Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The constant mean curvature hypersurfaces with prescribed gradient image

Rongli Huang, Dayan Wei, Yunhua Ye

Abstract

In this paper, we consider the existence of constant mean curvature hypersurfaces with prescribed gradient image. Let $Ω$ and $\tildeΩ$ 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.

The constant mean curvature hypersurfaces with prescribed gradient image

Abstract

In this paper, we consider the existence of constant mean curvature hypersurfaces with prescribed gradient image. Let and be uniformly convex bounded domains in with smooth boundary. We show that there exists unique convex solutions for the second boundary value problem of constant mean curvature equations.

Paper Structure

This paper contains 6 sections, 19 theorems, 237 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The strict obliqueness estimate
  4. The global $C^2$ estimate
  5. The invertibility of linearized operators
  6. Existence and uniqueness of solutions

Key Result

Theorem 1.1

Suppose that $\Omega$, $\tilde{\Omega}$ are uniformly convex bounded domains with smooth boundary in $\mathbb{R}^n$ and $\tilde{\Omega}\subset\subset B_1(0)$. Then there exists a uniformly convex solution $u\in C^{\infty}(\bar{\Omega})$ and a unique constant $c$ solving e1.1.3 and e1.1.4. Here $u$ i

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Corollary 3.1
  • Corollary 3.2
  • Definition 3.3
  • ...and 21 more