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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

Mengni Li

Department of Mathematics and Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China

Received  January 2020 Revised  September 2020 Published  January 2021 Early access  November 2020

Fund Project: The author is supported by Yau Mathematical Sciences Center, Tsinghua University

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

A. Figalli, The Monge-Ampère Equation and Its Applications, European Mathematical Society (EMS), Zürich, 2017. doi: 10.4171/170.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

H. Y. Jian, Y. Li and X. S. Tu, On a class of degenerate and singular Monge-Ampère equations, arXiv: 1908.06396. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

J. I. E. Urbas, Global Hölder estimates for equations of Monge-Ampère type, Invent. Math., 91 (1988), 1-29.  doi: 10.1007/BF01404910.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

A. Figalli, The Monge-Ampère Equation and Its Applications, European Mathematical Society (EMS), Zürich, 2017. doi: 10.4171/170.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

H. Y. Jian, Y. Li and X. S. Tu, On a class of degenerate and singular Monge-Ampère equations, arXiv: 1908.06396. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

J. I. E. Urbas, Global Hölder estimates for equations of Monge-Ampère type, Invent. Math., 91 (1988), 1-29.  doi: 10.1007/BF01404910.  Google Scholar

Figure 1.  The parameter $ a $
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