# American Institute of Mathematical Sciences

May  2019, 18(3): 1247-1259. doi: 10.3934/cpaa.2019060

## Symmetry of solutions to a class of Monge-Ampère equations

 Department of Mathematics, Tsinghua University, Beijing 100084, China

Received  April 2018 Revised  April 2018 Published  November 2018

Fund Project: The second author is supported by NSFC 11771237 and 41390452.

We study the symmetry of solutions to a class of Monge-Ampère type equations from a few geometric problems. We use a new transform to analyze the asymptotic behavior of the solutions near the infinity. By this and a moving plane method, we prove the radially symmetry of the solutions.

Citation: Fan Cui, Huaiyu Jian. Symmetry of solutions to a class of Monge-Ampère equations. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1247-1259. doi: 10.3934/cpaa.2019060
##### References:
 [1] H. Berestycki and L. Nirenberg, Monotonicity, symmetry and anti-symmetry of solutions of semilinear elliptic equations, J. Geom. Phys., 5 (1988), 237-275.  doi: 10.1016/0393-0440(88)90006-X. [2] H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brasil. Mat. (N.S.), 22 (1991), 1-37.  doi: 10.1007/BF01244896. [3] E. Calabi, Improper affine hypersurfaces of convex type and a generalization of a theorem by K. Jorgens, Michigan Math. J., 5 (1958), 105-126. [4] L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. pure. Appl. Math., 42 (1989), 271-297.  doi: 10.1002/cpa.3160420304. [5] X. Chen and H. Y. Jian, The radial solutions of Monge-Ampère equations and the semi-geostrophic system, Adv. Nonlinear Stud., 5 (2005), 587-600.  doi: 10.1515/ans-2005-0407. [6] W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.  doi: 10.1215/S0012-7094-91-06325-8. [7] L. Caffarelli, Y. Y. Li and L. Nirenberg, Some remarks on singular solutions of nonlinear elliptic equations. I, J. Fixed Point Theory Appl., 5 (2009), 353-395.  doi: 10.1007/s11784-009-0107-8. [8] S. Y. Cheng and S. T. Yau, On the regularity of the Monge-Ampere equation $\det ((\partial^2u/\partial x^ix^j)) = F(x,u)$, Comm. Pure Appl. Math., 30 (1977), 41-68.  doi: 10.1002/cpa.3160300104. [9] K. S. Chou and X. J. Wang, The Lp-Minkowski problem and the Minkowski problem in centroaffine geometry, Adv. Math., 205 (2006), 33-83.  doi: 10.1016/j.aim.2005.07.004. [10] K. S. Chou and X. J. Wang, Minkowski problems for complete noncompact convex hypersurfaces, Topol. Methods Nonlinear Anal., 6 (1995), 151-162.  doi: 10.12775/TMNA.1995.037. [11] M. Dou, A direct method of moving planes for fractorial Laplacian equations in the unit ball, Comm. pure Appl. Anal., 15 (2016), 1797-1807.  doi: 10.3934/cpaa.2016015. [12] L. Damascelli, F. Pacella and M. Ramaswamy, Symmetry of ground states of p-Laplace equations via the moving plane method, Arch. Rat. Mech. Anal., 148 (1999), 291-308.  