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The Neumann Problem for Parabolic Hessian Quotient Equations

Chuanqiang Chen, Xi-Nan Ma, Dekai Zhang

Abstract

In this paper, we consider the Neumann problem for parabolic Hessian quotient equations. We show that the $k$-admissible solution of the parabolic Hessian quotient equation exists for all time and converges to the smooth solution of elliptic Hessian quotient equations. Also the solutions of the classical Neumann problem converge to a translating solution.

The Neumann Problem for Parabolic Hessian Quotient Equations

Abstract

In this paper, we consider the Neumann problem for parabolic Hessian quotient equations. We show that the -admissible solution of the parabolic Hessian quotient equation exists for all time and converges to the smooth solution of elliptic Hessian quotient equations. Also the solutions of the classical Neumann problem converge to a translating solution.

Paper Structure

This paper contains 18 sections, 19 theorems, 205 equations.

Table of Contents

  1. Introduction
  2. preliminary
  3. Basic properties of elementary symmetric functions
  4. Key Lemmas
  5. $u_t$-estimate
  6. $C^0$ estimate
  7. $C^1$ estimates
  8. Interior gradient estimate
  9. Near boundary gradient estimate
  10. $C^2$-estimate
  11. Global $C^2$ estimates can be reduced to the double normal estimates
  12. Lower estimate of double normal second derivatives on boundary
  13. Upper estimate of double normal second derivatives on boundary
  14. Proof of Theorem 1.1
  15. Proof of Theorem 1.2
  16. ...and 3 more sections

Key Result

Theorem \oldthetheorem

Assume that $\Omega$ is a strictly convex bounded domain in $\mathbb{R}^n$, $n \ge 2$, with smooth boundary. Let $f, \varphi:\overline\Omega\times\mathbb{R}\rightarrow \mathbb{R}$, be smooth functions which satisfy 1.2 and 1.3. Suppose there is a smooth, $k$-convex function $u_0$ satisfying the comp where $\nu$ is outer unit normal vector of $\partial \Omega$. The rate of convergence is exponentia

Theorems & Definitions (31)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • ...and 21 more