The Neumann problem for a class of degenerate Hessian quotient type equations

TL;DR

This paper develops inequalities for Hessian quotient operators and uses them to establish existence and estimates for the Neumann problem of a class of degenerate Hessian quotient equations.

Contribution

It introduces new inequalities for Hessian quotient operators and extends the admissible range for solving the Neumann problem for degenerate equations.

Findings

Established important inequalities for Hessian quotient operators.

Proved global a priori estimates for the degenerate Hessian quotient equations.

Proved an existence theorem for the Neumann problem in an extended parameter range.

Abstract

In this paper, we obtain some important inequalities for a class of Hessian quotient type operators $\frac{\sigma_k(\Lambda(D^2u))}{\sigma_l(\Lambda(D^2u))}$, which can be regarded as a generalization of the classical Hessian quotient operators. As an application, we establish global a priori estimates and prove an existence theorem for the Neumann problem of the corresponding degenerate Hessian quotient type equation, in which the admissible range of $k$ is extended to $0< k \leq C^\mathbf{p}_n$ with $1 \leq \mathbf{p} \leq n-1$.