doi: 10.1007/s002050050163. [13] B. Franchi and E. Lanconelli, Radial symmetry of the ground states for a class of quasilinear elliptic equations, in Nonlinear Diffusion Equations and Their Equilibrium States (eds. W.-M. Ni, L. A. Peletier and James Serrin), Springer-Verlag, (1988), 287–292. doi: 10.1007/978-1-4613-9605-5_17. [14] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and relatedproperties via the maximum principle, Commun. Math. Phys., 68 (1979), 209-243. [15] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1997. [16] K. Jorgens, Uber die Losunger der Differentialgeichung rt-s2 = 1, Math. Anna., 127 (1954), 130-134.  doi: 10.1007/BF01361114. [17] H. Y. Jian and Y. Li, Optimal boundary regularity for a Singular Monge-Ampère equation, Journal of Differential Equations, 264 (2018), 6873-6890.  doi: 10.1016/j.jde.2018.01.051. [18] H. Y. Jian, J. Lu and X.-J. Wang, Nonuniqueness of solutions to the LP-Minkowski problem, Adv. Math., 281 (2015), 845-856.  doi: 10.1016/j.aim.2015.05.010. [19] H. Y. Jian, J. Lu and X.-J. Wang, A priori estimates and existences of solutions to the prescribed centroaffine curvature problem, J. Funct. Anal., 274 (2018), 826-862.  doi: 10.1016/j.jfa.2017.08.024. [20] H. Y. Jian, J. Lu and G. Zhang, Mirror symmetric solutions to the cetro-affine Minkowski prblem, Calc. Var. Partial Differential Equations, 55 (2016).  doi: 10.1007/s00526-016-0976-9. [21] H. Y. Jian and X.-J. Wang, Bernstein theorem and regularity for a class of Monge-Ampère equation, J. Diff. Geom., 93 (2013), 431-469. [22] H. Y. Jian and X.-J. Wang, Existence of entire solutions to the Monge-Ampère equation, Amer. J. Math., 136 (2014), 1093-1106.  doi: 10.1353/ajm.2014.0029. [23] H. Y. Jian, X.-J. Wang and Y. W. Zhao, Global smoothness for a singular Monge-Ampère equation, Journal of Differential Equations, 263 (2017), 7250-7262.  doi: 10.1016/j.jde.2017.08.004. [24] C. Li, Monotonocity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains, Commu. Partial Differ. Equations, 16 (1991), 491-526.  doi: 10.1080/03605309108820766. [25] Y. Li and W.-M. Ni, On the asymptotic behavior and radial symmetry of positive solutions of semilinear elliptic equations in $R^n$ II. Radial symmetry, Arch. Rat. Mech. Anal., 118 (1992), 223-243.  doi: 10.1007/BF00387896. [26] Y. Li and W.-M. Ni, Radial symmetry of positive solutions of nonlinear elliptic equations in Rn, Comm. Part. Diff. Eqs., 189 (1993), 104-397.  doi: 10.1080/03605309308820960. [27] E. Lutwak, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Diff. Geom., 38 (1993), 131-150. [28] A. V. Pogorelov, The Minkowski Multidimensional Problem, J. Wiley, New York, 1978. [29] J. Serrin and H.-H. Zou, Symmetry of Ground states of quasilinear elliptic equations, Arch. Rat. Mech. Anal., 148 (1999), 265-290.  doi: 10.1007/s002050050162. [30] J. Urbas, Complete noncompact self-similar solutions of Gauss curvature flows I. Positive powers, Math. Ann., 311 (1998), 251-274.  doi: 10.1007/s002080050187.

