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Geometric properties of the sections of solutions to the Monge-Ampère equation
Author(s):
Cristian
E.
Gutiérrez;
Qingbo
Huang
Journal:
Trans. Amer. Math. Soc.
352
(2000),
4381-4396.
MSC (1991):
Primary 35J60, 35D10;
Secondary 26B25
Posted:
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  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.
References:
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- A. D. Aleksandrov, Majorization of solutions of second-order linear equations, Amer. Math. Soc. Transl. 2(68) (1968), 120-143.
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- -, Properties of the solutions of the linearized Monge-Ampère equation, American J. of Math. 119(2) (1997), 423-465.MR 98e:35060
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 , Comm. on Pure and App. Math. 30 (1977), 41-68.MR 55:10727 - [H]
- Q. Huang, Harnack inequality for the linearized parabolic Monge-Ampère equation, Trans. Amer. Math. Soc., 351 (1999), 2025-2054. CMP 97:17
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- E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Math. Series 43, Princeton Univ. Press, Princeton, NJ, 1993. MR 95c:42002
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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:
10.1090/S0002-9947-00-02491-0
PII:
S 0002-9947(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
Posted:
May 12, 2000
Additional Notes:
The first author was partially supported by NSF grant DMS-9706497
Copyright of article:
Copyright
2000,
American Mathematical Society
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