Figure 1.
The parameter $ a $
-
Previous Article
Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional
- CPAA Home
- This Issue
-
Next Article
Periodic bouncing solutions for Hill's type sub-linear oscillators with obstacles
January
2021, 20(1): 301-317.
doi: 10.3934/cpaa.2020267
Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions
Department of Mathematics and Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China |
In this paper we study the boundary regularity of solutions to the Dirichlet problem for a class of Monge-Ampère type equations with nonzero boundary conditions. We construct global Hölder estimates for convex solutions to the problem and emphasize that the boundary regularity essentially depends on the convexity of the domain. The proof is based on a careful study of the concept of $ (a,\eta) $ type convex domain and a family of auxiliary functions.
Keywords:
Monge-Ampère type equation,
Dirichlet problem,
convex domain,
Hölder estimate,
boundary regularity.
Mathematics Subject Classification:
Primary: 35J96, 35B65; Secondary: 52A20.
Citation:
Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure and Applied Analysis,
2021, 20
(1)
: 301-317.
doi: 10.3934/cpaa.2020267
![]()
References:
| [1] |
L. Caffarelli, L. Nirenberg and J. Spruck,
The Dirichlet problem for nonlinear second-order elliptic equations. I. Monge-Ampère equation, Commun. Pure Appl. Math., 37 (1984), 369-402.
doi: 10.1002/cpa.3160370306. |
| [2] |
S. Y. Cheng and S. T. Yau,
On the regularity of the Monge-Ampère equation $\det\frac{\partial^2u}{\partial x_i\partial x_j} = F(x, u)$, Commun. Pure Appl. Math., 30 (1977), 41-68.
doi: 10.1002/cpa.3160300104. |
| [3] |
S. Y. Cheng and S. T. Yau,
Complete affine hypersurfaces. Ⅰ. The completeness of affine metrics, Commun. Pure Appl. Math., 39 (1986), 839-866.
doi: 10.1002/cpa.3160390606. |
| [4] |
K. S. Chou and X. J. Wang,
The $L_p$-Minkowski problem and the Minkowski problem in centroaffine geometry, Adv. Math., 205 (2006), 33-83.
doi: 10.1016/j.aim.2005.07.004. |
| [5] |
A. Figalli, The Monge-Ampère Equation and Its Applications, European Mathematical Society (EMS), Zürich, 2017.
doi: 10.4171/170. |
| [6] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 2001.
doi: 10.1007/978-3-642-61798-0. |
| [7] |
P. F. 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. |
| [8] |
Y. He, Q. R. Li and X. J. Wang, Multiple solutions of the $L_p$-Minkowski problem, Calc. Var. Partial Differ. Equ., 55 (2016), 13 pp.
doi: 10.1007/s00526-016-1063-y. |
| [9] |
Y. Huang, F. D. Jiang and J. K. Liu,
Boundary $C^{2, \alpha}$ estimates for Monge-Ampère type equations, Adv. Math., 281 (2015), 706-733.
doi: 10.1016/j.aim.2014.12.043. |
| [10] |
H. Y. Jian and Y. Li,
Optimal boundary regularity for a singular Monge-Ampère equation, J. Differ. Equ., 264 (2018), 6873-6890.
doi: 10.1016/j.jde.2018.01.051. |
| [11] |
H. Y. Jian, Y. Li and X. S. Tu, On a class of degenerate and singular Monge-Ampère equations, arXiv: 1908.06396. |
| [12] |
H. Y. Jian and X. J. Wang,
Bernstein theorem and regularity for a class of Monge-Ampère equations, J. Differ. Geom., 93 (2013), 431-469.
doi: 10.4310/jdg/1361844941. |
| [13] |
H. Y. Jian, X. J. Wang and Y. W. Zhao,
Global smoothness for a singular Monge-Ampère equation, J. Differ. Equ., 263 (2017), 7250-7262.
doi: 10.1016/j.jde.2017.08.004. |
| [14] |
N. Q. Le and O. Savin,
Schauder estimates for degenerate Monge-Ampère equations and smoothness of the eigenfunctions, Invent. Math., 207 (2017), 389-423.
doi: 10.1007/s00222-016-0677-1. |
| [15] |
M. N. Li and Y. Li, Global regularity for a class of Monge-Ampère type equations, Sci. China Math., (2020), 16pp.
doi: 10.1007/s11425-019-1691-1. |
| [16] |
C. Loewner and L. Nirenberg, Partial differential equations invariant under conformal or projective transformations, in Contributions to Analysis, Academic Press, New York, (1974), 245-272. |
| [17] |
N. S. Trudinger and J. I. E. Urbas,
The Dirichlet problem for the equation of prescribed Gauss curvature, Bull. Austral. Math. Soc., 28 (1983), 217-231.
