The mean curvature type hypersurfaces with prescribed gradient image

TL;DR

This paper proves the existence and uniqueness of convex hypersurfaces with prescribed gradient images, solving a boundary value problem related to mean curvature equations in convex domains.

Contribution

It establishes the existence and uniqueness of convex solutions for a second boundary value problem of mean curvature type equations in convex domains.

Findings

Unique convex solutions exist for the prescribed gradient image problem.

Solutions are obtained for the second boundary value problem in smooth, uniformly convex domains.

The work advances understanding of mean curvature hypersurfaces with specified gradient images.

Abstract

In this paper, we consider the existence of mean curvature type hypersurfaces with prescribed gradient image. Let $\Omega$ and $\tilde{\Omega}$ be uniformly convex bounded domains in $\mathbb{R}^n$ with smooth boundary. We show that there exists unique convex solutions for the second boundary value problem of mean curvature type equations.