# American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2022081
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## Asymptotic expansion of 2-dimensional gradient graph with vanishing mean curvature at infinity

 School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China

*Corresponding author

Received  November 2021 Revised  March 2022 Early access April 2022

Fund Project: The second author is supported by National Natural Science Foundation of China 11871102 and 11631002

In this paper, we establish the asymptotic expansion at infinity of gradient graph in dimension 2 with vanishing mean curvature at infinity. This corresponds to our previous results in higher dimensions and generalizes the results for minimal gradient graph on exterior domain in dimension 2. Different from the strategies for higher dimensions, instead of the equivalence of Green's function on unbounded domains, we apply a version of iteration methods from Bao–Li–Zhang [Calc.Var PDE, 52(2015), pp. 39-63] that is refined by spherical harmonic expansions to provide a more explicit asymptotic behavior than known results.

Citation: Zixiao Liu, Jiguang Bao. Asymptotic expansion of 2-dimensional gradient graph with vanishing mean curvature at infinity. Communications on Pure and Applied Analysis, doi: 10.3934/cpaa.2022081
##### References:
 [1] J. Bao, H. Li and L. Zhang, Monge-Ampère equation on exterior domains, Calc. Var. Partial Differ. Equ., 52 (2015), 39-63.  doi: 10.1007/s00526-013-0704-7. [2] A. Bhattacharya, Hessian estimates for Lagrangian mean curvature equation, Calc. Var. Partial Differ. Equ., 60 (2021), 23 pp. doi: 10.1007/s00526-021-02097-0. [3] L. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, American Mathematical Society, Providence, RI, 1995. doi: 10.1090/coll/043. [4] L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Commun. Pure Appl. Math., 42 (1989), 271-297.  doi: 10.1002/cpa.3160420304. [5] L. Caffarelli and 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. [6] E. Calabi, Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens, Michigan Math. J., 5 (1958), 105-126.  doi: 10.1307/mmj/1028998055. [7] H. Flanders, On certain functions with positive definite Hessian, Ann. Math., 71 (1960), 153-156.  doi: 10.2307/1969882. [8] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001 [9] M. Günther, Conformal normal coordinates, Ann. Global Anal. Geom., 11 (1993), 173-184.  doi: 10.1007/BF00773455. [10] Q. Han, X. Li and Y. Li, Asymptotic expansions of solutions of the Yamabe equation and the $\sigma_k$-Yamabe equation near isolated singular points, Commun. Pure Appl. Math., 74 (2021), 1915-1970.  doi: 10.1002/cpa.21943. [11] Z. C. Han, Y. Li and E. V. Teixeira, Asymptotic behavior of solutions to the $\sigma_k$-Yamabe equation near isolated singularities, Invent. Math., 182 (2010), 635-684.  doi: 10.1007/s00222-010-0274-7. [12] G. Hong, A Remark on Monge-Ampère equation over exterior domains, arXiv. 2007.12479. doi: 10.1007/s00229-019-01139-4. [13] R. Huang and Z. Wang, On the entire self-shrinking solutions to Lagrangian mean curvature flow, Calc. Var. Partial Differential Equations, 41 (2011), 321-339.  doi: 10.1007/s00526-010-0364-9. [14] K. Jörgens, Über die Lösungen der Differentialgleichung $rt-s^2 = 1$, Math. Ann., 127 (1954), 130-134.  doi: 10.1007/BF01361114. [15] N. Korevaar, R. Mazzeo, F. Pacard and R. Schoen, Refined asymptotics for constant scalar curvature metrics with isolated singularities, Invent. Math., 135 (1999), 233-272.  doi: 10.1007/s002220050285. [16] D. Li, Z. Li and Y. Yuan, A Bernstein problem for special Lagrangian equations in exterior domains, Adv. Math., 361 (2020), 106927, 29 pp. doi: 10.1016/j.aim.2019.106927. [17] Z. Liu and J. Bao, Asymptotic expansion at infinity of solutions of Monge-Ampère type equations, Nonlinear Analysis., 212 (2021), 17 pp. doi: 10.1016/j.na.2021.112450. [18] Z. Liu and J. Bao, Asymptotic expansion and optimal symmetry of minimal gradient graph equations in dimension 2, Commun. Contemp. Math., (2022), 25 pp. doi: 10.1142/S0219199721501108. [19] Z. Liu and J. Bao, Asymptotic expansion at infinity of solutions of special Lagrangian equations, J. Geom. Anal., 32 (2022), 34 pp. doi: 10.1007/s12220-021-00841-8. [20] A. V. Pogorelov, On the improper convex affine hyperspheres, Geometriae Dedicata, 1 (1972), 33-46.  doi: 10.1007/BF00147379. [21] C. Wang, R. Huang and J. Bao, On the second boundary value problem for Lagrangian mean curvature equation, arXiv: 1808.01139. [22] M. Warren, Calibrations associated to Monge-Ampère equations, Trans. Amer. Math. Soc., 362 (2010), 3947-3962.  doi: 10.1090/S0002-9947-10-05109-3. [23] M. Yan, Extension of convex function, J. Convex Anal., 21 (2014), 965-987. [24] Y. Yuan, A Bernstein problem for special Lagrangian equations, Invent. Math., 150 (2002), 117-125.  doi: 10.1007/s00222-002-0232-0. [25] Y. Yuan, Global solutions to special Lagrangian equations, Proc. Amer. Math. Soc., 134 (2006), 1355-1358.  doi: 10.1090/S0002-9939-05-08081-0.

