Global $C^{1+\alpha,\frac{1+\alpha}{2}}$ regularity on the linearized parabolic Monge-Amp$\grave{e}$re equation

TL;DR

This paper proves global $C^{1+eta,rac{1+eta}{2}}$ regularity estimates for solutions to the linearized parabolic Monge-Ampère equation, extending regularity theory under specific geometric and boundary conditions.

Contribution

It establishes the first global regularity estimates for solutions of the linearized parabolic Monge-Ampère equation under broad conditions.

Findings

Proves global $C^{1+eta,rac{1+eta}{2}}$ regularity for solutions.

Provides conditions on domain, boundary data, and right-hand side for regularity.

Extends regularity results to a class of degenerate parabolic equations.

Abstract

In this paper, we establish global $C^{1+\alpha,\frac{1+\alpha}{2}}$ estimates for solutions of the linearized parabolic Monge-Amp$\grave{e}$re equation $$\mathcal{L}_\phi u(x,t):=-u_t\,\mathrm{det}D^2\phi(x)+\mathrm{tr}[\Phi(x) D^2 u]=f(x,t)$$ under appropriate conditions on the domain, Monge-Amp$\grave{e}$re measures, boundary data and $f$, where $\Phi:=\mathrm{det}(D^2\phi)(D^2\phi)^{-1}$ is the cofactor of the Hessian of $D^2\phi$.