doi: 10.1017/S000497270002089X. |
| [18] |
N. S. Trudinger and X. J. Wang,
Boundary regularity for the Monge-Ampère and affine maximal surface equations, Ann. Math., 167 (2008), 993-1028.
doi: 10.4007/annals.2008.167.993. |
| [19] |
N. S. Trudinger and X. J. Wang, The Monge-Ampère equation and its geometric applications, in Handbook of Geometric Analysis, International Press, Somerville, MA, (2008), 467-524. |
| [20] |
J. I. E. Urbas,
Global Hölder estimates for equations of Monge-Ampère type, Invent. Math., 91 (1988), 1-29.
doi: 10.1007/BF01404910. |
show all references
References:
| [1] |
L. Caffarelli, L. Nirenberg and J. Spruck,
The Dirichlet problem for nonlinear second-order elliptic equations. I. Monge-Ampère equation, Commun. Pure Appl. Math., 37 (1984), 369-402.
doi: 10.1002/cpa.3160370306. |
| [2] |
S. Y. Cheng and S. T. Yau,
On the regularity of the Monge-Ampère equation $\det\frac{\partial^2u}{\partial x_i\partial x_j} = F(x, u)$, Commun. Pure Appl. Math., 30 (1977), 41-68.
doi: 10.1002/cpa.3160300104. |
| [3] |
S. Y. Cheng and S. T. Yau,
Complete affine hypersurfaces. Ⅰ. The completeness of affine metrics, Commun. Pure Appl. Math., 39 (1986), 839-866.
doi: 10.1002/cpa.3160390606. |
| [4] |
K. S. Chou and X. J. Wang,
The $L_p$-Minkowski problem and the Minkowski problem in centroaffine geometry, Adv. Math., 205 (2006), 33-83.
doi: 10.1016/j.aim.2005.07.004. |
| [5] |
A. Figalli, The Monge-Ampère Equation and Its Applications, European Mathematical Society (EMS), Zürich, 2017.
doi: 10.4171/170. |
| [6] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 2001.
doi: 10.1007/978-3-642-61798-0. |
| [7] |
P. F. 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. |
| [8] |
Y. He, Q. R. Li and X. J. Wang, Multiple solutions of the $L_p$-Minkowski problem, Calc. Var. Partial Differ. Equ., 55 (2016), 13 pp.
doi: 10.1007/s00526-016-1063-y. |
| [9] |
Y. Huang, F. D. Jiang and J. K. Liu,
Boundary $C^{2, \alpha}$ estimates for Monge-Ampère type equations, Adv. Math., 281 (2015), 706-733.
doi: 10.1016/j.aim.2014.12.043. |
| [10] |
H. Y. Jian and Y. Li,
Optimal boundary regularity for a singular Monge-Ampère equation, J. Differ. Equ., 264 (2018), 6873-6890.
doi: 10.1016/j.jde.2018.01.051. |
| [11] |
H. Y. Jian, Y. Li and X. S. Tu, On a class of degenerate and singular Monge-Ampère equations, arXiv: 1908.06396. |
| [12] |
H. Y. Jian and X. J. Wang,
Bernstein theorem and regularity for a class of Monge-Ampère equations, J. Differ. Geom., 93 (2013), 431-469.
doi: 10.4310/jdg/1361844941. |
| [13] |
H. Y. Jian, X. J. Wang and Y. W. Zhao,
Global smoothness for a singular Monge-Ampère equation, J. Differ. Equ., 263 (2017), 7250-7262.
doi: 10.1016/j.jde.2017.08.004. |
| [14] |
N. Q. Le and O. Savin,
Schauder estimates for degenerate Monge-Ampère equations and smoothness of the eigenfunctions, Invent. Math., 207 (2017), 389-423.
doi: 10.1007/s00222-016-0677-1. |
| [15] |
M. N. Li and Y. Li, Global regularity for a class of Monge-Ampère type equations, Sci. China Math., (2020), 16pp.
doi: 10.1007/s11425-019-1691-1. |
| [16] |
C. Loewner and L. Nirenberg, Partial differential equations invariant under conformal or projective transformations, in Contributions to Analysis, Academic Press, New York, (1974), 245-272. |
| [17] |
N. S. Trudinger and J. I. E. Urbas,
The Dirichlet problem for the equation of prescribed Gauss curvature, Bull. Austral. Math. Soc., 28 (1983), 217-231.
doi: 10.1017/S000497270002089X. |
| [18] |
N. S. Trudinger and X. J. Wang,
Boundary regularity for the Monge-Ampère and affine maximal surface equations, Ann. Math., 167 (2008), 993-1028.
