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

Abstract

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

24 pages