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February  2021, 20(2): 915-931. doi: 10.3934/cpaa.2020297

The degenerate Monge-Ampère equations with the Neumann condition

Juhua Shi 1, and  Feida Jiang 2,,

1. 

School of Science, Nanjing University of Science and Technology, Nanjing 210094, China
2. 

School of Mathematics and Shing-Tung Yau Center of Southeast University, Southeast University, Nanjing 211189, China

* Corresponding author

Received  August 2020 Revised  October 2020 Published  December 2020

Fund Project: This work was supported by the National Natural Science Foundation of China (No. 11771214, No. 12001276).

In this paper, we study a priori derivative estimates (up to the second order) of solutions for the Monge-Ampère equation $ \det D^{2}u = f(x) $ with the Neumann boundary value condition, which are independent of $ \inf f $. Based on these uniform estimates, the existence and uniqueness of the global $ C^{1,1} $ solution to the Neumann problem of the degenerate Monge-Ampère equation are established under the assumption $ f^{\frac{1}{n-1}}\in C^{1,1}(\bar{\Omega}) $.

Keywords: Monge-Ampère equations, Neumann problem, degenerate equations, global second order derivative estimates.
Mathematics Subject Classification: Primary: 35J96, 35J25; Secondary: 35J70, 35B65.
Citation: Juhua Shi, Feida Jiang. The degenerate Monge-Ampère equations with the Neumann condition. Communications on Pure & Applied Analysis, 2021, 20 (2) : 915-931. doi: 10.3934/cpaa.2020297
References:
[1]

Z. Błocki, Interior regularity of the degenerate Monge-Ampère equations, Bull. Austral. Math. Soc., 68 (2003), 81-92.  doi: 10.1017/S0004972700037436.  Google Scholar

[2]

Z. Błocki, Regularity of the degenerate Monge-Ampère equation on compact Kähler manifolds, Math. Z., 244 (2003), 153-161.  doi: 10.1007/s00209-002-0483-x.  Google Scholar

[3]

J. M. Bony, Principe du maximum dans les escapes de sobolev, C. R. Acad. Sci. Pair Sér, 265 (1967), A333–A336.  Google Scholar

[4]

L. A. Caffarelli, J. Kohn, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations, II: Complex Monge-Ampère equation and uniformly elliptic equations, Commun. Pure Appl. Math., 38 (1985) 209–252. doi: 10.1002/cpa.3160380206.  Google Scholar

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L. A. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations I: Monge-Ampère equation, Comm. Pure Appl. Math., 37 (1984), 369-402.  doi: 10.1002/cpa.3160370306.  Google Scholar

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L. A. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for the degenerate Monge-Ampère equation, Revista Math. Iberoamericana, 2 (1986), 19-27.  doi: 10.4171/RMI/23.  Google Scholar

[7]

S. Dinew, S. Pliś and X. Zhang, Regularity of degenerate Hessian equations, Calc. Var. Partial Differ. Equ., 58 (2019), 21pp. doi: 10.1007/s00526-019-1574-4.  Google Scholar

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D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equation of Second Order, 2$^nd$ edition, Springer, Berlin, 1983.  Google Scholar

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P. Guan, $C^{2}$ a priori estimates for degenerate Monge-Ampère equations, Duke Math. J., 86 (1997), 323-346.  doi: 10.1215/S0012-7094-97-08610-5.  Google Scholar

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P. Guan, N. S. Trudinger and X. J. Wang, On the Dirichlet problem for degenerate Monge-Ampère equations, Acta Math., 182 (1999), 87-104.  doi: 10.1007/BF02392824.  Google Scholar

[11]

H. Ishii and P. L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, J. Differ. Equ., 83 (1990), 26-78.  doi: 10.1016/0022-0396(90)90068-Z.  Google Scholar

[12]

F. Jiang and N. S. Trudinger, Oblique boundary value problems for augmented Hessian equation III, Commun. Partial Differ. Equ., 44 (2019), 708-748.  doi: 10.1080/03605302.2019.1597113.  Google Scholar

[13]

F. Jiang, N. S. Trudinger and N. Xiang, On the Neumann problem for Monge-Ampère type equations, Canad. J. Math., 68 (2016), 1334-1361.  doi: 10.4153/CJM-2016-001-3.  Google Scholar

