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 & 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.  Google Scholar

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

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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.   Google Scholar

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

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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.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.   Google Scholar

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

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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.  Google Scholar

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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.   Google Scholar

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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  Google Scholar

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

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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.  Google Scholar

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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.  Google Scholar

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

[21]

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

[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.  Google Scholar

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

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

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

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[13]

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

[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.  Google Scholar

[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.   Google Scholar

[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  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[21]

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

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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