The Neumann problem for a class of Hessian quotient type equations

TL;DR

This paper establishes interior gradient estimates and existence results for the Neumann problem involving a class of Hessian quotient equations with gradient terms, advancing the understanding of solutions' regularity and solvability.

Contribution

It provides new interior gradient estimates and global a priori bounds for the Neumann problem of Hessian quotient equations with gradient dependence.

Findings

Derived interior gradient estimates for solutions.

Established global a priori estimates.

Proved existence of solutions under growth conditions.

Abstract

In this paper, we consider the Neumann problem for a class of Hessian quotient equations involving a gradient term on the right-hand side in Euclidean space. More precisely, we derive the interior gradient estimates for the $(\Lambda, k)$-convex solution of Hessian quotient equation $\frac{\sigma_k(\Lambda(D^2 u))}{\sigma_l(\Lambda(D^2 u))}=\psi(x,u,D u)$ with $0\leq l<k\leq C^{p-1}_{n-1}$ under the assumption of the growth condition. As an application, we obtain the global a priori estimates and the existence theorem for the Neumann problem of this Hessian quotient type equation.