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Transactions of the American Mathematical Society

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Geometric properties of the sections of solutions to the Monge-Ampère equation


Authors: Cristian E. Gutiérrez and Qingbo Huang
Journal: Trans. Amer. Math. Soc. 352 (2000), 4381-4396
MSC (1991): Primary 35J60, 35D10; Secondary 26B25
DOI: https://doi.org/10.1090/S0002-9947-00-02491-0
Published electronically: May 12, 2000
MathSciNet review: 1665332
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Abstract: In this paper we establish several geometric properties of the cross sections of generalized solutions $\phi $ to the Monge-Ampère equation $ \det D^{2}\phi = \mu $, when the measure $\mu $ satisfies a doubling property. A main result is a characterization of the doubling measures $\mu$in terms of a geometric property of the cross sections of $\phi $. This is used to obtain estimates of the shape and invariance properties of the cross sections that are valid under appropriate normalizations.


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Additional Information

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

Qingbo Huang
Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122
Address at time of publication: Department of Mathematics, University of Texas at Austin, Austin, Texas 78712
Email: qhuang@math.utexas.edu

DOI: https://doi.org/10.1090/S0002-9947-00-02491-0
Keywords: Real Monge-Amp\`{e}re equation, Aleksandrov's solutions, doubling measures, strict convexity, spaces of homogenous type
Received by editor(s): June 9, 1997
Published electronically: May 12, 2000
Additional Notes: The first author was partially supported by NSF grant DMS-9706497
Article copyright: © Copyright 2000 American Mathematical Society

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