Weighted Eigenvalue Problem for a Class of Hessian Equations

2025/05/06 by Renjie He, He, Rongxun, Genggeng Huang +1 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2505.03231

openalex publication_date 2025/05/06 · openalex created_date 2025/10/16 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the existence and uniqueness of solutions to the weighted eigenvalue problem for k-Hessian equation. To achieve this, we establish the uniform a priori estimates for gradient and second derivatives of solutions to Hessian equation with weight |x|^2sk on the right-hand-side. We also prove that the eigenfunction is a minimizer of the corresponding functional among all k-admissible functions vanishing on the boundary.

Citations

• On the Hessian Hardy-Sobolev Inequality and Related Variational Problems
• Second order estimates for convex solutions of degenerate k-Hessian equations
• A spectral characterization and an approximation scheme for the Hessian eigenvalue
• Schauder estimates for degenerate Monge-Ampère equations and smoothness of the eigenfunctions
• New maximum principles for linear elliptic equations
• Hessian measures II
• The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian

Cited by

• On the Hessian Hardy-Sobolev Inequality and Related Variational Problems

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