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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 & Continuous Dynamical Systems, 2015, 35 (12) : 6069-6084. doi: 10.3934/dcds.2015.35.6069
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show all references

References:
[1]

SIAM J. Math. Anal., 27 (1996), 1661-1679. doi: 10.1137/S0036141094265936.  Google Scholar

[2]

Ann. Math., 131 (1990), 135-150. doi: 10.2307/1971510.  Google Scholar

[3]

Revista Math. Iberoamericana, 2 (1986), 19-27. doi: 10.4171/RMI/23.  Google Scholar

[4]

J. Diff. Geom., 56 (2000), 189-234.  Google Scholar

[5]

Int. Math. Res. Not. IMRN, (2011), 967-1009. doi: 10.1093/imrn/rnq099.  Google Scholar

[6]

Trans. Amer. Math. Soc., 367 (2015), 4407-4422. doi: 10.1090/S0002-9947-2014-06306-X.  Google Scholar

[7]

Northern California Symplectic Geometry Seminar, Amer. Math. Soc. Transl. Ser. 2, 196 (1999), 13-33.  Google Scholar

[8]

Advances in Math., 132 (1997), 24-45. doi: 10.1006/aima.1997.1677.  Google Scholar

[9]

Trans. Amer. Math. Soc., 361 (2009), 4581-4591. doi: 10.1090/S0002-9947-09-04640-6.  Google Scholar

[10]

Acta. Math., 182 (1999), 87-104. doi: 10.1007/BF02392824.  Google Scholar

[11]

Math. Z., 209 (1992), 289-306. doi: 10.1007/BF02570835.  Google Scholar

[12]

Comm. PDE, 36 (2011), 635-656. doi: 10.1080/03605302.2010.514171.  Google Scholar

[13]

C. R. Acad. Sci. Paris Ser. I Math., 333 (2001), 725-728. doi: 10.1016/S0764-4442(01)02117-6.  Google Scholar

[14]

Trans. Amer. Math. Soc., 363 (2011), 5871-5886. doi: 10.1090/S0002-9947-2011-05240-2.  Google Scholar

[15]

J. Wiley, New York, 1978.  Google Scholar

[16]

Rocky Mountain J. Math., 7 (1977), 345-364. doi: 10.1216/RMJ-1977-7-2-345.  Google Scholar

[17]

Mem. Amer. Math. Soc., 180 (2006), x+157 pp. doi: 10.1090/memo/0847.  Google Scholar

[18]

J. Amer. Math. Soc., 26 (2013), 63-99. doi: 10.1090/S0894-0347-2012-00747-4.  Google Scholar

[19]

Proc. Amer. Math. Soc., 141 (2013), 3573-3578. doi: 10.1090/S0002-9939-2013-11748-X.  Google Scholar

[20]

J. Differential Equations, 256 (2014), 327-388. doi: 10.1016/j.jde.2013.08.019.  Google Scholar

[21]

Amer. J. Math., 114 (1992), 495-550. doi: 10.2307/2374768.  Google Scholar

[22]

Adv. Math., 193 (2005), 373-415. doi: 10.1016/j.aim.2004.05.009.  Google Scholar

[23]

Advances in Math., 217 (2008), 967-1026. doi: 10.1016/j.aim.2007.07.004.  Google Scholar

[24]

Bull. Austral. Math. Soc., 30 (1984), 321-334. doi: 10.1017/S0004972700002069.  Google Scholar

[25]

Ann. of Math. (2), 167 (2008), 993-1028. doi: 10.4007/annals.2008.167.993.  Google Scholar

[26]

Handbook of Geometric Analysis, International Press, 7 (2008), 467-524.  Google Scholar

[27]

Proc. Amer. Math. Soc., 123 (1995), 841-845. doi: 10.2307/2160809.  Google Scholar

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