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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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


Authors: Cristian E. Gutiérrez and Federico Tournier
Journal: Trans. Amer. Math. Soc. 358 (2006), 4843-4872
MSC (2000): Primary 35B45, 35J60, 35J70
DOI: https://doi.org/10.1090/S0002-9947-06-04189-4
Published electronically: June 9, 2006
MathSciNet review: 2231875
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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 $L_{\Phi }u=f$.


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Additional Information

Cristian E. Gutiérrez
Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122
Email: gutierrez@math.temple.edu

Federico Tournier
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-2067
Address at time of publication: Instituto Argentino de Matemática, Saavedra 15, 1038 Buenos Aires, Argentina
Email: fedeleti@aol.com

Keywords: A priori estimates of second derivatives, cross sections of solutions, viscosity solutions, nonuniformly elliptic equations
Received by editor(s): August 19, 2004
Published electronically: June 9, 2006
Additional Notes: The first author was partially supported by NSF grant DMS–0300004.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.