May  2022, 42(5): 2199-2214. doi: 10.3934/dcds.2021188

Optimal boundary regularity for some singular Monge-Ampère equations on bounded convex domains

Department of Mathematics, Indiana University, Bloomington, 831 E 3rd St, Bloomington, IN 47405, USA

*Corresponding author: Nam Q. Le

Received  April 2021 Published  May 2022 Early access  December 2021

Fund Project: The author is supported by NSF grant DMS-2054686

By constructing explicit supersolutions, we obtain the optimal global Hölder regularity for several singular Monge-Ampère equations on general bounded open convex domains including those related to complete affine hyperbolic spheres, and proper affine hyperspheres. Our analysis reveals that certain singular-looking equations, such as $ \det D^2 u = |u|^{-n-2-k} (x\cdot Du -u)^{-k} $ with zero boundary data, have unexpected degenerate nature.

Citation: 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
References:
[1]

K. J. BöröczkyE. LutwakD. Yang and G. Zhang, The logarithmic Minkowski problem, J. Amer. Math. Soc., 26 (2013), 831-852.  doi: 10.1090/S0894-0347-2012-00741-3.

[2]

L. A. Caffarelli, A localization property of viscosity solutions to the Monge-Ampère equation and their strict convexity, Ann. of Math., 131 (1990), 129-134.  doi: 10.2307/1971509.

[3]

L. CaffarelliL. 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.

[4] E. Calabi, Complete affine hyperspheres. I. Symposia Mathematica, Vol. X (Convegno di Geometria Differenziale, INDAM, Rome, 1971), Academic Press, London, 1972. 
[5]

H. Chen and G. Huang, Existence and regularity of the solutions of some singular Monge-Ampère equations, J. Differential Equations, 267 (2019), 866-878.  doi: 10.1016/j.jde.2019.01.030.

[6]

S. Y. Cheng and S. T. Yau, On the regularity of the Monge-Ampère equation det(2u = ∂xi∂xj) = F (x, u), Comm. Pure Appl. Math., 30 (1977), 41-68.  doi: 10.1002/cpa.3160300104.

[7]

S. Y. Cheng and S.-T. Yau, Complete affine hypersurfaces. I. The completeness of affine metrics, Comm. Pure Appl. Math., 39 (1986), 839-866.  doi: 10.1002/cpa.3160390606.

[8]

K.-S. Chou and X.-J. Wang, The Lp-Minkowski problem and the Minkowski problem in centroaffine geometry, Adv. Math., 205 (2006), 33-83.  doi: 10.1016/j.aim.2005.07.004.

[9]

A. Figalli, The Monge-Ampère Equation and its Applications, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), 2017. doi: 10.4171/170.

[10]

C. E. Gutiérrez, The Monge-Ampère Equation, Second edition. Birkhäuser, Boston, 2001. doi: 10.1007/978-1-4612-0195-3.

[11]

H. Jian and Y. Li, Optimal boundary regularity for a singular Monge-Ampère equation, J. Differential Equations, 264 (2018), 6873-6890.  doi: 10.1016/j.jde.2018.01.051.

[12]

H. Jian and Y. Li, A singular Monge-Ampère equation on unbounded domains, Sci. China Math., 61 (2018), 1473-1480.  doi: 10.1007/s11425-018-9351-1.

[13]

H. Jian and X.-J. Wang, Bernstein theorem and regularity for a class of Monge-Ampère equations, J. Differential Geom., 93 (2013), 431-469.  doi: 10.4310/jdg/1361844941.

[14]

N. Q. Le, The eigenvalue problem for the Monge-Ampère operator on general bounded convex domains, Ann. Sc. Norm. Super. Pisa Cl. Sci., 18 (2018), 1519-1559. 

[15]

F. H. Lin and L. Wang, A class of fully nonlinear elliptic equations with singularity at the boundary, J. Geom. Anal., 8 (1998), 583-598.  doi: 10.1007/BF02921713.

