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Regularity of weak solutions to the Monge-Ampère equation

Authors: Cristian E. Gutiérrez and David Hartenstine
Journal: Trans. Amer. Math. Soc. 355 (2003), 2477-2500
MSC (2000): Primary 35D10, 35J65, 35J60
Published electronically: January 14, 2003
MathSciNet review: 1973999
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Abstract: We study the properties of generalized solutions to the Monge-Ampère equation $\det D^2 u = \nu$, where the Borel measure $\nu$ satisfies a condition, introduced by Jerison, that is weaker than the doubling property. When $\nu = f \, dx$, this condition, which we call $D_{\epsilon}$, admits the possibility of $f$ vanishing or becoming infinite. Our analysis extends the regularity theory (due to Caffarelli) available when $0 < \lambda \leq f \leq \Lambda < \infty$, which implies that $\nu = f \, dx$is doubling. The main difference between the $D_{\epsilon}$ case and the case when $f$ is bounded between two positive constants is the need to use a variant of the Aleksandrov maximum principle (due to Jerison) and some tools from convex geometry, in particular the Hausdorff metric.

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Additional Information

Cristian E. Gutiérrez
Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122

David Hartenstine
Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112

Keywords: Aleksandrov solutions, strict convexity, Hausdorff metric, doubling property, H\"older estimates
Received by editor(s): March 3, 2002
Received by editor(s) in revised form: October 7, 2002
Published electronically: January 14, 2003
Additional Notes: The first author was partially supported by NSF grant DMS–0070648.
Article copyright: © Copyright 2003 American Mathematical Society

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