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

TL;DR

This paper proves global $W^{2,p}$ regularity estimates for solutions to the linearized Monge-Ampère equation under VMO-type conditions on the measure density, advancing understanding of regularity in degenerate elliptic PDEs.

Contribution

It establishes the first global $W^{2,p}$ estimates for the linearized Monge-Ampère equation with VMO-type coefficients, broadening regularity theory in degenerate elliptic equations.

Findings

Proved global $W^{2,p}$ estimates for solutions.

Extended regularity results to VMO-type coefficient conditions.

Enhanced understanding of degenerate elliptic PDE regularity.

Abstract

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