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Liouville theorem and sharp solvability for solutions of the parabolic Monge-Ampère equation with periodic data

Kui Yan, Jiguang Bao

Abstract

We prove a Liouville Theorem for ancient solutions of the parabolic Monge-Ampère equation with smooth periodic data, generalizing Caffarelli-Li's result \cite{cl04} in 2004 to the parabolic background. To achieve this, we obtain a necessary and sufficient condition for the existence of the smooth periodic solution of the equation $\left(1-u_t\right)\det \left(D_x^2u+I\right)=f$ in $\mathbb{R}^{n+1}$, where $f$ is smooth and periodic in both spatial and temporal variables. This parabolic existence theorem parallels the elliptic counterpart established by Li \cite{l90} in 1990.

Liouville theorem and sharp solvability for solutions of the parabolic Monge-Ampère equation with periodic data

Abstract

We prove a Liouville Theorem for ancient solutions of the parabolic Monge-Ampère equation with smooth periodic data, generalizing Caffarelli-Li's result \cite{cl04} in 2004 to the parabolic background. To achieve this, we obtain a necessary and sufficient condition for the existence of the smooth periodic solution of the equation in , where is smooth and periodic in both spatial and temporal variables. This parabolic existence theorem parallels the elliptic counterpart established by Li \cite{l90} in 1990.

Paper Structure

This paper contains 5 sections, 16 theorems, 256 equations.

Table of Contents

  1. introduction
  2. a calculus lemma related to the periodicity
  3. proof of Theorem \ref{['existence']}
  4. asymptotic behavior of $u$ at infinity
  5. proof of Theorem \ref{['liouville']}

Key Result

Theorem 1.1

Let $f\in C^{2+\alpha,\frac{2+\alpha}{2}}\left(\mathbb{R}^{n+1}_-\right)$ satisfy that for every $(x,t)\in\mathbb{R}^{n+1}_-$, $k,l\in\mathbb{N}$ and $1\le i\le n$, and $u\in C^{2,1}\left(\mathbb{R}^{n+1}_-\right)$ be a p-convex ancient solution to pma1 with where $m_1,m_2>0$ are two constants. Then there exist a constant $\tau>0$, a vector $b\in\mathbb{R}^n$ and an $n\times n$ positive definite

Theorems & Definitions (35)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Lemma 2.1
  • proof
  • proof
  • Proposition 3.1
  • ...and 25 more