$W^\{2,p\}$–estimates for the linearized Monge–Ampère equation

Abstract:

Let $\Omega \subseteq \mathbb {R}^n$ be a strictly convex domain and let $\phi \in C^2(\Omega )$ be a convex function such that $\lambda \leq \text {det}D^2\phi \leq \Lambda$ in $\Omega$. The linearized Monge–Ampère equation is \begin{equation*} L_{\Phi }u=\textrm {trace}(\Phi D^2u)=f, \end{equation*} where $\Phi = (\text {det}D^2\phi )(D^2\phi )^{-1}$ is the matrix of cofactors of $D^2\phi$. We prove that there exist $p>0$ and $C>0$ depending only on $n,\lambda ,\Lambda$, and $\textrm {dist}(\Omega ^\prime ,\Omega )$ such that \begin{equation*} \|D^2u\|_{L^p(\Omega ^\prime )}\leq C(\|u\|_{L^\infty (\Omega )}+\|f\|_{L^n(\Omega )}) \end{equation*} for all solutions $u\in C^2(\Omega )$ to the equation $L_{\Phi }u=f$.