The constant mean curvature hypersurfaces with prescribed gradient image

TL;DR

This paper proves the existence and uniqueness of convex solutions to a boundary value problem for constant mean curvature hypersurfaces with a prescribed gradient image in convex domains.

Contribution

It establishes the existence and uniqueness of solutions for the second boundary value problem in the context of constant mean curvature hypersurfaces with prescribed gradient images.

Findings

Existence of convex solutions in convex domains.

Uniqueness of solutions for the boundary value problem.

Solution properties depend on domain convexity.

Abstract

In this paper, we consider the existence of constant mean curvature 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 constant mean curvature equations.