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December  2015, 35(12): 6069-6084. doi: 10.3934/dcds.2015.35.6069

Regularity of the homogeneous Monge-Ampère equation

1. 

Centre for Mathematics and Its Applications, the Australian National University, Canberra, ACT 0200, Australia

2. 

Centre for Mathematics and Its Applications, Australian National University, Canberra, ACT 0200

Received  September 2013 Revised  February 2014 Published  May 2015

In this paper, we study the regularity of convex solutions to the Dirichlet problem of the homogeneous Monge-Ampère equation $\det D^2 u=0$. We prove that if the domain is a strip region and the boundary functions are locally uniformly convex and $C^{k+2,\alpha}$ smooth, then the solution is $C^{k+2,\alpha}$ smooth up to boundary. By an example, we show the solution may fail to be $C^{2}$ smooth if boundary functions are not locally uniformly convex. Similar results have also been obtained for the Dirichlet problem on bounded convex domains.
Citation: 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
References:
[1]

J. Benoist and J. B. Hiriart-Urruty, What is the subdifferential of the closed convex hull of a function?, SIAM J. Math. Anal., 27 (1996), 1661-1679. doi: 10.1137/S0036141094265936.

[2]

L. Caffarelli, Interior $W^{2,p}$ estimates for solutions of Monge-Ampère equations, Ann. Math., 131 (1990), 135-150. doi: 10.2307/1971510.

[3]

L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for the degenerate Monge-Ampère equation, Revista Math. Iberoamericana, 2 (1986), 19-27. doi: 10.4171/RMI/23.

[4]

X. Chen, Complex Monge-Ampère and symplectic manifolds, J. Diff. Geom., 56 (2000), 189-234.

[5]

X. Chen and W. He, The space of volume forms, Int. Math. Res. Not. IMRN, (2011), 967-1009. doi: 10.1093/imrn/rnq099.

[6]

G. De Philippis and A. Figallli, Optimal regularity of the convex envelope, Trans. Amer. Math. Soc., 367 (2015), 4407-4422. doi: 10.1090/S0002-9947-2014-06306-X.

[7]

S. K. Donaldson, Symmetric spaces, Kähler geometry and Hamiltonian dynamics, Northern California Symplectic Geometry Seminar, Amer. Math. Soc. Transl. Ser. 2, 196 (1999), 13-33.

[8]

P. Guan, Regularity of a class of quasilinear degenerate elliptic equations, Advances in Math., 132 (1997), 24-45. doi: 10.1006/aima.1997.1677.

[9]

P. Guan and E. T. Sawyer, Regularity of subelliptic Monge-Ampère equations in the plane, Trans. Amer. Math. Soc., 361 (2009), 4581-4591. doi: 10.1090/S0002-9947-09-04640-6.

[10]

P. 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.

[11]

J. X. Hong, Dirichlet problems for general Monge-Ampère equations, Math. Z., 209 (1992), 289-306. doi: 10.1007/BF02570835.

[12]

J. X. Hong, G. Huang and W. Wang, Existence of global smooth solutions to Dirichlet problem for degenrate elliptic Monge-Ampère equations, Comm. PDE, 36 (2011), 635-656. doi: 10.1080/03605302.2010.514171.

[13]

B. Kirchheim and J. Kristensen, Differentiability of convex envelopes, C. R. Acad. Sci. Paris Ser. I Math., 333 (2001), 725-728. doi: 10.1016/S0764-4442(01)02117-6.

[14]

A. Oberman and L. Silvestre, The Dirichlet problem for the convex envelope, Trans. Amer. Math. Soc., 363 (2011), 5871-5886. doi: 10.1090/S0002-9947-2011-05240-2.

[15]

A. V. Pogorelov, The Minkowski Multidimensional Problem, J. Wiley, New York, 1978.

[16]

J. Rauch and B. A. Taylor, The Dirichlet problem for the multi-dimensional Monge-Ampère equation, Rocky Mountain J. Math., 7 (1977), 345-364. doi: 10.1216/RMJ-1977-7-2-345.

