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A Liouville theorem for ancient solutions of the parabolic Monge-Ampère equation with periodic data

Kui Yan, Jiguang Bao

Abstract

This article is concerned with the parabolic Monge-Ampère equation $-u_t\det D_x^2u=f$, where $f=f_1(x)f_2(t)$ and $f_1,f_2$ are positive periodic functions. We prove that any classical parabolically convex ancient solution $u$ must be of the form $-τt+p(x)+v(x,t)$, where $τ$ is a positive constant, $p(x)$ is a convex quadratic polynomial, and $v$ inherits both the spatial and temporal periodicity from $f$. This work extends previous contributions by Caffarelli-Li \cite{cl04} on periodic frameworks for the elliptic Monge-Ampère equations, and generalizes Zhang-Bao \cite{zb18}'s Liouville theorem for $f_2\equiv1$ in parabolic case.

A Liouville theorem for ancient solutions of the parabolic Monge-Ampère equation with periodic data

Abstract

This article is concerned with the parabolic Monge-Ampère equation , where and are positive periodic functions. We prove that any classical parabolically convex ancient solution must be of the form , where is a positive constant, is a convex quadratic polynomial, and inherits both the spatial and temporal periodicity from . This work extends previous contributions by Caffarelli-Li \cite{cl04} on periodic frameworks for the elliptic Monge-Ampère equations, and generalizes Zhang-Bao \cite{zb18}'s Liouville theorem for in parabolic case.

Paper Structure

This paper contains 7 sections, 27 theorems, 373 equations.

Table of Contents

  1. Introduction and main results
  2. Interior estimates when $f=1$ in bounded domains
  3. A homogenization estimate
  4. Asymptotic behavior at infinity
  5. Positive upper and lower bounds of $D_x^2u$
  6. Capture of $\Delta_e^2u$ and $\Delta_k^tu$
  7. Proof of Theorem \ref{['mainthm1']}

Key Result

Theorem 1.1

Let $n\ge1$ and $u\in C^{2,1}\left(\mathbb{R}^{n+1}_-\right)$ be a parabolically convex ancient solution to pma1 with u_t, where $f:=f_1f_2$ and $f_1,f_2$ satisfy f1f2. Then there exist a constant $\tau>0$, an $n\times n$ symmetric positive definite matrix $A$ with and a vector $b\in\mathbb{R}^n$, such that is $a_i$-periodic in $x_i$ variable for each $1\le i\le n$ and $a_0$-periodic in $t$ vari

Theorems & Definitions (54)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • ...and 44 more