doi: 10.3934/cpaa.2022081
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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. BaoH. 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. CaffarelliB. 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. HanX. 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. HanY. 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. KorevaarR. MazzeoF. 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. BaoH. 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. CaffarelliB. 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. HanX. 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. HanY. 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. KorevaarR. MazzeoF. 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.

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