[17]

E. T. Sawyer and R. L. Wheeden, Hölder continuity of weak solutions to subelliptic equations with rough coefficients, Mem. Amer. Math. Soc., 180 (2006), x+157 pp. doi: 10.1090/memo/0847.

[18]

O. Savin, Pointwise $C^{2,\alpha}$ estimates at the boundary for the Monge-Ampère equation, J. Amer. Math. Soc., 26 (2013), 63-99. doi: 10.1090/S0894-0347-2012-00747-4.

[19]

O. Savin, Global $W^{2,p}$ estimates for the Monge-Ampère equation, Proc. Amer. Math. Soc., 141 (2013), 3573-3578. doi: 10.1090/S0002-9939-2013-11748-X.

[20]

O. Savin, A localisation theorem and boundary regularity for a class of degenerate Monge-Ampère equations, J. Differential Equations, 256 (2014), 327-388. doi: 10.1016/j.jde.2013.08.019.

[21]

S. Semmes, Complex Monge-Ampère and symplectic manifolds, Amer. J. Math., 114 (1992), 495-550. doi: 10.2307/2374768.

[22]

C. Rios, E. T. Sawyer and R. L. Wheeden, A higher-dimensional partial Legendre transform, and regularity of degenerate Monge-Ampère equations, Adv. Math., 193 (2005), 373-415. doi: 10.1016/j.aim.2004.05.009.

[23]

C. Rios, E. T. Sawyer and R. L. Wheeden, Regularity of subelliptic Monge-Ampère equations, Advances in Math., 217 (2008), 967-1026. doi: 10.1016/j.aim.2007.07.004.

[24]

N. S. Trudinger and J. Urbas, On the second derivative estimates for equations of Monge-Ampère type, Bull. Austral. Math. Soc., 30 (1984), 321-334. doi: 10.1017/S0004972700002069.

[25]

N. S. Trudinger and X. J. Wang, Boundary regularity for the Monge-Ampère and affine maximal surface equations, Ann. of Math. (2), 167 (2008), 993-1028. doi: 10.4007/annals.2008.167.993.

[26]

N. S. Trudinger and X. J. Wang, The Monge-Ampère equation and its geometric applications, Handbook of Geometric Analysis, International Press, 7 (2008), 467-524.

[27]

X. J. Wang, Some counterexamples to the regularity of Monge-Ampère equations, Proc. Amer. Math. Soc., 123 (1995), 841-845. doi: 10.2307/2160809.

show all references

References:
[1]

J. Benoist and J. B. Hiriart-Urruty, What is the subdifferential of the closed convex hull of a function?, SIAM J. Math. Anal., 27 (1996), 1661-1679. doi: 10.1137/S0036141094265936.

[2]

L. Caffarelli, Interior $W^{2,p}$ estimates for solutions of Monge-Ampère equations, Ann. Math., 131 (1990), 135-150. doi: 10.2307/1971510.

[3]

L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for the degenerate Monge-Ampère equation, Revista Math. Iberoamericana, 2 (1986), 19-27. doi: 10.4171/RMI/23.

[4]

X. Chen, Complex Monge-Ampère and symplectic manifolds, J. Diff. Geom., 56 (2000), 189-234.

[5]

X. Chen and W. He, The space of volume forms, Int. Math. Res. Not. IMRN, (2011), 967-1009. doi: 10.1093/imrn/rnq099.

[6]

G. De Philippis and A. Figallli, Optimal regularity of the convex envelope, Trans. Amer. Math. Soc., 367 (2015), 4407-4422. doi: 10.1090/S0002-9947-2014-06306-X.

[7]

S. K. Donaldson, Symmetric spaces, Kähler geometry and Hamiltonian dynamics, Northern California Symplectic Geometry Seminar, Amer. Math. Soc. Transl. Ser. 2, 196 (1999), 13-33.

[8]

P. Guan, Regularity of a class of quasilinear degenerate elliptic equations, Advances in Math., 132 (1997), 24-45. doi: 10.1006/aima.1997.1677.

