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Interior second derivative estimates for solutions to the linearized Monge-Ampère equation


Authors: Cristian E. Gutiérrez and Truyen Nguyen
Journal: Trans. Amer. Math. Soc. 367 (2015), 4537-4568
MSC (2010): Primary 35J60, 35J70, 35J96
Published electronically: March 13, 2015
MathSciNet review: 3335393
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Abstract: Let $ \Omega \subset \mathbb{R}^n$ be a bounded convex domain and $ \phi \in C(\overline \Omega )$ be a convex function such that $ \phi $ is sufficiently smooth on $ \partial \Omega $ and the Monge-Ampère measure $ \det D^2\phi $ is bounded away from zero and infinity in $ \Omega $. The corresponding linearized Monge-Ampère equation is

$\displaystyle \mathrm {trace}(\Phi D^2 u) =f, $

where $ \Phi := \det D^2 \phi ~ (D^2\phi )^{-1}$ is the matrix of cofactors of $ D^2\phi $. We prove a conjecture about the relationship between $ L^p$ estimates for $ D^2 u$ and the closeness between $ \det D^2\phi $ and one. As a consequence, we obtain interior $ W^{2,p}$ estimates for solutions to such equations whenever the measure $ \det D^2\phi $ is given by a continuous density and the function $ f$ belongs to $ L^q(\Omega )$ for some $ q> \max {\{p,n\}}$.

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Additional Information

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

Truyen Nguyen
Affiliation: Department of Mathematics, The University of Akron, Akron, Ohio 44325
Email: tnguyen@uakron.edu

DOI: https://doi.org/10.1090/S0002-9947-2015-06048-6
Received by editor(s): August 13, 2012
Received by editor(s) in revised form: December 12, 2012
Published electronically: March 13, 2015
Additional Notes: The first author gratefully acknowledges the support provided by NSF grant DMS-1201401
The second author gratefully acknowledges the support provided by NSF grant DMS-0901449
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.