On the Hessian Hardy-Sobolev Inequality and Related Variational Problems

2025/05/06 by Renjie He, Wei Ke, He, Rongxun +1 · 1 citation
Computer Science · Engineering · Mathematics · #Analysis of PDEs (math.AP) #Contact Mechanics and Variational Inequalities #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in engineering

paper · pdf · doi:10.48550/arxiv.2505.03245

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

Abstract

In this paper, we first prove the Hardy-Sobolev inequality for the Hessian integral by means of a descent gradient flow of certain Hessian functionals. As an application, we study the existence and regularity results of solutions to related variational problems. Our results extend the variational theory of the Hessian equation in \citeCW01variational.

Citations

• Weighted Eigenvalue Problem for a Class of Hessian Equations
• Second order estimates for convex solutions of degenerate k-Hessian equations
• The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian
• Global and local behavior of positive solutions of nonlinear elliptic equations

Cited by

• Weighted Eigenvalue Problem for a Class of Hessian Equations

Related

• A fully nonlinear Sobolev trace inequality
• Entire solutions of the generalized Hessian inequality
• A quantitative result for the k-Hessian equation
• Sobolev inequalities and regularity of the linearized complex Monge-Ampere and Hessian equations
• Variational integrals on Hessian spaces: partial regularity for critical points