Interior second derivative estimates for solutions to the linearized Monge-Ampère equation

Abstract:

Let $\Omega \subset \mathbb {R}^n$ be a bounded convex domain and $\phi \in C(\overline \Omega )$ be a convex function such that $\phi$ is sufficiently smooth on $\partial \Omega$ and the Monge-Ampère measure $\det D^2\phi$ is bounded away from zero and infinity in $\Omega$. The corresponding linearized Monge-Ampère equation is \[ \mathrm {trace}(\Phi D^2 u) =f, \] where $\Phi := \det D^2 \phi ~ (D^2\phi )^{-1}$ is the matrix of cofactors of $D^2\phi$. We prove a conjecture 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 equations whenever the measure $\det D^2\phi$ is given by a continuous density and the function $f$ belongs to $L^q(\Omega )$ for some $q> \max {\{p,n\}}$.