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On the regularity of the complex Monge-Ampère equations


Author: Weiyong He
Journal: Proc. Amer. Math. Soc. 140 (2012), 1719-1727
MSC (2010): Primary 35J60, 35J96
DOI: https://doi.org/10.1090/S0002-9939-2011-11178-X
Published electronically: September 9, 2011
MathSciNet review: 2869156
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Abstract: We shall consider the regularity of solutions for the complex Monge-Ampère equations in $ \mathbb{C}^n$ or a bounded domain. First we prove interior $ C^2$ estimates of solutions in a bounded domain for the complex Monge-Ampère equations with the assumption of an $ L^p$ bound for $ \triangle u$, $ p>n^2$, and of a Lipschitz condition on the right-hand side. Then we shall construct a family of Pogorelov-type examples for the complex Monge-Ampère equations. These examples give generalized entire solutions (as well as viscosity solutions) of the complex Monge-Ampère equation $ \det(u_{i\bar j})=1$ in $ \mathbb{C}^n$.


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Additional Information

Weiyong He
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403
Email: whe@uoregon.edu

DOI: https://doi.org/10.1090/S0002-9939-2011-11178-X
Keywords: The complex Monge-Ampère equation, regularity
Received by editor(s): June 7, 2010
Received by editor(s) in revised form: January 25, 2011
Published electronically: September 9, 2011
Additional Notes: The author is partially supported by an NSF grant.
Communicated by: Chuu-Lian Terng
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.