[9]

P. Guan and E. T. Sawyer, Regularity of subelliptic Monge-Ampère equations in the plane, Trans. Amer. Math. Soc., 361 (2009), 4581-4591. doi: 10.1090/S0002-9947-09-04640-6.

[10]

P. 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.

[11]

J. X. Hong, Dirichlet problems for general Monge-Ampère equations, Math. Z., 209 (1992), 289-306. doi: 10.1007/BF02570835.

[12]

J. X. Hong, G. Huang and W. Wang, Existence of global smooth solutions to Dirichlet problem for degenrate elliptic Monge-Ampère equations, Comm. PDE, 36 (2011), 635-656. doi: 10.1080/03605302.2010.514171.

[13]

B. Kirchheim and J. Kristensen, Differentiability of convex envelopes, C. R. Acad. Sci. Paris Ser. I Math., 333 (2001), 725-728. doi: 10.1016/S0764-4442(01)02117-6.

[14]

A. Oberman and L. Silvestre, The Dirichlet problem for the convex envelope, Trans. Amer. Math. Soc., 363 (2011), 5871-5886. doi: 10.1090/S0002-9947-2011-05240-2.

[15]

A. V. Pogorelov, The Minkowski Multidimensional Problem, J. Wiley, New York, 1978.

[16]

J. Rauch and B. A. Taylor, The Dirichlet problem for the multi-dimensional Monge-Ampère equation, Rocky Mountain J. Math., 7 (1977), 345-364. doi: 10.1216/RMJ-1977-7-2-345.

[17]

E. T. Sawyer and R. L. Wheeden, Hölder continuity of weak solutions to subelliptic equations with rough coefficients, Mem. Amer. Math. Soc., 180 (2006), x+157 pp. doi: 10.1090/memo/0847.

[18]

O. Savin, Pointwise $C^{2,\alpha}$ estimates at the boundary for the Monge-Ampère equation, J. Amer. Math. Soc., 26 (2013), 63-99. doi: 10.1090/S0894-0347-2012-00747-4.

[19]

O. Savin, Global $W^{2,p}$ estimates for the Monge-Ampère equation, Proc. Amer. Math. Soc., 141 (2013), 3573-3578. doi: 10.1090/S0002-9939-2013-11748-X.

[20]

O. Savin, A localisation theorem and boundary regularity for a class of degenerate Monge-Ampère equations, J. Differential Equations, 256 (2014), 327-388. doi: 10.1016/j.jde.2013.08.019.

[21]

S. Semmes, Complex Monge-Ampère and symplectic manifolds, Amer. J. Math., 114 (1992), 495-550. doi: 10.2307/2374768.

[22]

C. Rios, E. T. Sawyer and R. L. Wheeden, A higher-dimensional partial Legendre transform, and regularity of degenerate Monge-Ampère equations, Adv. Math., 193 (2005), 373-415. doi: 10.1016/j.aim.2004.05.009.

[23]

C. Rios, E. T. Sawyer and R. L. Wheeden, Regularity of subelliptic Monge-Ampère equations, Advances in Math., 217 (2008), 967-1026. doi: 10.1016/j.aim.2007.07.004.

[24]

N. S. Trudinger and J. Urbas, On the second derivative estimates for equations of Monge-Ampère type, Bull. Austral. Math. Soc., 30 (1984), 321-334. doi: 10.1017/S0004972700002069.

[25]

N. S. Trudinger and X. J. Wang, Boundary regularity for the Monge-Ampère and affine maximal surface equations, Ann. of Math. (2), 167 (2008), 993-1028. doi: 10.4007/annals.2008.167.993.

[26]

N. S. Trudinger and X. J. Wang, The Monge-Ampère equation and its geometric applications, Handbook of Geometric Analysis, International Press, 7 (2008), 467-524.

[27]

X. J. Wang, Some counterexamples to the regularity of Monge-Ampère equations, Proc. Amer. Math. Soc., 123 (1995), 841-845. doi: 10.2307/2160809.

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