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 | 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.

**[A-F-T]**H. Aimar, L. Forzani and R. Toledano,*Balls and quasi-metrics: a space of homogeneous type modeling the real analysis related to the Monge-Ampère equation*, J. Fourier Anal. Appl.**4**, (1998), 377-381.MR**99j:35043****[A]**A. D. Aleksandrov,*Majorization of solutions of second-order linear equations*, Amer. Math. Soc. Transl.**2(68)**(1968), 120-143.**[C1]**L. A. Caffarelli,*Boundary regularity of maps with convex potentials*, Comm. on Pure and Appl. Math.**45**(1992), 1141-1151. MR**93k:35054****[C2]**-,*Some regularity properties of solutions of Monge-Ampère equation*, Comm. on Pure and Appl. Math.**44**(1991), 965-969.MR**92h:35088****[C-G1]**-,*Real analysis related to the Monge-Ampère equation*, Trans. Amer. Math. Soc.**348**(1996), 1075-1092. MR**96h:35047****[C-G2]**-,*Properties of the solutions of the linearized Monge-Ampère equation*, American J. of Math.**119(2)**(1997), 423-465.MR**98e:35060****[Ch-Y]**Skiu Y. Cheng and Shing T. Yau,*On regularity of the Monge-Ampère equation*, 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****[R-T]**J. Rauch and B. A. Taylor,*The Dirichlet problem for the multidimensional Monge-Ampère equation*, Rocky Mountain J. of Math.**7(2)**(1977), 345-364. MR**56:12582****[S]**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:
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