Global $W^{2,p}$ regularity on the linearized Monge-Amp$\grave{e}$re equation with $\mathrm{VMO}$ type coefficients

Abstract

In this paper, we establish global $W^{2,p}$ estimates for solutions of the linearized Monge-Amp$\grave{e}$re equation $${\mathcal{L}}_{\varphi }u:=\mathrm{tr}[\Phi D^{2}u]=f,$$ where the density of the Monge-Amp$\grave{e}$re measure $g:=\det D^2\varphi$ satisfies a $\mathrm{VMO}$-type condition, and $\Phi :=(\det D^2\varphi )(D^2\varphi {)}^{-1}$ is the cofactor matrix of $D^2\varphi$.

20 pages