The Monge-Amp\`ere equation for strictly $(n-1)$-convex functions with Neumann condition

TL;DR

This paper establishes global $C^2$ estimates and proves the existence of solutions for the Monge-Ampère equation involving strictly $(n-1)$-convex functions with Neumann boundary conditions.

Contribution

It introduces a novel approach to obtain global estimates and existence results for a class of Monge-Ampère equations with $(n-1)$-convexity and Neumann boundary conditions.

Findings

Established global $C^2$ estimates for the equation.

Proved existence of solutions using the method of continuity.

Extended the theory to strictly $(n-1)$-convex functions with Neumann conditions.

Abstract

A $C^2$ function on $\mathbb{R}^n$ is called strictly $(n-1)$-convex if the sum of any $n-1$ eigenvalues of its Hessian is positive. In this paper, we establish a global $C^2$ estimates to the Monge-Amp\`ere equation for strictly $(n-1)$-convex functions with Neumann condition. By the method of continuity, we prove an existence theorem for strictly $(n-1)$-convex solutions of the Neumann problems.