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The parameter $ a $
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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
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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. |
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