Transactions of the American Mathematical Society

Author(s): Pengfei Guan; Eric Sawyer
Journal: Trans. Amer. Math. Soc. 361 (2009), 4581-4591.
MSC (2000): Primary 35J60, 35B65
Posted: April 14, 2009
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Abstract: We establish a $C^\infty$ regularity result for $C^{1,1}$ solutions of degenerate Monge-Ampère equation in $\mathbb{R}^2$ , under the assumption that the trace of the Hessian is bounded from below.

1.: Y.D. BURAGO AND S.Z. SHEFEL, The geometry of surfaces in Euclidean spaces, Geometry, III, Encyclopaedia Math. Sci. 48, Springer, Berlin, 1992. MR 1306734
2.: L. CAFFARELLI, L. NIRENBERG AND J. SPRUCK, The Dirichlet problem for nonlinear second order elliptic equations, I. Monge-Amp ère equations, Comm. Pure Appl. Math. 37 (1984), 369-402. MR 739925 (87f:35096)
3.: P. DASKALOPOULOS AND O. SAVIN, On Monge-Ampère equations with homogeneous right hand side, preprint, June 2007, arXiv:0706.3748.
4.: P. GUAN, Regularity of a class of quasilinear degenerate elliptic equations, Advances in Mathematics 132 (1997), 24-45. MR 1488238 (99a:35068)
5.: P. GUAN, $C^{2}$ a priori estimates for degenerate Monge-Ampère equations, Duke Math. J. 86 (1997), 323-346. MR 1430436 (98d:35074)
6.: P. GUAN AND Y.Y LI, On Weyl problem with nonnegative Gauss curvature, Journal of Differential Geometry 39 (1994), 331-342. MR 1267893 (95c:53051)
7.: P. GUAN, N. S. TRUDINGER AND X.-J. WANG, On the Dirichlet problem for degenerate Monge-Ampère equations, Acta Math. 182 (1999), 87-104. MR 1687172 (2000h:35051)
8.: J. HONG AND C. ZUILY, Isometric embedding of the -sphere with nonnegative curvature in $\mathbb{R}^{3}$ , Math. Z. 219 (1995), no. 3, 323-334. MR 1339708 (96e:53005)
9.: J. IAIA, Isometric embeddings of surfaces with nonnegative curvature in $\mathbb{R}^{3}$ , Duke Math. J. 67 (1992), 423-459. MR 1177314 (93i:53004)
10.: L. NIRENBERG, The Weyl and Minkowski problems in differential geometry in the large, Comm. Pure and Appl. Math. 6 (1953), 337-394. MR 0058265 (15:347b)
11.: C. RIOS, E. SAWYER AND R. L. WHEEDEN, A higher-dimensional partial Legendre transform, and regularity of degenerate Monge-Ampère equations, Advances in Mathematics 193 (2005), 373-415. MR 2137289 (2006f:35099)
12.: C. RIOS, E. SAWYER AND R. L. WHEEDEN, Regularity of subelliptic Monge-Ampère equations, to appear in Advances in Mathematics.

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35J60, 35B65

Pengfei Guan
Affiliation: Department of Mathematics and Statistics, McGill University, Montreal, Quebec, Canada H3A 2K6
Email: guan@math.mcgill.ca

Eric Sawyer
Affiliation: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1
Email: sawyer@mcmaster.ca

DOI: 10.1090/S0002-9947-09-04640-6
PII: S 0002-9947(09)04640-6
Received by editor(s): April 26, 2007
Posted: April 14, 2009
Additional Notes: Research of the authors was supported in part by NSERC Discovery Grants.
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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