February  2020, 19(2): 1129-1145. doi: 10.3934/cpaa.2020053

Optimal global asymptotic behavior of the solution to a singular monge-ampère equation

School of Mathematics and Information Science, Yantai University, Yantai 264005, Shandong, China

Received  April 2019 Revised  June 2019 Published  October 2019

Fund Project: The author is supported by NSF grant of P. R. China under grant 11571295.

This paper is mainly concerned with the optimal global asymptotic behavior of the unique convex solution to a singular Dirichlet problem for the Monge-Ampère equation $ {\rm det} \ D^2 u = b(x)g(-u), \ u<0, \ x \in \Omega, \ u|_{\partial \Omega} = 0, $ where $ \Omega $ is a strict convex and bounded smooth domain in $ \mathbb R^n $ with $ n\geq 2 $, $ g\in C^1((0,\infty)) $ is positive and decreasing in $ (0, \infty) $ with $ \lim_{s \rightarrow 0^+}g(s) = \infty $, $ b \in C^{\infty}(\Omega) $ is positive in $ \Omega $, but may vanish or blow up on the boundary properly. Our approach is based on the construction of suitable sub- and super-solutions.

Citation: 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
References:
[1]

N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Encyclopediaof Mathematics and its Applications 27, Cambridge University Press, 1987.

[2]

E. Calabi, Complete affine hypersurfaces I, Symposia Math., 10 (1972), 19-38. 

[3]

S. Y. Cheng and S.-T. Yau, On the regularity of the Monge-Ampère equation det $(({\partial^2u}/{\partial x^i\partial x^j})) = F(x, u)$, Comm. Pure Appl. Math., 30 (1977), 41-68. 

[4]

S. Y. Cheng and S.-T. Yau, Complete affine hypersurfaces Ⅰ: The completeness of affine metrics, Comm. Pure Appl. Math., 39 (1986), 839-866. 

[5]

Kai-Seng Chou and Xu-Jia Wang, The $L_p$-Minkowski problem and the Minkowski problem in centroaffine geometry, Adv. Math., 205 (2006), 33-83. 

[6]

F.-C. Cȋrstea and C. Trombetti, On the Monge-Ampère equation with boundary blow-up: existence, muniqueness and asymptotics, Cal. Var. Partial Diff. Equations, 31 (2008), 167-186.  doi: 10.1007/s00526-007-0108-7.

[7]

F. CuiH. Y. Jian and Y. Li, Boundary Hölder estimates for nonlinear singular elliptic equations, J. Math. Anal. Appl., 470 (2019), 1185-1193.  doi: 10.1016/j.jmaa.2018.10.059.

[8]

H. Y. Jian and Xu-Jia Wang, Bernstein theorem and regularity for a class of Monge-Ampère equation, J. Diff. Geom., 93 (2013), 431-469. 

[9]

H. Y. Jian and Xu-Jia Wang, Optimal boundary regularity for nonlinear singular elliptic equations, Adv. Math., 251 (2014), 111-126.  doi: 10.1016/j.aim.2013.10.009.

[10]

H. Y. JianXu-Jia Wang and Y. W. Zhao, Global smoothness for a singular Monge-Ampère equation, J. Diff. Equations, 263 (2017), 7250-7262.  doi: 10.1016/j.jde.2017.08.004.

[11]

H. Y. Jian and Y. Li, Optimal boundary regularity for a singular Monge-Ampère equation, J. Diff. Equations, 264 (2018), 6873-6890.  doi: 10.1016/j.jde.2018.01.051.

[12]

A. C. Lazer and P. J. McKenna, On singular boundary value problems for the Monge-Ampère Operator, J. Math. Anal. Appl., 197 (1996), 341-362. 

[13]

A. C. Lazer and P. J. McKenna, On a singular elliptic boundary value problem, Proc. Amer. Math. Soc., 111 (1991), 721-730. 

[14]

D. S. Li and S. S. Ma, Boundary behavior of solutions of Monge-Ampère equations with singular righthand sides, J. Math. Anal. Appl., 454 (2017), 79-93.  doi: 10.1016/j.jmaa.2017.04.074.

