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The Neumann Problem for Hessian Equations

Xi-Nan Ma, Guohuan Qiu

Abstract

In this paper, we prove the existence of a classical solution to a Neumann boundary problem for Hessian equations in uniformly convex domain. The methods depend upon the established of a priori derivative estimates up to second order. So we give a affirmative answer to a conjecture of N. Trudinger in 1986.

The Neumann Problem for Hessian Equations

Abstract

In this paper, we prove the existence of a classical solution to a Neumann boundary problem for Hessian equations in uniformly convex domain. The methods depend upon the established of a priori derivative estimates up to second order. So we give a affirmative answer to a conjecture of N. Trudinger in 1986.

Paper Structure

This paper contains 9 sections, 15 theorems, 180 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $C^0$ and $C^1$ estimates
  4. Proof of Theorem \ref{['MQX']}
  5. Case I: boundary estimates
  6. Case II: Near boundary estimates
  7. $C^2$ priori estimates
  8. Second Normal Derivative Bounds On The Boundary
  9. Existence of the boundary problem

Key Result

Theorem \oldthetheorem

Let $\Omega$ be a $C^4$ bounded uniformly convex domain in $\mathbb{R}^n$. Where $f\in C^{2}(\overline{\Omega})$ is positive function and $\varphi \in C^{3}(\overline{\Omega})$. Then there exists a unique $k$ admissible solution $u \in C^{3,\alpha}(\overline{\Omega})$ of the boundary value problem,

Theorems & Definitions (26)

  • Theorem \oldthetheorem
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Theorem \oldthetheorem
  • ...and 16 more