Hessian eigenvalue problem on a smooth, bounded, (k-1)-convex domain in 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≤ k≤ 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.