Global $W^{1,p}$ estimates for solutions to the linearized Monge--Amp\`ere equations

TL;DR

This paper establishes global $W^{1,p}$ regularity estimates for solutions to the linearized Monge-Ampère equations with low integrability right-hand sides, extending known results to a broader class of equations.

Contribution

It provides affine invariant global $W^{1,p}$ estimates for solutions with right-hand side in $L^q$, for a range of $q$, under natural geometric and boundary conditions.

Findings

Established $W^{1,p}$ estimates for $p<rac{nq}{n-q}$

Extended regularity results to low integrability right-hand sides

Provided affine invariant estimates analogous to classical results

Abstract

In this paper, we investigate regularity for solutions to the linearized Monge-Amp\`ere equations when the nonhomogeneous term has low integrability. We establish global $W^{1,p}$ estimates for all $p<\frac{nq}{n-q}$ for solutions to the equations with right hand side in $L^q$ where $n/2<q\leq n$. These estimates hold under natural assumptions on the domain, Monge-Amp\`ere measures and boundary data. Our estimates are affine invariant analogues of the global $W^{1,p}$ estimates of N. Winter for fully nonlinear, uniformly elliptic equations.