[15]

F. H. Lin and L. H. Wang, A class of fully nonlinear elliptic equations with singularity at the boundary, J. Geom. Anal., 8 (1998), 583-598. 

[16]

C. Loewner and L. Nirenberg, Partial differential equations invariant under conformal or projective transformations, in Contributions to Analysis (A Collection of Papers Dedicated to Lipman Bers), Academic Press, New York, 1974, 245-274

[17]

E. Lutwak, The Brunn-Minkowski-Firey theory I, Mixed volumes and the Minkowski problem, J. Diff. Geom., 38 (1993), 131-150. 

[18]

A. Mohammed, Existence and estimates of solutions to a singular Dirichlet problem for the Monge-Ampère equation, J. Math. Anal. Appl., 340 (2008), 1226-1234.  doi: 10.1016/j.jmaa.2007.09.014.

[19]

A. Mohammed, Singular boundary value problems for the Monge-Ampère equation, Nonlinear Anal., 70 (2009), 457-464.  doi: 10.1016/j.na.2007.12.017.

[20]

L. Nirenberg, Monge-Ampère equations and some associated problems in geometry, in, Proc. Internat. Congress of Mathematicians, vol. 2, Vancouver, 1974, 275–279.

[21]

E. Seneta, Regular Varying Functions, Lecture Notes in Math., vol. 508, Springer-Verlag, 1976.

[22]

H. Sun and M. Q. Feng, Boundary behavior of $k$-convex solutions for singular $k$-Hessian equations, Nonlinear Anal., 176 (2018), 141-156.  doi: 10.1016/j.na.2018.06.010.

[23]

Kaising Tso, On a real Monge-Ampère functional, Invert. Math., 101 (1990), 425-448. 

[24]

H. T. Yang and Y. B. Chang, On the blow-up boundary solutions of the Monge-Ampère equation with singular weights, Commun. Pure Appl. Anal., 11 (2012), 697-708.  doi: 10.3934/cpaa.2012.11.697.

[25]

X. M. Zhang and Y. Du, Sharp conditions for the existence of boundary blow-up solutions to the Monge-Ampère equation, Cal. Var. Partial Diff. Equations, 57 (2018), 1-24.  doi: 10.1007/s00526-018-1312-3.

[26]

Z. J. Zhang, Refined boundary behavior of the unique convex solution to a singular Dirichlet problem for the Monge-Ampère equation, Adv. Nonlinear studies, 18 (2018), 289-302.  doi: 10.1515/ans-2017-6045.

show all references

References:
[1]

N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Encyclopediaof Mathematics and its Applications 27, Cambridge University Press, 1987.

[2]

E. Calabi, Complete affine hypersurfaces I, Symposia Math., 10 (1972), 19-38. 

[3]

S. Y. Cheng and S.-T. Yau, On the regularity of the Monge-Ampère equation det $(({\partial^2u}/{\partial x^i\partial x^j})) = F(x, u)$, Comm. Pure Appl. Math., 30 (1977), 41-68. 

[4]

S. Y. Cheng and S.-T. Yau, Complete affine hypersurfaces Ⅰ: The completeness of affine metrics, Comm. Pure Appl. Math., 39 (1986), 839-866. 

[5]

Kai-Seng Chou and Xu-Jia Wang, The $L_p$-Minkowski problem and the Minkowski problem in centroaffine geometry, Adv. Math., 205 (2006), 33-83. 

[6]

F.-C. Cȋrstea and C. Trombetti, On the Monge-Ampère equation with boundary blow-up: existence, muniqueness and asymptotics, Cal. Var. Partial Diff. Equations, 31 (2008), 167-186.  doi: 10.1007/s00526-007-0108-7.

[7]

F. CuiH. Y. Jian and Y. Li, Boundary Hölder estimates for nonlinear singular elliptic equations, J. Math. Anal. Appl., 470 (2019), 1185-1193.  doi: 10.1016/j.jmaa.2018.10.059.

