# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2020331

## A Liouville theorem of parabolic Monge-AmpÈre equations in half-space

 1 School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China 2 School of Mathematics and Statistics, Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing 100081, China

* Corresponding author: Bo Wang

Received  February 2020 Revised  June 2020 Published  September 2020

Fund Project: The first and second author are partially supported by NSFC 11871102 and 11631002. The third author is partially supported by NSFC 11701027 and Beijing Institute of Technology Research Fund Program for Young Scholars

In this paper, we establish the gradient and second derivative estimates for solutions to two kinds of parabolic Monge-Ampère equations in half-space under certain boundary data and growth condition. We also use such estimates to obtain the Liouville theorems for these two kinds of parabolic Monge-Ampère equations and one kind of elliptic Monge-Ampère equation.

Citation: Ziwei Zhou, Jiguang Bao, Bo Wang. A Liouville theorem of parabolic Monge-AmpÈre equations in half-space. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020331
##### References:
 [1] J. Bao, H. Li and L. Zhang, Monge-Ampère equation on exterior domains, Calc. Var. Partial Differential Equations, 52 (2015), 39-63.  doi: 10.1007/s00526-013-0704-7.  Google Scholar [2] L. Caffarelli, Topics in PDEs: The Monge-Ampère Equation, Graduate course, Courant Institute, New York University, 1995. Google Scholar [3] L. Caffarelli and Y. Y. Li, An extension to a theorem of Jörgens, Calabi, and Pogorelov, Commun. Pure Appl. Math., 56 (2003), 549-583.  doi: 10.1002/cpa.10067.  Google Scholar [4] E. Calabi, Improper affine hyperspheres of convex type and a generalization of a theorem by K.Jörgens, Mich. Math. J., 5 (1958), 105-126.  doi: 10.1307/mmj/1028998055.  Google Scholar [5] S. Y. Cheng and S. T. Yau, Complete affine hypersurfaces Ⅰ. The completeness of affine metrics, Commun. Pure Appl. Math., 39 (1986), 839-866.  doi: 10.1002/cpa.3160390606.  Google Scholar [6] C. E. Gutiérrez and Q. Huang, Geometric properties of the sections of solutions to the Monge-Ampère equation, Trans. Amer. Math. Soc., 352 (2000), 4381-4396.  doi: 10.1090/S0002-9947-00-02491-0.  Google Scholar [7] C. E. Gutiérrez and Q. Huang, A generalization of a theorem by Calabi to the parabolic Monge-Ampère equation, Indiana Univ. Math. J., 47 (1998), 1459-1480.  doi: 10.1512/iumj.1998.47.1563.  Google Scholar [8] X. Jia, D. Li and Z. Li, Asymptotic behavior at infinity of solutions of Monge-Ampère equations in half spaces, J. Differential Equations, 269 (2020), 326–348, arXiv: 1808.02643. doi: 10.1016/j.jde.2019.12.007.  Google Scholar [9] K. Jörgens, Über die Lösungen der Differentialgleichung $rt-s^2 = 1$, Math. Ann., 127 (1954), 130-134.  doi: 10.1007/BF01361114.  Google Scholar [10] J. Jost and Y. L. Xin, Some aspects of the global geometry of entire space-like submanifolds, Dedicated to Shiing-Shen Chern on His 90th Birthday, Results Math., 40 (2001), 233-245.  doi: 10.1007/BF03322708.  Google Scholar [11] N. V. Krylov, Sequences of convex functions and estimates of the maximum of the solution of a parabolic equation, (Russian) Sibirsk. Mat. Ž., 17 (1976), 290–303.  Google Scholar [12] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific. 1996. doi: 10.1142/3302.  Google Scholar [13] A. V. Pogorelov, On the improper affine hyperspheres, Geom. Dedic., 1 (1972), 33-46.  doi: 10.1007/BF00147379.  Google Scholar [14] O. Savin, Pointwise $C^{2, \alpha}$ estimates at the boundary for the Monge-Ampère equation, J. Amer. Math. Soc., 26 (2013), 63-99.  doi: 10.1090/S0894-0347-2012-00747-4.  Google Scholar [15] O. Savin, A localization theorem and boundary regularity for a class of degenerate Monge-Ampère equations, J. Differential Equations, 256 (2014), 327-388.  doi: 10.1016/j.jde.2013.08.019.  Google Scholar [16] K. Tso, Deforming a hypersurface by its Gauss-Kronecker curvature, Comm.pure Appl.math, 38 (1985), 867-882.  doi: 10.1002/cpa.3160380615.  Google Scholar [17] B. Wang and J. Bao, Asymptotic behavior on a kind of parabolic Monge-Ampère equation, J. Differential Equations, 259 (2015), 344-370.  doi: 10.1016/j.jde.2015.02.029.  Google Scholar [18] R. Wang and G. Wang, On existence, uniqueness and regularity of viscosity solutions for the first initial boundary value problems to parabolic Monge-Ampère equation, Northeast. Math. J., 8 (1992), 417-446.   Google Scholar [19] R. Wang and G. Wang, The geometric measure theoretical characterization of viscosity solutions to parabolic Monge-Ampère type equation, J. Partial Diff. Eqs., 6 (1993), 237-254.   Google Scholar [20] R. Wang and G. Wang, On another kind of parabolic Monge-Ampère equation: The existence, uniqueness and regularity of the viscosity solution, Northeastern Mathematical Journal, 10 (1994), 434-454.   Google Scholar [21] J. Xiong and J. Bao, On Jögens, Calabi, and Pogorelov type theorem and isolated singularities of parabolic Monge-Ampère equations, J. Differ. Equ., 250 (2011), 367-385.  doi: 10.1016/j.jde.2010.08.024.  Google Scholar [22] W. Zhang, J. Bao and B. Wang, An extension of Jörgens-Calabi-Pogorelov theorem to parabolic Monge-Ampère equation, Calc. Var. Partial Differential Equations, 57 (2018), Paper No. 90, 36 pp. doi: 10.1007/s00526-018-1363-5.  Google Scholar

