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

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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.

Keywords: Singular Monge-Ampère equation, optimal global Hölder regularity, supersolution, complete affine hyperbolic spheres, proper affine hyperspheres.

Mathematics Subject Classification: Primary: 35J96, 35J75; Secondary: 53A15.

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:

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[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. 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.
[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∂x^j) = 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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References:

[1]	K. J. Böröczky, E. Lutwak, D. 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. 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.
[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∂x^j) = 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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