[16]

P.-L. Lions, Two remarks on Monge-Ampère equations, Ann. Mat. Pura Appl., 142 (1985), 263-275.  doi: 10.1007/BF01766596.

[17]

E. Lutwak, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom., 38 (1993), 131-150.  doi: 10.4310/jdg/1214454097.

[18]

K. Tso, On a real Monge-Ampère functional, Invent. Math., 101 (1990), 425-448.  doi: 10.1007/BF01231510.

show all references

References:
[1]

K. J. BöröczkyE. LutwakD. Yang and G. Zhang, The logarithmic Minkowski problem, J. Amer. Math. Soc., 26 (2013), 831-852.  doi: 10.1090/S0894-0347-2012-00741-3.

[2]

L. A. Caffarelli, A localization property of viscosity solutions to the Monge-Ampère equation and their strict convexity, Ann. of Math., 131 (1990), 129-134.  doi: 10.2307/1971509.

[3]

L. CaffarelliL. 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.

[4] E. Calabi, Complete affine hyperspheres. I. Symposia Mathematica, Vol. X (Convegno di Geometria Differenziale, INDAM, Rome, 1971), Academic Press, London, 1972. 
[5]

H. Chen and G. Huang, Existence and regularity of the solutions of some singular Monge-Ampère equations, J. Differential Equations, 267 (2019), 866-878.  doi: 10.1016/j.jde.2019.01.030.

[6]

S. Y. Cheng and S. T. Yau, On the regularity of the Monge-Ampère equation det(2u = ∂xi∂xj) = F (x, u), Comm. Pure Appl. Math., 30 (1977), 41-68.  doi: 10.1002/cpa.3160300104.

[7]

S. Y. Cheng and S.-T. Yau, Complete affine hypersurfaces. I. The completeness of affine metrics, Comm. Pure Appl. Math., 39 (1986), 839-866.  doi: 10.1002/cpa.3160390606.

[8]

K.-S. Chou and X.-J. Wang, The Lp-Minkowski problem and the Minkowski problem in centroaffine geometry, Adv. Math., 205 (2006), 33-83.  doi: 10.1016/j.aim.2005.07.004.

[9]

A. Figalli, The Monge-Ampère Equation and its Applications, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), 2017. doi: 10.4171/170.

[10]

C. E. Gutiérrez, The Monge-Ampère Equation, Second edition. Birkhäuser, Boston, 2001. doi: 10.1007/978-1-4612-0195-3.

[11]

H. Jian and Y. Li, Optimal boundary regularity for a singular Monge-Ampère equation, J. Differential Equations, 264 (2018), 6873-6890.  doi: 10.1016/j.jde.2018.01.051.

[12]

H. Jian and Y. Li, A singular Monge-Ampère equation on unbounded domains, Sci. China Math., 61 (2018), 1473-1480.  doi: 10.1007/s11425-018-9351-1.

[13]

H. Jian and X.-J. Wang, Bernstein theorem and regularity for a class of Monge-Ampère equations, J. Differential Geom., 93 (2013), 431-469.  doi: 10.4310/jdg/1361844941.

[14]

N. Q. Le, The eigenvalue problem for the Monge-Ampère operator on general bounded convex domains, Ann. Sc. Norm. Super. Pisa Cl. Sci., 18 (2018), 1519-1559. 

[15]

F. H. Lin and L. Wang, A class of fully nonlinear elliptic equations with singularity at the boundary, J. Geom. Anal., 8 (1998), 583-598.  doi: 10.1007/BF02921713.

[16]

P.-L. Lions, Two remarks on Monge-Ampère equations, Ann. Mat. Pura Appl., 142 (1985), 263-275.  doi: 10.1007/BF01766596.

[17]

E. Lutwak, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom., 38 (1993), 131-150.  doi: 10.4310/jdg/1214454097.

[18]

K. Tso, On a real Monge-Ampère functional, Invent. Math., 101 (1990), 425-448.  doi: 10.1007/BF01231510.

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