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

Published electronically:
May 12, 2000

MathSciNet review:
1665332

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we establish several geometric properties of the cross sections of generalized solutions to the Monge-Ampère equation , when the measure satisfies a doubling property. A main result is a characterization of the doubling measures in terms of a geometric property of the cross sections of . 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:
http://dx.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