Pointwise 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

Published electronically:
August 7, 2012

MathSciNet review:
2983006

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 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****[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**, 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**, 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**, 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****[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**, 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****[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**, 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**, 10.1524/anly.1996.16.1.101

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:
https://doi.org/10.1090/S0894-0347-2012-00747-4

Received by editor(s):
January 28, 2011

Received by editor(s) in revised form:
January 5, 2012

Published electronically:
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.