[14]

F. Jiang, N. Xiang and J. Xu, Gradient estimates for Neumann boundary value problem of Monge-Ampère type equations, Commun. Contemp. Math., 19 (2017), 1650041, 16pp. doi: 10.1142/S0219199716500413.  Google Scholar

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N. V. Krylov, Smoothness of the payoff function for a controllable diffusion process in a domain, Math. USSR Izv., 34 (1990), 65-95.  doi: 10.1070/IM1990v034n01ABEH000603.  Google Scholar

[16]

P. L. Lions, N. S. Trudinger and J. Urbas, The Neumann problem for equations of Monge-Ampère type, Commun. Pure Appl. Math., 39 (1986), 539-563.  doi: 10.1002/cpa.3160390405.  Google Scholar

[17]

S. Y. Li, Oblique boundary value problems for Monge-Ampère equations, Pacific J. Math., 190 (1999), 155-172.  doi: 10.2140/pjm.1999.190.155.  Google Scholar

[18]

X. Ma, G. Qiu and J. Xu, Gradient estimates on Hessian equations for Neumann problem, Sci. Sin. Math. (Chinese), 46 (2016), 1117-1126.   Google Scholar

[19]

X. Ma and G. Qiu, The Neumann problem for Hessian equations, Commun. Math. Phys., 36 (2019), 1-28.  doi: 10.1007/s00220-019-03339-1.  Google Scholar

[20]

N. S. Trudinger, On degenerate fully nonlinear elliptic equations in balls, Bull. Aust. Math. Soc., 35 (1987), 299-307.  doi: 10.1017/S0004972700013253.  Google Scholar

[21]

N. S. Trudinger and J. Urbas, On the second derivative estimates for equations of Monge-Ampère type, Bull. Austral. Math. Soc., 30 (1984), 321-334.  doi: 10.1017/S0004972700002069.  Google Scholar

[22]

J. Urbas, The oblique boundary value problem for equations of Monge-Ampère type in $\mathbb{R}^2$, Miniconference on geometry and partial differential equations, Austral. Nat. Univ., 12 (1987), 171-195.   Google Scholar

[23]

J. Urbas, Oblique boundary value problems for Monge-Ampère type, Calc. Var. Partial Differ. Equ., 7 (1998), 13-39.  doi: 10.1007/s005260050097.  Google Scholar

[24]

X. J. Wang, Some counterexamples to the regularity of Monge-Ampère equations, Proc. Amer. Math. Soc., 123 (1995), 841-845.  doi: 10.2307/2160809.  Google Scholar

[25]

X. J. Wang, Oblique derivative problem for equations of Monge-Ampère type, Chinese J. Contemp. Math., 13 (1992), 13-22.   Google Scholar

show all references

References:
[1]

Z. Błocki, Interior regularity of the degenerate Monge-Ampère equations, Bull. Austral. Math. Soc., 68 (2003), 81-92.  doi: 10.1017/S0004972700037436.  Google Scholar

[2]

Z. Błocki, Regularity of the degenerate Monge-Ampère equation on compact Kähler manifolds, Math. Z., 244 (2003), 153-161.  doi: 10.1007/s00209-002-0483-x.  Google Scholar

[3]

J. M. Bony, Principe du maximum dans les escapes de sobolev, C. R. Acad. Sci. Pair Sér, 265 (1967), A333–A336.  Google Scholar

[4]

L. A. Caffarelli, J. Kohn, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations, II: Complex Monge-Ampère equation and uniformly elliptic equations, Commun. Pure Appl. Math., 38 (1985) 209–252. doi: 10.1002/cpa.3160380206.  Google Scholar

[5]

L. A. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations I: Monge-Ampère equation, Comm. Pure Appl. Math., 37 (1984), 369-402.  doi: 10.1002/cpa.3160370306.  Google Scholar

[6]

L. A. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for the degenerate Monge-Ampère equation, Revista Math. Iberoamericana, 2 (1986), 19-27.  doi: 10.4171/RMI/23.  Google Scholar

[7]

