Pointwise 𝐶^{2,𝛼} estimates at the boundary for the Monge-Ampère equation

Savin, O.

doi:10.1090/S0894-0347-2012-00747-4

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-6834(online) ISSN 0894-0347(print)

Pointwise $C^{2,\alpha}$ estimates at the boundary for the Monge-Ampère equation

Author: O. Savin
Journal: J. Amer. Math. Soc. 26 (2013), 63-99
MSC (2010): Primary 35J96
Posted: August 7, 2012
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove a localization property of boundary sections for solutions to the Monge-Ampère equation. As a consequence we obtain pointwise $C^{2,\alpha }$ estimates at boundary points under appropriate local conditions on the right-hand side and boundary data.

[CC] Luis A. Caffarelli and Xavier Cabré, Fully nonlinear elliptic equations, American Mathematical Society Colloquium Publications, vol. 43, American Mathematical Society, Providence, RI, 1995. MR 1351007 (96h:35046)
[CNS] 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), no. 3, 369–402. MR 739925 (87f:35096), http://dx.doi.org/10.1002/cpa.3160370306
[C1] L. A. Caffarelli, A localization property of viscosity solutions to the Monge-Ampère equation and their strict convexity, Ann. of Math. (2) 131 (1990), no. 1, 129–134. MR 1038359 (91f:35058), http://dx.doi.org/10.2307/1971509
[C2] Luis A. Caffarelli, Interior 𝑊^{2,𝑝} estimates for solutions of the Monge-Ampère equation, Ann. of Math. (2) 131 (1990), no. 1, 135–150. MR 1038360 (91f:35059), http://dx.doi.org/10.2307/1971510
[I] N. M. Ivočkina, A priori estimate of \𝑣𝑒𝑟𝑡𝑢\𝑣𝑒𝑟𝑡_{𝐶₂(\𝑜𝑣𝑒𝑟𝑙𝑖𝑛𝑒Ω)} of convex solutions of the Dirichlet problem for the Monge-Ampère equation, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 96 (1980), 69–79, 306 (Russian). Boundary value problems of mathematical physics and related questions in the theory of functions, 12. MR 579472 (82b:35056)
[JW] Huai-Yu Jian and Xu-Jia Wang, Continuity estimates for the Monge-Ampère equation, SIAM J. Math. Anal. 39 (2007), no. 2, 608–626. MR 2338423 (2009g:35073), http://dx.doi.org/10.1137/060669036
[K] N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain, Izv. Akad. Nauk SSSR Ser. Mat. 47 (1983), no. 1, 75–108 (Russian). MR 688919 (85g:35046)
[LS] Le N., Savin O., Some minimization problems in the class of convex functions with prescribed determinant, preprint, arXiv:1109.5676.
[S] Savin O., A localization property at the boundary for the Monge-Ampere equation, preprint arXiv:1010.1745.
[TW] Neil S. Trudinger and Xu-Jia Wang, Boundary regularity for the Monge-Ampère and affine maximal surface equations, Ann. of Math. (2) 167 (2008), no. 3, 993–1028. MR 2415390 (2010h:35168), http://dx.doi.org/10.4007/annals.2008.167.993
[W] Xu-Jia Wang, Regularity for Monge-Ampère equation near the boundary, Analysis 16 (1996), no. 1, 101–107. MR 1384356 (96m:35114)

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 35J96

Retrieve articles in all journals with MSC (2010): 35J96

Additional Information

O. Savin
Affiliation: Department of Mathematics, Columbia University, New York, New York 10027
Email: savin@math.columbia.edu

DOI: http://dx.doi.org/10.1090/S0894-0347-2012-00747-4
PII: S 0894-0347(2012)00747-4
Received by editor(s): January 28, 2011
Received by editor(s) in revised form: January 5, 2012
Posted: August 7, 2012
Additional Notes: The author was partially supported by NSF grant 0701037.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|