Global $W^{2,p}$ estimates for solutions to the linearized Monge--Amp\`ere equations

TL;DR

This paper proves global $W^{2,p}$ estimates for solutions to the linearized Monge-Ampère equations, extending previous results to an affine invariant setting and under natural assumptions.

Contribution

It introduces affine invariant global $W^{2,p}$ estimates for the linearized Monge-Ampère equations, bridging fully nonlinear elliptic and Monge-Ampère theory.

Findings

Establishes affine invariant $W^{2,p}$ estimates

Extends Winter's estimates to the linearized Monge-Ampère setting

Provides linearized counterparts of Savin's estimates

Abstract

In this paper, we establish global $W^{2,p}$ estimates for solutions to the linearized Monge-Amp\`ere equations under natural assumptions on the domain, Monge-Amp\`ere measures and boundary data. Our estimates are affine invariant analogues of the global $W^{2,p}$ estimates of Winter for fully nonlinear, uniformly elliptic equations, and also linearized counterparts of Savin's global $W^{2,p}$ estimates for the Monge-Amp\`ere equations.