-
Previous Article
The anisotropic fractional isoperimetric problem with respect to unconditional unit balls
- CPAA Home
- This Issue
-
Next Article
On the Cahn-Hilliard equation with mass source for biological applications
The degenerate Monge-Ampère equations with the Neumann condition
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 |
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}) $.
References:
[1] |
Z. Błocki,
Interior regularity of the degenerate Monge-Ampère equations, Bull. Austral. Math. Soc., 68 (2003), 81-92.
doi: 10.1017/S0004972700037436. |
[2] |
Z. Błocki,
Regularity of the degenerate Monge-Ampère equation on compact Kähler manifolds, Math. Z., 244 (2003), 153-161.
doi: 10.1007/s00209-002-0483-x. |
[3] |
J. M. Bony, Principe du maximum dans les escapes de sobolev, C. R. Acad. Sci. Pair Sér, 265 (1967), A333–A336. |
[4] |
L. A. Caffarelli, J. Kohn, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations, II: Complex Monge-Ampère equation and uniformly elliptic equations, Commun. Pure Appl. Math., 38 (1985) 209–252.
doi: 10.1002/cpa.3160380206. |
[5] |
L. A. Caffarelli, L. Nirenberg and J. Spruck,
The Dirichlet problem for nonlinear second-order elliptic equations I: Monge-Ampère equation, Comm. Pure Appl. Math., 37 (1984), 369-402.
doi: 10.1002/cpa.3160370306. |
[6] |
L. A. Caffarelli, L. Nirenberg and J. Spruck,
The Dirichlet problem for the degenerate Monge-Ampère equation, Revista Math. Iberoamericana, 2 (1986), 19-27.
doi: 10.4171/RMI/23. |
[7] |
S. Dinew, S. Pliś and X. Zhang, Regularity of degenerate Hessian equations, Calc. Var. Partial Differ. Equ., 58 (2019), 21pp.
doi: 10.1007/s00526-019-1574-4. |
[8] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equation of Second Order, 2$^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. 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. |
[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. 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. |
[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.
|
[1] |
Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (1) : 301-317. doi: 10.3934/cpaa.2020267 |
[2] |
Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020377 |
[3] |
Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272 |
[4] |
Liupeng Wang, Yunqing Huang. Error estimates for second-order SAV finite element method to phase field crystal model. Electronic Research Archive, 2021, 29 (1) : 1735-1752. doi: 10.3934/era.2020089 |
[5] |
Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020340 |
[6] |
Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249 |
[7] |
Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168 |
[8] |
Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163 |
[9] |
Olivier Ley, Erwin Topp, Miguel Yangari. Some results for the large time behavior of Hamilton-Jacobi equations with Caputo time derivative. Discrete & Continuous Dynamical Systems - A, 2021 doi: 10.3934/dcds.2021007 |
[10] |
Jonathan J. Wylie, Robert M. Miura, Huaxiong Huang. Systems of coupled diffusion equations with degenerate nonlinear source terms: Linear stability and traveling waves. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 561-569. doi: 10.3934/dcds.2009.23.561 |
[11] |
Md. Masum Murshed, Kouta Futai, Masato Kimura, Hirofumi Notsu. Theoretical and numerical studies for energy estimates of the shallow water equations with a transmission boundary condition. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1063-1078. doi: 10.3934/dcdss.2020230 |
[12] |
Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020354 |
[13] |
Nguyen Huu Can, Nguyen Huy Tuan, Donal O'Regan, Vo Van Au. On a final value problem for a class of nonlinear hyperbolic equations with damping term. Evolution Equations & Control Theory, 2021, 10 (1) : 103-127. doi: 10.3934/eect.2020053 |
[14] |
Xiaoping Zhai, Yongsheng Li. Global large solutions and optimal time-decay estimates to the Korteweg system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1387-1413. doi: 10.3934/dcds.2020322 |
[15] |
Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2020164 |
[16] |
Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020103 |
[17] |
Elimhan N. Mahmudov. Infimal convolution and duality in convex optimal control problems with second order evolution differential inclusions. Evolution Equations & Control Theory, 2021, 10 (1) : 37-59. doi: 10.3934/eect.2020051 |
[18] |
Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Ground state homoclinic orbits for a class of asymptotically periodic second-order Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1627-1652. doi: 10.3934/dcdsb.2020176 |
[19] |
Mathew Gluck. Classification of solutions to a system of $ n^{\rm th} $ order equations on $ \mathbb R^n $. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246 |
[20] |
Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056 |
2019 Impact Factor: 1.105
Tools
Metrics
Other articles
by authors
[Back to Top]