On the Hessian Hardy-Sobolev Inequality and Related Variational Problems

TL;DR

This paper establishes a Hardy-Sobolev inequality for Hessian integrals using gradient flow methods and explores the existence and regularity of solutions to related variational problems, extending previous Hessian equation theories.

Contribution

It introduces a new Hardy-Sobolev inequality for Hessian integrals and applies gradient flow techniques to study associated variational problems, expanding the existing Hessian equation framework.

Findings

Proved Hardy-Sobolev inequality for Hessian integrals

Demonstrated existence of solutions to related variational problems

Established regularity results for these solutions

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 \cite{CW01variational}.