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

Savin, O.

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

Pointwise $C^{2,\alpha }$ estimates at the boundary for the Monge-Ampère equation
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by O. Savin;

J. Amer. Math. Soc. 26 (2013), 63-99

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

Published electronically: August 7, 2012

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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.

References

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, DOI 10.1090/coll/043
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, DOI 10.1002/cpa.3160370306
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, DOI 10.2307/1971509
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, DOI 10.2307/1971510
N. M. Ivočkina, A priori estimate of $|u|_{C_2(\overline \Omega )}$ 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). MR 579472
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, DOI 10.1137/060669036
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
Le N., Savin O., Some minimization problems in the class of convex functions with prescribed determinant, preprint, arXiv:1109.5676.
Savin O., A localization property at the boundary for the Monge-Ampere equation, preprint arXiv:1010.1745.
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, DOI 10.4007/annals.2008.167.993
Xu-Jia Wang, Regularity for Monge-Ampère equation near the boundary, Analysis 16 (1996), no. 1, 101–107. MR 1384356, DOI 10.1524/anly.1996.16.1.101