show all references

##### References:
 [1] J. Bao, H. Li and L. Zhang, Monge-Ampère equation on exterior domains, Calc. Var. Partial Differ. Equ., 52 (2015), 39-63.  doi: 10.1007/s00526-013-0704-7. [2] A. Bhattacharya, Hessian estimates for Lagrangian mean curvature equation, Calc. Var. Partial Differ. Equ., 60 (2021), 23 pp. doi: 10.1007/s00526-021-02097-0. [3] L. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, American Mathematical Society, Providence, RI, 1995. doi: 10.1090/coll/043. [4] L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Commun. Pure Appl. Math., 42 (1989), 271-297.  doi: 10.1002/cpa.3160420304. [5] L. Caffarelli and 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. [6] E. Calabi, Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens, Michigan Math. J., 5 (1958), 105-126.  doi: 10.1307/mmj/1028998055. [7] H. Flanders, On certain functions with positive definite Hessian, Ann. Math., 71 (1960), 153-156.  doi: 10.2307/1969882. [8] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001 [9] M. Günther, Conformal normal coordinates, Ann. Global Anal. Geom., 11 (1993), 173-184.  doi: 10.1007/BF00773455. [10] Q. Han, X. Li and Y. Li, Asymptotic expansions of solutions of the Yamabe equation and the $\sigma_k$-Yamabe equation near isolated singular points, Commun. Pure Appl. Math., 74 (2021), 1915-1970.  doi: 10.1002/cpa.21943. [11] Z. C. Han, Y. Li and E. V. Teixeira, Asymptotic behavior of solutions to the $\sigma_k$-Yamabe equation near isolated singularities, Invent. Math., 182 (2010), 635-684.  doi: 10.1007/s00222-010-0274-7. [12] G. Hong, A Remark on Monge-Ampère equation over exterior domains, arXiv. 2007.12479. doi: 10.1007/s00229-019-01139-4. [13] R. Huang and Z. Wang, On the entire self-shrinking solutions to Lagrangian mean curvature flow, Calc. Var. Partial Differential Equations, 41 (2011), 321-339.  doi: 10.1007/s00526-010-0364-9. [14] K. Jörgens, Über die Lösungen der Differentialgleichung $rt-s^2 = 1$, Math. Ann., 127 (1954), 130-134.  doi: 10.1007/BF01361114. [15] N. Korevaar, R. Mazzeo, F. Pacard and R. Schoen, Refined asymptotics for constant scalar curvature metrics with isolated singularities, Invent. Math., 135 (1999), 233-272.  doi: 10.1007/s002220050285. [16] D. Li, Z. Li and Y. Yuan, A Bernstein problem for special Lagrangian equations in exterior domains, Adv. Math., 361 (2020), 106927, 29 pp. doi: 10.1016/j.aim.2019.106927. [17] Z. Liu and J. Bao, Asymptotic expansion at infinity of solutions of Monge-Ampère type equations, Nonlinear Analysis., 212 (2021), 17 pp. doi: 10.1016/j.na.2021.112450. [18] Z. Liu and J. Bao, Asymptotic expansion and optimal symmetry of minimal gradient graph equations in dimension 2, Commun. Contemp. Math., (2022), 25 pp. doi: 10.1142/S0219199721501108. [19] Z. Liu and J. Bao, Asymptotic expansion at infinity of solutions of special Lagrangian equations, J. Geom. Anal., 32 (2022), 34 pp. doi: 10.1007/s12220-021-00841-8. [20] A. V. Pogorelov, On the improper convex affine hyperspheres, Geometriae Dedicata, 1 (1972), 33-46.  doi: 10.1007/BF00147379. [21] C. Wang, R. Huang and J. Bao, On the second boundary value problem for Lagrangian mean curvature equation, arXiv: 1808.01139. [22] M. Warren, Calibrations associated to Monge-Ampère equations, Trans. Amer. Math. Soc., 362 (2010), 3947-3962.  doi: 10.1090/S0002-9947-10-05109-3. [23] M. Yan, Extension of convex function, J. Convex Anal., 21 (2014), 965-987. [24] Y. Yuan, A Bernstein problem for special Lagrangian equations, Invent. Math., 150 (2002), 117-125.  doi: 10.1007/s00222-002-0232-0. [25] Y. Yuan, Global solutions to special Lagrangian equations, Proc. Amer. Math. Soc., 134 (2006), 1355-1358.  doi: 10.1090/S0002-9939-05-08081-0.
