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Transactions of the American Mathematical Society
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
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$   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 $ p>0$ and $ C>0$ 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$.


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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

DOI: http://dx.doi.org/10.1090/S0002-9947-06-04189-4
PII: S 0002-9947(06)04189-4
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.