Liouville theorem and sharp solvability for solutions of the parabolic Monge-Amp\`ere equation with periodic data

TL;DR

This paper establishes a Liouville theorem for ancient solutions of the parabolic Monge-Ampère equation with periodic data and provides a necessary and sufficient condition for the existence of smooth periodic solutions, extending classical results to a parabolic setting.

Contribution

It generalizes the Liouville theorem to the parabolic Monge-Ampère equation with periodic data and characterizes the existence of smooth periodic solutions.

Findings

Liouville theorem for ancient solutions with periodic data

Necessary and sufficient condition for smooth periodic solutions

Extension of elliptic results to parabolic equations

Abstract

We prove a Liouville Theorem for ancient solutions of the parabolic Monge-Amp\`ere 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.