Interior second derivative estimates for solutions to the linearized Monge--Amp\`ere equation

TL;DR

This paper establishes interior second derivative estimates for solutions to the linearized Monge--Ampère equation, linking the regularity of solutions to the closeness of the Monge--Ampère measure to one and the integrability of the source term.

Contribution

It proves a conjecture relating $L^p$ estimates of second derivatives to the proximity of the Monge--Ampère measure to one, leading to new interior $W^{2,p}$ estimates.

Findings

Established interior $W^{2,p}$ estimates under continuous density conditions.

Connected $L^p$ regularity of solutions to the closeness of $ ext{det} D^2 $ to 1.

Extended regularity results to solutions with $f ot o 0$ near boundary.

Abstract

Let $\Omega\subset \R^n$ be a bounded convex domain and $\phi\in C(\bar\Omega)$ be a convex function such that $\phi$ is sufficiently smooth on $\partial\Omega$ and the Monge--Amp\`ere measure $\det D^2\phi$ is bounded away from zero and infinity in $\Omega$. The corresponding linearized Monge--Amp\`ere equation is \[ \trace(\Phi D^2 u) =f, \] where $\Phi := \det D^2 \phi ~ (D^2\phi)^{-1}$ is the matrix of cofactors of $D^2\phi$. We prove a conjecture in \cite{GT} about the relationship between $L^p$ estimates for $D^2 u$ and the closeness between $\det D^2\phi$ and one. As a consequence, we obtain interior $W^{2,p}$ estimates for solutions to such equation whenever the measure $\det D^2\phi$ is given by a continuous density and the function $f$ belongs to $L^q(\Omega)$ for some $q> \max{\{p,n\}}$.