The degenerate Monge-Ampère equations with the Neumann condition

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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 February 2021 Early access 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 and 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.
[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.
[3]	J. M. Bony, Principe du maximum dans les escapes de sobolev, C. R. Acad. Sci. Pair Sér, 265 (1967), A333–A336.
[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.
[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.
[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.
[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.
[8]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equation of Second Order, 2$^nd$ edition, Springer, Berlin, 1983.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[18]	X. Ma, G. Qiu and J. Xu, Gradient estimates on Hessian equations for Neumann problem, Sci. Sin. Math. (Chinese), 46 (2016), 1117-1126.
[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.
[20]	N. S. Trudinger, On degenerate fully nonlinear elliptic equations in balls, Bull. Aust. Math. Soc., 35 (1987), 299-307. doi: 10.1017/S0004972700013253.
[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.
[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.
[23]	J. Urbas, Oblique boundary value problems for Monge-Ampère type, Calc. Var. Partial Differ. Equ., 7 (1998), 13-39. doi: 10.1007/s005260050097.
[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.
[25]	X. J. Wang, Oblique derivative problem for equations of Monge-Ampère type, Chinese J. Contemp. Math., 13 (1992), 13-22.

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.
[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.
[3]	J. M. Bony, Principe du maximum dans les escapes de sobolev, C. R. Acad. Sci. Pair Sér, 265 (1967), A333–A336.
[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.
[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.
[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.
[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.
[8]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equation of Second Order, 2$^nd$ edition, Springer, Berlin, 1983.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[18]	X. Ma, G. Qiu and J. Xu, Gradient estimates on Hessian equations for Neumann problem, Sci. Sin. Math. (Chinese), 46 (2016), 1117-1126.
[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.
[20]	N. S. Trudinger, On degenerate fully nonlinear elliptic equations in balls, Bull. Aust. Math. Soc., 35 (1987), 299-307. doi: 10.1017/S0004972700013253.
[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.
[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.
[23]	J. Urbas, Oblique boundary value problems for Monge-Ampère type, Calc. Var. Partial Differ. Equ., 7 (1998), 13-39. doi: 10.1007/s005260050097.
[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.
[25]	X. J. Wang, Oblique derivative problem for equations of Monge-Ampère type, Chinese J. Contemp. Math., 13 (1992), 13-22.

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