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

Gutiérrez, Cristian; Huang, Qingbo

doi:10.1090/S0002-9947-00-02491-0

Geometric properties of the sections of solutions to the Monge-Ampère equation
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by Cristian E. Gutiérrez and Qingbo Huang PDF

Trans. Amer. Math. Soc. 352 (2000), 4381-4396 Request permission

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.

References

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), no. 4-5, 377–381. MR 1658608, DOI 10.1007/BF02498215
A. D. Aleksandrov, Majorization of solutions of second-order linear equations, Amer. Math. Soc. Transl. 2(68) (1968), 120-143.
Luis A. Caffarelli, Boundary regularity of maps with convex potentials, Comm. Pure Appl. Math. 45 (1992), no. 9, 1141–1151. MR 1177479, DOI 10.1002/cpa.3160450905
Luis A. Caffarelli, Some regularity properties of solutions of Monge Ampère equation, Comm. Pure Appl. Math. 44 (1991), no. 8-9, 965–969. MR 1127042, DOI 10.1002/cpa.3160440809
Luis A. Caffarelli and Cristian E. Gutiérrez, Real analysis related to the Monge-Ampère equation, Trans. Amer. Math. Soc. 348 (1996), no. 3, 1075–1092. MR 1321570, DOI 10.1090/S0002-9947-96-01473-0
Luis A. Caffarelli and Cristian E. Gutiérrez, Properties of the solutions of the linearized Monge-Ampère equation, Amer. J. Math. 119 (1997), no. 2, 423–465. MR 1439555
Shiu Yuen Cheng and Shing Tung Yau, On the regularity of the Monge-Ampère equation $\textrm {det}(\partial ^{2}u/\partial x_{i}\partial sx_{j})=F(x,u)$, Comm. Pure Appl. Math. 30 (1977), no. 1, 41–68. MR 437805, DOI 10.1002/cpa.3160300104
Q. Huang, Harnack inequality for the linearized parabolic Monge-Ampère equation, Trans. Amer. Math. Soc., 351 (1999), 2025–2054.
Jeffrey Rauch and B. A. Taylor, The Dirichlet problem for the multidimensional Monge-Ampère equation, Rocky Mountain J. Math. 7 (1977), no. 2, 345–364. MR 454331, DOI 10.1216/RMJ-1977-7-2-345
Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192