S. Dinew, S. Pliś and X. Zhang, Regularity of degenerate Hessian equations, Calc. Var. Partial Differ. Equ., 58 (2019), 21pp. doi: 10.1007/s00526-019-1574-4.  Google Scholar

[8]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equation of Second Order, 2$^nd$ edition, Springer, Berlin, 1983.  Google Scholar

[9]

P. Guan, $C^{2}$ a priori estimates for degenerate Monge-Ampère equations, Duke Math. J., 86 (1997), 323-346.  doi: 10.1215/S0012-7094-97-08610-5.  Google Scholar

[10]

P. Guan, N. S. Trudinger and X. J. Wang, On the Dirichlet problem for degenerate Monge-Ampère equations, Acta Math., 182 (1999), 87-104.  doi: 10.1007/BF02392824.  Google Scholar

[11]

H. Ishii and P. L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, J. Differ. Equ., 83 (1990), 26-78.  doi: 10.1016/0022-0396(90)90068-Z.  Google Scholar

[12]

F. Jiang and N. S. Trudinger, Oblique boundary value problems for augmented Hessian equation III, Commun. Partial Differ. Equ., 44 (2019), 708-748.  doi: 10.1080/03605302.2019.1597113.  Google Scholar

[13]

F. Jiang, N. S. Trudinger and N. Xiang, On the Neumann problem for Monge-Ampère type equations, Canad. J. Math., 68 (2016), 1334-1361.  doi: 10.4153/CJM-2016-001-3.  Google Scholar

[14]

F. Jiang, N. Xiang and J. Xu, Gradient estimates for Neumann boundary value problem of Monge-Ampère type equations, Commun. Contemp. Math., 19 (2017), 1650041, 16pp. doi: 10.1142/S0219199716500413.  Google Scholar

[15]

N. V. Krylov, Smoothness of the payoff function for a controllable diffusion process in a domain, Math. USSR Izv., 34 (1990), 65-95.  doi: 10.1070/IM1990v034n01ABEH000603.  Google Scholar

[16]

P. L. Lions, N. S. Trudinger and J. Urbas, The Neumann problem for equations of Monge-Ampère type, Commun. Pure Appl. Math., 39 (1986), 539-563.  doi: 10.1002/cpa.3160390405.  Google Scholar

[17]

S. Y. Li, Oblique boundary value problems for Monge-Ampère equations, Pacific J. Math., 190 (1999), 155-172.  doi: 10.2140/pjm.1999.190.155.  Google Scholar

[18]

X. Ma, G. Qiu and J. Xu, Gradient estimates on Hessian equations for Neumann problem, Sci. Sin. Math. (Chinese), 46 (2016), 1117-1126.   Google Scholar

[19]

X. Ma and G. Qiu, The Neumann problem for Hessian equations, Commun. Math. Phys., 36 (2019), 1-28.  doi: 10.1007/s00220-019-03339-1.  Google Scholar

[20]

N. S. Trudinger, On degenerate fully nonlinear elliptic equations in balls, Bull. Aust. Math. Soc., 35 (1987), 299-307.  doi: 10.1017/S0004972700013253.  Google Scholar

[21]

N. S. Trudinger and J. Urbas, On the second derivative estimates for equations of Monge-Ampère type, Bull. Austral. Math. Soc., 30 (1984), 321-334.  doi: 10.1017/S0004972700002069.  Google Scholar

[22]

J. Urbas, The oblique boundary value problem for equations of Monge-Ampère type in $\mathbb{R}^2$, Miniconference on geometry and partial differential equations, Austral. Nat. Univ., 12 (1987), 171-195.   Google Scholar

[23]

J. Urbas, Oblique boundary value problems for Monge-Ampère type, Calc. Var. Partial Differ. Equ., 7 (1998), 13-39.  doi: 10.1007/s005260050097.  Google Scholar

[24]

X. J. Wang, Some counterexamples to the regularity of Monge-Ampère equations, Proc. Amer. Math. Soc., 123 (1995), 841-845.  doi: 10.2307/2160809.  Google Scholar

[25]

X. J. Wang, Oblique derivative problem for equations of Monge-Ampère type, Chinese J. Contemp. Math., 13 (1992), 13-22.   Google Scholar

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