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##### References:
 [1] H. Berestycki and L. Nirenberg, Monotonicity, symmetry and anti-symmetry of solutions of semilinear elliptic equations, J. Geom. Phys., 5 (1988), 237-275.  doi: 10.1016/0393-0440(88)90006-X. [2] H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brasil. Mat. (N.S.), 22 (1991), 1-37.  doi: 10.1007/BF01244896. [3] E. Calabi, Improper affine hypersurfaces of convex type and a generalization of a theorem by K. Jorgens, Michigan Math. J., 5 (1958), 105-126. [4] L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. pure. Appl. Math., 42 (1989), 271-297.  doi: 10.1002/cpa.3160420304. [5] X. Chen and H. Y. Jian, The radial solutions of Monge-Ampère equations and the semi-geostrophic system, Adv. Nonlinear Stud., 5 (2005), 587-600.  doi: 10.1515/ans-2005-0407. [6] W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.  doi: 10.1215/S0012-7094-91-06325-8. [7] L. Caffarelli, Y. Y. Li and L. Nirenberg, Some remarks on singular solutions of nonlinear elliptic equations. I, J. Fixed Point Theory Appl., 5 (2009), 353-395.  doi: 10.1007/s11784-009-0107-8. [8] S. Y. Cheng and S. T. Yau, On the regularity of the Monge-Ampere equation $\det ((\partial^2u/\partial x^ix^j)) = F(x,u)$, Comm. Pure Appl. Math., 30 (1977), 41-68.  doi: 10.1002/cpa.3160300104. [9] K. S. Chou and X. J. Wang, The Lp-Minkowski problem and the Minkowski problem in centroaffine geometry, Adv. Math., 205 (2006), 33-83.  doi: 10.1016/j.aim.2005.07.004. [10] K. S. Chou and X. J. Wang, Minkowski problems for complete noncompact convex hypersurfaces, Topol. Methods Nonlinear Anal., 6 (1995), 151-162.  doi: 10.12775/TMNA.1995.037. [11] M. Dou, A direct method of moving planes for fractorial Laplacian equations in the unit ball, Comm. pure Appl. Anal., 15 (2016), 1797-1807.  doi: 10.3934/cpaa.2016015. [12] L. Damascelli, F. Pacella and M. Ramaswamy, Symmetry of ground states of p-Laplace equations via the moving plane method, Arch. Rat. Mech. Anal., 148 (1999), 291-308.  doi: 10.1007/s002050050163. [13] B. Franchi and E. Lanconelli, Radial symmetry of the ground states for a class of quasilinear elliptic equations, in Nonlinear Diffusion Equations and Their Equilibrium States (eds. W.-M. Ni, L. A. Peletier and James Serrin), Springer-Verlag, (1988), 287–292. doi: 10.1007/978-1-4613-9605-5_17. [14] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and relatedproperties via the maximum principle, Commun. Math. Phys., 68 (1979), 209-243. [15] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1997. [16] K. Jorgens, Uber die Losunger der Differentialgeichung rt-s2 = 1, Math. Anna., 127 (1954), 130-134.  doi: 10.1007/BF01361114. [17] H. Y. Jian and Y. Li, Optimal boundary regularity for a Singular Monge-Ampère equation, Journal of Differential Equations, 264 (2018), 6873-6890.  doi: 10.1016/j.jde.2018.01.051. [18] H. Y. Jian, J. Lu and X.-J. Wang, Nonuniqueness of solutions to the LP-Minkowski problem, Adv. Math., 281 (2015), 845-856.  doi: 10.1016/j.aim.2015.05.010. [19] H. Y. Jian, J. Lu and X.-J. Wang, A priori estimates and existences of solutions to the prescribed centroaffine curvature problem, J. Funct. Anal., 274 (2018), 826-862.  doi: 10.1016/j.jfa.2017.08.024. [20] H. Y. Jian, J. Lu and G. Zhang, Mirror symmetric solutions to the cetro-affine Minkowski prblem, Calc. Var. Partial Differential Equations, 55 (2016).  doi: 10.1007/s00526-016-0976-9. [21] H. Y. Jian and X.-J. Wang, Bernstein theorem and regularity for a class of Monge-Ampère equation, J. Diff. Geom., 93 (2013), 431-469. [22] H. Y. Jian and X.-J. Wang, Existence of entire solutions to the Monge-Ampère equation, Amer. J. Math., 136 (2014), 1093-1106.  doi: 10.1353/ajm.2014.0029. [23] H. Y. Jian, X.-J. Wang and Y. W. Zhao, Global smoothness for a singular Monge-Ampère equation, Journal of Differential Equations, 263 (2017), 7250-7262.  doi: 10.1016/j.jde.2017.08.004. [24] C. Li, Monotonocity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains, Commu. Partial Differ. Equations, 16 (1991), 491-526.  doi: 10.1080/03605309108820766. [25] Y. Li and W.-M. Ni, On the asymptotic behavior and radial symmetry of positive solutions of semilinear elliptic equations in $R^n$ II. Radial symmetry, Arch. Rat. Mech. Anal., 118 (1992), 223-243.  doi: 10.1007/BF00387896. [26] Y. Li and W.-M. Ni, Radial symmetry of positive solutions of nonlinear elliptic equations in Rn, Comm. Part. Diff. Eqs., 189 (1993), 104-397.  doi: 10.1080/03605309308820960. [27] E. Lutwak, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Diff. Geom., 38 (1993), 131-150. [28] A. V. Pogorelov, The Minkowski Multidimensional Problem, J. Wiley, New York, 1978. [29] J. Serrin and H.-H. Zou, Symmetry of Ground states of quasilinear elliptic equations, Arch. Rat. Mech. Anal., 148 (1999), 265-290.  doi: 10.1007/s002050050162. [30] J. Urbas, Complete noncompact self-similar solutions of Gauss curvature flows I. Positive powers, Math. Ann., 311 (1998), 251-274.  doi: 10.1007/s002080050187.
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