doi: 10.4007/annals.2008.167.993. |
| [19] |
N. S. Trudinger and X. J. Wang, The Monge-Ampère equation and its geometric applications, in Handbook of Geometric Analysis, International Press, Somerville, MA, (2008), 467-524. |
| [20] |
J. I. E. Urbas,
Global Hölder estimates for equations of Monge-Ampère type, Invent. Math., 91 (1988), 1-29.
doi: 10.1007/BF01404910. |
| [1] |
Qi-Rui Li, Xu-Jia Wang. Regularity of the homogeneous Monge-Ampère equation. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 6069-6084. doi: 10.3934/dcds.2015.35.6069 |
| [2] |
Nam Q. Le. Optimal boundary regularity for some singular Monge-Ampère equations on bounded convex domains. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2199-2214. doi: 10.3934/dcds.2021188 |
| [3] |
Jingang Xiong, Jiguang Bao. The obstacle problem for Monge-Ampère type equations in non-convex domains. Communications on Pure and Applied Analysis, 2011, 10 (1) : 59-68. doi: 10.3934/cpaa.2011.10.59 |
| [4] |
Luca Codenotti, Marta Lewicka. Visualization of the convex integration solutions to the Monge-Ampère equation. Evolution Equations and Control Theory, 2019, 8 (2) : 273-300. doi: 10.3934/eect.2019015 |
| [5] |
Shouchuan Hu, Haiyan Wang. Convex solutions of boundary value problem arising from Monge-Ampère equations. Discrete and Continuous Dynamical Systems, 2006, 16 (3) : 705-720. doi: 10.3934/dcds.2006.16.705 |
| [6] |
Alessio Figalli, Young-Heon Kim. Partial regularity of Brenier solutions of the Monge-Ampère equation. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 559-565. doi: 10.3934/dcds.2010.28.559 |
| [7] |
Bo Guan, Qun Li. A Monge-Ampère type fully nonlinear equation on Hermitian manifolds. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1991-1999. doi: 10.3934/dcdsb.2012.17.1991 |
| [8] |
Barbara Brandolini, Carlo Nitsch, Cristina Trombetti. Shape optimization for Monge-Ampère equations via domain derivative. Discrete and Continuous Dynamical Systems - S, 2011, 4 (4) : 825-831. doi: 10.3934/dcdss.2011.4.825 |
| [9] |
Limei Dai, Hongyu Li. Entire subsolutions of Monge-Ampère type equations. Communications on Pure and Applied Analysis, 2020, 19 (1) : 19-30. doi: 10.3934/cpaa.2020002 |
| [10] |
Jiakun Liu, Neil S. Trudinger. On Pogorelov estimates for Monge-Ampère type equations. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 1121-1135. doi: 10.3934/dcds.2010.28.1121 |
| [11] |
Shuyu Gong, Ziwei Zhou, Jiguang Bao. Existence and uniqueness of viscosity solutions to the exterior problem of a parabolic Monge-Ampère equation. Communications on Pure and Applied Analysis, 2020, 19 (10) : 4921-4936. doi: 10.3934/cpaa.2020218 |
| [12] |
Yahui Niu. Monotonicity of solutions for a class of nonlocal Monge-Ampère problem. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5269-5283. doi: 10.3934/cpaa.2020237 |
| [13] |
Adam M. Oberman. Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian. Discrete and Continuous Dynamical Systems - B, 2008, 10 (1) : 221-238. doi: 10.3934/dcdsb.2008.10.221 |
| [14] |
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 |
| [15] |
Diego Maldonado. On interior $C^2$-estimates for the Monge-Ampère equation. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1427-1440. doi: 10.3934/dcds.2018058 |
| [16] |
Fan Cui, Huaiyu Jian. Symmetry of solutions to a class of Monge-Ampère equations. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1247-1259. doi: 10.3934/cpaa.2019060 |
| [17] |
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 |
| [18] |
Gregorio Díaz, Jesús Ildefonso Díaz. On the free boundary associated with the stationary Monge--Ampère operator on the set of non strictly convex functions. Discrete and Continuous Dynamical Systems, 2015, 35 (4) : 1447-1468. doi: 10.3934/dcds.2015.35.1447 |
| [19] |
Haitao Yang, Yibin Chang. On the blow-up boundary solutions of the Monge -Ampére equation with singular weights. Communications on Pure and Applied Analysis, 2012, 11 (2) : 697-708. doi: 10.3934/cpaa.2012.11.697 |
| [20] |
Limei Dai. Multi-valued solutions to a class of parabolic Monge-Ampère equations. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1061-1074. doi: 10.3934/cpaa.2014.13.1061 |
2021 Impact Factor: 1.273
Tools
Metrics
Other articles
by authors
[Back to Top]