A Liouville theorem for ancient solutions of the parabolic Monge-Amp\`ere equation with periodic data

TL;DR

This paper proves a Liouville theorem characterizing ancient solutions of a periodic parabolic Monge-Ampère equation, showing they must have a specific quadratic and periodic structure, extending previous elliptic and parabolic results.

Contribution

It extends Liouville theorems for the parabolic Monge-Ampère equation with periodic data, providing a precise form for ancient solutions.

Findings

Ancient solutions are of the form $- au t + p(x) + v(x,t)$ with specific structure.

Solutions inherit periodicity from the data functions.

Generalizes previous Liouville theorems to the periodic parabolic setting.

Abstract

This article is concerned with the parabolic Monge-Amp\`ere 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 $-\tau t+p(x)+v(x,t)$, where $\tau$ 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-Amp\`ere equations, and generalizes Zhang-Bao \cite{zb18}'s Liouville theorem for $f_2\equiv1$ in parabolic case.