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

References in corpus (1)

• Interior estimates for solutions of Abreu's equation