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##### References:
 [1] J. Bao, H. Li and L. Zhang, Monge-Ampère equation on exterior domains, Calc. Var. Partial Differential Equations, 52 (2015), 39-63.  doi: 10.1007/s00526-013-0704-7.  Google Scholar [2] L. Caffarelli, Topics in PDEs: The Monge-Ampère Equation, Graduate course, Courant Institute, New York University, 1995. Google Scholar [3] L. Caffarelli and Y. Y. Li, An extension to a theorem of Jörgens, Calabi, and Pogorelov, Commun. Pure Appl. Math., 56 (2003), 549-583.  doi: 10.1002/cpa.10067.  Google Scholar [4] E. Calabi, Improper affine hyperspheres of convex type and a generalization of a theorem by K.Jörgens, Mich. Math. J., 5 (1958), 105-126.  doi: 10.1307/mmj/1028998055.  Google Scholar [5] S. Y. Cheng and S. T. Yau, Complete affine hypersurfaces Ⅰ. The completeness of affine metrics, Commun. Pure Appl. Math., 39 (1986), 839-866.  doi: 10.1002/cpa.3160390606.  Google Scholar [6] C. E. Gutiérrez and Q. Huang, Geometric properties of the sections of solutions to the Monge-Ampère equation, Trans. Amer. Math. Soc., 352 (2000), 4381-4396.  doi: 10.1090/S0002-9947-00-02491-0.  Google Scholar [7] C. E. Gutiérrez and Q. Huang, A generalization of a theorem by Calabi to the parabolic Monge-Ampère equation, Indiana Univ. Math. J., 47 (1998), 1459-1480.  doi: 10.1512/iumj.1998.47.1563.  Google Scholar [8] X. Jia, D. Li and Z. Li, Asymptotic behavior at infinity of solutions of Monge-Ampère equations in half spaces, J. Differential Equations, 269 (2020), 326–348, arXiv: 1808.02643. doi: 10.1016/j.jde.2019.12.007.  Google Scholar [9] K. Jörgens, Über die Lösungen der Differentialgleichung $rt-s^2 = 1$, Math. Ann., 127 (1954), 130-134.  doi: 10.1007/BF01361114.  Google Scholar [10] J. Jost and Y. L. Xin, Some aspects of the global geometry of entire space-like submanifolds, Dedicated to Shiing-Shen Chern on His 90th Birthday, Results Math., 40 (2001), 233-245.  doi: 10.1007/BF03322708.  Google Scholar [11] N. V. Krylov, Sequences of convex functions and estimates of the maximum of the solution of a parabolic equation, (Russian) Sibirsk. Mat. Ž., 17 (1976), 290–303.  Google Scholar [12] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific. 1996. doi: 10.1142/3302.  Google Scholar [13] A. V. Pogorelov, On the improper affine hyperspheres, Geom. Dedic., 1 (1972), 33-46.  doi: 10.1007/BF00147379.  Google Scholar [14] O. Savin, Pointwise $C^{2, \alpha}$ estimates at the boundary for the Monge-Ampère equation, J. Amer. Math. Soc., 26 (2013), 63-99.  doi: 10.1090/S0894-0347-2012-00747-4.  Google Scholar [15] O. Savin, A localization theorem and boundary regularity for a class of degenerate Monge-Ampère equations, J. Differential Equations, 256 (2014), 327-388.  doi: 10.1016/j.jde.2013.08.019.  Google Scholar [16] K. Tso, Deforming a hypersurface by its Gauss-Kronecker curvature, Comm.pure Appl.math, 38 (1985), 867-882.  doi: 10.1002/cpa.3160380615.  Google Scholar [17] B. Wang and J. Bao, Asymptotic behavior on a kind of parabolic Monge-Ampère equation, J. Differential Equations, 259 (2015), 344-370.  doi: 10.1016/j.jde.2015.02.029.  Google Scholar [18] R. Wang and G. Wang, On existence, uniqueness and regularity of viscosity solutions for the first initial boundary value problems to parabolic Monge-Ampère equation, Northeast. Math. J., 8 (1992), 417-446.   Google Scholar [19] R. Wang and G. Wang, The geometric measure theoretical characterization of viscosity solutions to parabolic Monge-Ampère type equation, J. Partial Diff. Eqs., 6 (1993), 237-254.   Google Scholar [20] R. Wang and G. Wang, On another kind of parabolic Monge-Ampère equation: The existence, uniqueness and regularity of the viscosity solution, Northeastern Mathematical Journal, 10 (1994), 434-454.   Google Scholar [21] J. Xiong and J. Bao, On Jögens, Calabi, and Pogorelov type theorem and isolated singularities of parabolic Monge-Ampère equations, J. Differ. Equ., 250 (2011), 367-385.  doi: 10.1016/j.jde.2010.08.024.  Google Scholar [22] W. Zhang, J. Bao and B. Wang, An extension of Jörgens-Calabi-Pogorelov theorem to parabolic Monge-Ampère equation, Calc. Var. Partial Differential Equations, 57 (2018), Paper No. 90, 36 pp. doi: 10.1007/s00526-018-1363-5.  Google Scholar
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