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-estimates for the linearized Monge-Ampère equation

Authors:	Cristian E Gutié rrez Federico Tournier

Institution:	Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122 ; Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-2067

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 $\lambda \leq$ det $D^2\phi \leq\Lambda$ in $\Omega$ . The linearized Monge-Ampère equation is $\displaystyle L_{\Phi}u=\textrm{trace}(\Phi D^2u)=f,$ where $\Phi = ($ det $D^2\phi)(D^2\phi)^{-1}$ is the matrix of cofactors of $D^2\phi$ . We prove that there exist and depending only on $n,\lambda,\Lambda$ , and $\textrm{dist}(\Omega^\prime,\Omega)$ such that $\displaystyle \Vert D^2u\Vert _{L^p(\Omega^\prime)}\leq C(\Vert u\Vert _{L^\infty(\Omega)}+\Vert f\Vert _{L^n(\Omega)})$ for all solutions $u\in C^2(\Omega)$ to the equation $L_{\Phi}u=f$ .

Keywords:	A priori estimates of second derivatives cross sections of solutions viscosity solutions nonuniformly elliptic equations

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