[8]

H. Y. Jian and Xu-Jia Wang, Bernstein theorem and regularity for a class of Monge-Ampère equation, J. Diff. Geom., 93 (2013), 431-469. 

[9]

H. Y. Jian and Xu-Jia Wang, Optimal boundary regularity for nonlinear singular elliptic equations, Adv. Math., 251 (2014), 111-126.  doi: 10.1016/j.aim.2013.10.009.

[10]

H. Y. JianXu-Jia Wang and Y. W. Zhao, Global smoothness for a singular Monge-Ampère equation, J. Diff. Equations, 263 (2017), 7250-7262.  doi: 10.1016/j.jde.2017.08.004.

[11]

H. Y. Jian and Y. Li, Optimal boundary regularity for a singular Monge-Ampère equation, J. Diff. Equations, 264 (2018), 6873-6890.  doi: 10.1016/j.jde.2018.01.051.

[12]

A. C. Lazer and P. J. McKenna, On singular boundary value problems for the Monge-Ampère Operator, J. Math. Anal. Appl., 197 (1996), 341-362. 

[13]

A. C. Lazer and P. J. McKenna, On a singular elliptic boundary value problem, Proc. Amer. Math. Soc., 111 (1991), 721-730. 

[14]

D. S. Li and S. S. Ma, Boundary behavior of solutions of Monge-Ampère equations with singular righthand sides, J. Math. Anal. Appl., 454 (2017), 79-93.  doi: 10.1016/j.jmaa.2017.04.074.

[15]

F. H. Lin and L. H. Wang, A class of fully nonlinear elliptic equations with singularity at the boundary, J. Geom. Anal., 8 (1998), 583-598. 

[16]

C. Loewner and L. Nirenberg, Partial differential equations invariant under conformal or projective transformations, in Contributions to Analysis (A Collection of Papers Dedicated to Lipman Bers), Academic Press, New York, 1974, 245-274

[17]

E. Lutwak, The Brunn-Minkowski-Firey theory I, Mixed volumes and the Minkowski problem, J. Diff. Geom., 38 (1993), 131-150. 

[18]

A. Mohammed, Existence and estimates of solutions to a singular Dirichlet problem for the Monge-Ampère equation, J. Math. Anal. Appl., 340 (2008), 1226-1234.  doi: 10.1016/j.jmaa.2007.09.014.

[19]

A. Mohammed, Singular boundary value problems for the Monge-Ampère equation, Nonlinear Anal., 70 (2009), 457-464.  doi: 10.1016/j.na.2007.12.017.

[20]

L. Nirenberg, Monge-Ampère equations and some associated problems in geometry, in, Proc. Internat. Congress of Mathematicians, vol. 2, Vancouver, 1974, 275–279.

[21]

E. Seneta, Regular Varying Functions, Lecture Notes in Math., vol. 508, Springer-Verlag, 1976.

[22]

H. Sun and M. Q. Feng, Boundary behavior of $k$-convex solutions for singular $k$-Hessian equations, Nonlinear Anal., 176 (2018), 141-156.  doi: 10.1016/j.na.2018.06.010.

[23]

Kaising Tso, On a real Monge-Ampère functional, Invert. Math., 101 (1990), 425-448. 

[24]

H. T. Yang and Y. B. Chang, On the blow-up boundary solutions of the Monge-Ampère equation with singular weights, Commun. Pure Appl. Anal., 11 (2012), 697-708.  doi: 10.3934/cpaa.2012.11.697.

[25]

X. M. Zhang and Y. Du, Sharp conditions for the existence of boundary blow-up solutions to the Monge-Ampère equation, Cal. Var. Partial Diff. Equations, 57 (2018), 1-24.  doi: 10.1007/s00526-018-1312-3.

[26]

Z. J. Zhang, Refined boundary behavior of the unique convex solution to a singular Dirichlet problem for the Monge-Ampère equation, Adv. Nonlinear studies, 18 (2018), 289-302.  doi: 10.1515/ans-2017-6045.

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