 [1] Zhijun Zhang. Optimal global asymptotic behavior of the solution to a singular monge-ampère equation. Communications on Pure and Applied Analysis, 2020, 19 (2) : 1129-1145. doi: 10.3934/cpaa.2020053 [2] Qi-Rui Li, Xu-Jia Wang. Regularity of the homogeneous Monge-Ampère equation. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 6069-6084. doi: 10.3934/dcds.2015.35.6069 [3] Luca Codenotti, Marta Lewicka. Visualization of the convex integration solutions to the Monge-Ampère equation. Evolution Equations and Control Theory, 2019, 8 (2) : 273-300. doi: 10.3934/eect.2019015 [4] Bo Guan, Qun Li. A Monge-Ampère type fully nonlinear equation on Hermitian manifolds. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1991-1999. doi: 10.3934/dcdsb.2012.17.1991 [5] Alessio Figalli, Young-Heon Kim. Partial regularity of Brenier solutions of the Monge-Ampère equation. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 559-565. doi: 10.3934/dcds.2010.28.559 [6] Adam M. Oberman. Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian. Discrete and Continuous Dynamical Systems - B, 2008, 10 (1) : 221-238. doi: 10.3934/dcdsb.2008.10.221 [7] Shuyu Gong, Ziwei Zhou, Jiguang Bao. Existence and uniqueness of viscosity solutions to the exterior problem of a parabolic Monge-Ampère equation. Communications on Pure and Applied Analysis, 2020, 19 (10) : 4921-4936. doi: 10.3934/cpaa.2020218 [8] Diego Maldonado. On interior $C^2$-estimates for the Monge-Ampère equation. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1427-1440. doi: 10.3934/dcds.2018058 [9] Haitao Yang, Yibin Chang. On the blow-up boundary solutions of the Monge -Ampére equation with singular weights. Communications on Pure and Applied Analysis, 2012, 11 (2) : 697-708. doi: 10.3934/cpaa.2012.11.697 [10] Daehwan Kim, Juncheol Pyo. Existence and asymptotic behavior of helicoidal translating solitons of the mean curvature flow. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5897-5919. doi: 10.3934/dcds.2018256 [11] Cristian Enache. Maximum and minimum principles for a class of Monge-Ampère equations in the plane, with applications to surfaces of constant Gauss curvature. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1347-1359. doi: 10.3934/cpaa.2014.13.1347 [12] Jan Haskovec, Nader Masmoudi, Christian Schmeiser, Mohamed Lazhar Tayeb. The Spherical Harmonics Expansion model coupled to the Poisson equation. Kinetic and Related Models, 2011, 4 (4) : 1063-1079. doi: 10.3934/krm.2011.4.1063 [13] Thierry Paul, Mario Pulvirenti. Asymptotic expansion of the mean-field approximation. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 1891-1921. doi: 10.3934/dcds.2019080 [14] Minkyu Kwak, Kyong Yu. The asymptotic behavior of solutions of a semilinear parabolic equation. Discrete and Continuous Dynamical Systems, 1996, 2 (4) : 483-496. doi: 10.3934/dcds.1996.2.483 [15] Carmen Cortázar, Manuel Elgueta, Fernando Quirós, Noemí Wolanski. Asymptotic behavior for a nonlocal diffusion equation on the half line. Discrete and Continuous Dynamical Systems, 2015, 35 (4) : 1391-1407. doi: 10.3934/dcds.2015.35.1391 [16] Yongqin Liu. Asymptotic behavior of solutions to a nonlinear plate equation with memory. Communications on Pure and Applied Analysis, 2017, 16 (2) : 533-556. doi: 10.3934/cpaa.2017027 [17] Shota Sato, Eiji Yanagida. Asymptotic behavior of singular solutions for a semilinear parabolic equation. Discrete and Continuous Dynamical Systems, 2012, 32 (11) : 4027-4043. doi: 10.3934/dcds.2012.32.4027 [18] José Manuel Palacios. Orbital and asymptotic stability of a train of peakons for the Novikov equation. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2475-2518. doi: 10.3934/dcds.2020372 [19] Dominika Pilarczyk. Asymptotic stability of singular solution to nonlinear heat equation. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 991-1001. doi: 10.3934/dcds.2009.25.991 [20] Alexander Komech, Elena Kopylova, David Stuart. On asymptotic stability of solitons in a nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1063-1079. doi: 10.3934/cpaa.2012.11.1063

2021 Impact Factor: 1.273